2.1 Physical nature of pressure-driven growth

As alluded to above, Darcy’s law for flow in a porous medium describes flow down a gradient of pressure. However in a system for which the relative mobility of a particular phase is much lower in a restricted part of the flow domain than in any other part of the domain, almost all the loss in pressure is expected to occur across that low mobility zone. Indeed the pressure-driven growth model considers a limiting case (Shan & Rossen Reference Shan and Rossen2004) in which the entirety of the loss in pressure occurs across the neighbourhood of the foam front, the relative mobilities for gas in the coarsely textured foam upstream and for liquid in the surfactant solution downstream being assumed to be arbitrarily large compared to the relative mobility of gas in the finely textured foam at the front itself. Under these circumstances, all the non-trivial dynamics becomes focussed on the foam front itself: the foam front is pushed along by pressure, but is retarded by dissipation localised at the front, with pressure falling from a driving injection pressure upstream of the front to a background hydrostatic pressure downstream, the latter growing with depth. There is moreover a maximum depth to which the front can penetrate, corresponding to the depth at which the injection pressure is balanced by the hydrostatic pressure.

We are interested here, not just in the speed of foam front propagation, but also in the propagation direction. Since Darcy’s law describes flow down a pressure gradient, the direction of the flow necessarily matches the direction of the pressure gradient, which in the circumstances described above, must therefore be across the region of finely textured foam in the neighbourhood of the front (Shan & Rossen Reference Shan and Rossen2004). This implies the motion of the finely textured foam is normal to the front, i.e. the direction of propagation of the front is normal to the instantaneous front location. We consider that the gas is initially injected down a vertical injection well, so that the initial orientation of the front is vertical, and the initial propagation of the front is therefore horizontal. Nonetheless points higher up on the front propagate faster than those lower down, because the deeper one moves, the greater the hydrostatic pressure that opposes the pressure of gas injection. The front therefore reorients over time away from the vertical and propagation of the front thereby reorients away from the horizontal.

To give an idea of scale, in field applications, the foam front can over time (several days perhaps (Mas-Hernández et al. Reference Mas-Hernández, Grassia and Shokri2015a )), travel as much as several kilometres horizontally, penetrating several hundreds of metres below the surface (de Velde Harsenhorst et al. Reference de Velde Harsenhorst, Dharma, Andrianov and Rossen2014). In this work however, we shall focus on the comparatively early time evolution of the front, such that the horizontal displacement has still not managed to grow any larger than the penetration depth (i.e. several hundreds of metres). Meanwhile the zone of dissipative, finely textured foam at the foam front can be up to the order of a hundred metres thick, the thickness being measured along the direction of motion of the foam front (which is horizontal initially but which then reorients over time). The front thickness is rather less than either the penetration depth or the horizontal distance over which the front itself travels (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014). Under these circumstances the front can be idealised in the first instance as being arbitrarily thin. The thickness of the dissipative, finely textured foam zone also tends to be less, the less the distance that the front has travelled (Shi Reference Shi1996; Shan & Rossen Reference Shan and Rossen2004): the implication is that the front initially moves quickly, but then slows down over time as the dissipation grows.

2.2 Physics excluded from pressure-driven growth

In addition to describing the physics included in the pressure-driven growth model, as was done above, it is also worth considering the physical effects that are excluded. This is done in what follows, and helps us to appreciate the possible limitations of the model.

The model treats gas in the finely textured foam as a fluid of low mobility, but considers the flow to be Newtonian. Foam is in actual fact a non-Newtonian fluid, so that the slower the motion of the gas in the region containing foam, the more its relative mobility should decrease. Typically the way that such non-Newtonian behaviour is modelled (Ma et al. Reference Ma, Ren, Mateen, Morel and Cordelier2015; Zeng et al. Reference Zeng, Muthuswamy, Ma, Wang, Farajzadeh, Puerto, Vincent-Bonnieu, Eftekhari, Wang and Da2016) is to suppose that there is a maximum possible mobility reduction (the maximum amount that the gas mobility is reduced in the presence of foam compared to the situation without foam) and this is achieved for gas moving at or below a certain given speed, defined in terms of a capillary number. If the gas moves more quickly than this given speed however, the extent to which the mobility is reduced becomes less. It is worth remembering that the pressure-driven growth model applies specifically to the case in which the mobility at the finely textured foam front is much less than the mobility elsewhere in the system (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014). The pressure-driven growth model is therefore most likely to be applicable to systems in which the finely textured foam front experiences the maximum or close to the maximum mobility reduction. For this reason, within the pressure-driven growth model to be considered here, we choose to neglect the effect of non-Newtonian behaviour upon the mobility reduction. Another way of stating this is that we suppose that gas mobility is far more strongly affected by foam texture than by non-Newtonian effects, which is consistent with some of the cases considered in the study by Kovscek et al. (Reference Kovscek, Patzek and Radke1997).

Since the pressure-driven growth model relies so heavily upon having finely textured foam at the foam front, any effects that may compromise the integrity of that finely textured foam at the front may likewise impact upon the validity of the model. Although the model itself formally describes the process of surfactant alternating gas (i.e. gas injected into surfactant solution within a porous medium), the ultimate aim of the surfactant-alternating-gas process is to use foam to improve oil recovery. The presence of oil mixed with the surfactant solution might negatively affect the foam stability (see e.g. Farajzadeh et al. Reference Farajzadeh, Andrianov, Krastev, Hirasaki and Rossen2012; Osei-Bonsu, Shokri & Grassia Reference Osei-Bonsu, Shokri and Grassia2015, Reference Osei-Bonsu, Shokri and Grassia2016) and thereby influence the texture of the foam at the foam front. In addition to there being a requirement to maintain the integrity of the finely textured foam at the front itself, the validity of the model also demands that this finely textured foam be localised at the front (Shan & Rossen Reference Shan and Rossen2004). Failure of the foam to collapse abruptly and thereby coarsen as it dries out upstream, would affect the validity of the model in much the same fashion as oil destroying the integrity of the finely textured foam at the front.

Yet another aspect of the physics of foam in porous media that pressure-driven growth does not attempt to describe is the relative motion between liquid and gas within the foam. The model (Shan & Rossen Reference Shan and Rossen2004) does consider the gravity-induced hydrostatic pressure difference between liquid downstream of the foam front and gas upstream, but not any gravity-induced gas-to-liquid relative motion within the foam itself. Such relative motion within the foam with liquid draining downwards and gas rising upwards can lead to very low liquid content at the top of the foam. This dry out at the top makes the foam very coarsely textured and hence the gas mobility rises dramatically there. A tongue of highly mobile gas at the top of the system can then propagate above the liquid surfactant solution, far in advance of the remainder of the foam front which is much less mobile. This effect has been found by Kovscek et al. (Reference Kovscek, Patzek and Radke1997) to result in a concave corner forming at the point where the highly mobile tongue of gas meets the remainder of the front.

The pressure-driven growth model focusses attention solely on the finely textured low mobility zone at the foam front: therefore it is not designed to capture the aforementioned tongue of highly mobile gas at the top nor the concave corner that occurs where the tongue meets the rest of the front. Pressure-driven growth can however capture a related phenomenon associated with foam propagating into a surfactant solution with spatially non-uniform surfactant concentration, specifically with a lower surfactant concentration (and hence lower density) at the top, and a higher surfactant concentration (and hence higher density) lower down (Mas-Hernández et al. Reference Mas-Hernández, Grassia and Shokri2015a ). This again can lead to a concave corner in the foam front at the point where the lower surfactant concentration region (with less stable and hence less finely textured foam) meets the higher surfactant concentration region (with more stable and hence more finely textured foam). Our purpose in this work however is not to consider such systems with spatial non-uniformities in the foam texture measured along the front, but rather to investigate whether concave corners can occur even for foam fronts for which there is no variation in texture along the front itself: as we shall see later on, concave corners in front shape can develop even under these conditions.

Having now discussed the physics that the pressure-driven growth model includes, as well as the physics that it excludes, we proceed (in the next section) to introduce the governing equations for the model.

2.3 Governing equation for pressure-driven growth

The equation governing the pressure-driven growth model (Shan & Rossen Reference Shan and Rossen2004; Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) describes the motion of a material point $\boldsymbol{x}$ on the foam front via

(2.1) $$\begin{eqnarray}\frac{\text{d}\boldsymbol{x}}{\text{d}t}=\frac{k\unicode[STIX]{x1D706}_{r}}{(1-S_{w})\unicode[STIX]{x1D6F7}}\frac{\unicode[STIX]{x0394}P}{\unicode[STIX]{x1D70F}s}\boldsymbol{n},\end{eqnarray}$$

where $t$ is time, $\text{d}/\text{d}t$ represents a time derivative following a front material point, $\boldsymbol{n}$ is the front normal (which varies with distance ${\mathcal{S}}$ measured along the front, see figure 1(b)), $k$ is the permeability of the reservoir, $\unicode[STIX]{x1D706}_{r}$ is the relative mobility within the finely textured foam zone (relative mobility being the ratio between a relative permeability and an effective viscosity as already explained earlier), $S_{w}$ is the liquid fraction in the foam, $\unicode[STIX]{x1D6F7}$ is the reservoir porosity, $\unicode[STIX]{x0394}P$ is the driving pressure difference, $s$ is the distance the material point has travelled to date (again see figure 1 b), $\unicode[STIX]{x1D70F}$ (a small dimensionless parameter) is the ratio between the thickness of the finely textured foam front and the distance a material point has travelled: Grassia et al. (Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) employed a value $\unicode[STIX]{x1D70F}=0.01$ . As the distance $s$ through which the front has displaced grows, the thickness of the finely textured foam front $\unicode[STIX]{x1D70F}s$ likewise grows, and, since the dissipation is tied to the fine texture, the front slows down as a result. An additional slow down of the front over and above this could be incorporated into the governing equation due to non-Newtonian effects, but (for reasons already explained in § 2.2) such effects are neglected here.

The maximum depth to which the foam front can penetrate $d_{max}$ is given by balancing injection pressure with hydrostatic pressure and satisfies

(2.2) $$\begin{eqnarray}d_{max}=P_{inj}/(\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}g),\end{eqnarray}$$

where $P_{inj}$ is the injection pressure, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ is the liquid-to-gas density difference and $g$ is acceleration due to gravity.

We define our $x$ , $y$ coordinate system (figure 1 b) such that $x=0$ corresponds to the location of gas injection and $y=0$ corresponds to the maximum penetration depth. It follows that the net driving pressure (injection pressure less hydrostatic) grows linearly with coordinate $y$

(2.3) $$\begin{eqnarray}\unicode[STIX]{x0394}P=P_{inj}y/d_{max}.\end{eqnarray}$$

2.4 Dimensionless equations

We now make our governing equations dimensionless. Distances are non-dimensionalised on the scale $d_{max}$ , pressures are non-dimensionalised on the scale $P_{inj}$ and times are non-dimensionalised on the scale

(2.4) $$\begin{eqnarray}t_{scale}=\frac{(1-S_{w})\unicode[STIX]{x1D719}}{k\unicode[STIX]{x1D706}_{r}}\frac{d_{max}^{2}}{P_{inj}}\unicode[STIX]{x1D70F}.\end{eqnarray}$$

If we now use the symbols $\boldsymbol{x}$ , $s$ and $t$ to denote dimensionless (instead of dimensional) variables, we deduce the dimensionless analogue of (2.1)

(2.5) $$\begin{eqnarray}\frac{\text{d}\boldsymbol{x}}{\text{d}t}=\frac{y}{s}\boldsymbol{n}.\end{eqnarray}$$

Taking the dot product of this with $\boldsymbol{n}$ , it can be deduced that $s$ (the dimensionless path length over which the material point has travelled in dimensionless time $t$ ) evolves according to

(2.6) $$\begin{eqnarray}\text{d}s/\text{d}t=y/s.\end{eqnarray}$$

These dimensionless equations must be solved with suitable initial and boundary conditions. Regarding initial conditions, at $t=0$ , we want the front to be at $x=0$ for all $y$ values in the solution domain, with additionally $s=0$ initially for all $y$ . This however predicts infinite $\text{d}\boldsymbol{x}/\text{d}t$ initially. In numerical computations, to avoid having infinite velocities initially, typically we put the front at some location $x=s_{0}$ (where $s_{0}$ is a dimensionless parameter much smaller than unity) with in addition $s=s_{0}$ initially. Typically $s_{0}$ is chosen (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) with a value of the order of 10 $^{-2}$ . Regarding boundary conditions, at the top of the solution domain $y=1$ , we require that the front material points move parallel to the top boundary: in other words the normal $\boldsymbol{n}$ is horizontal at that point. Suppose we use the symbol $\unicode[STIX]{x1D6FC}$ (as sketched in figure 1 b) to denote the angle that the normal makes to the horizontal at any point on the front. By definition

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}\equiv \arctan (\text{d}x/\text{d}y),\end{eqnarray}$$

where $\text{d}/\text{d}y$ denotes a spatial derivative with respect to vertical location along the foam front at any given instant in time (a contrast from $\text{d}/\text{d}t$ which denotes the motion of material points on trajectories that are normal to the instantaneous foam front). In view of (2.7), another way of writing the top boundary condition is as $\unicode[STIX]{x1D6FC}=0$ at $y=1$ .

Corroborating a claim made back in § 2.1, one thing we immediately notice here based on (2.5)–(2.6) is that points higher up advance faster and hence further than points lower down. The conventional hypothesis in the literature is that this leads to a convex front shape as seen from downstream (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014; Mas-Hernández et al. Reference Mas-Hernández, Grassia and Shokri2015a ,Reference Mas-Hernández, Grassia and Shokri b ). One of our aims here is to investigate whether that conventional hypothesis is correct, or whether the shape might admit a concavity. The role that concavities can play from a mathematical point of view is described in the next section.

2.5 Mathematical nature of pressure-driven growth and the behaviour of concavities

In order to understand the possible significance of concavities in pressure-driven growth, it is useful to consider the nature of the model in general mathematical terms.

A key mathematical feature of pressure-driven growth is that it assigns no energy cost to the overall length of the front, meaning that the solutions can admit some really quite unusual front shapes, including shapes containing various types of singularities. In particular it is known that concave regions in a front tend to focus down into corners which are ‘arbitrarily’ sharp (‘arbitrarily’ sharp in the current context implying a radius of curvature comparable with the thickness of the dissipative, finely textured foam zone, which is much less than the distance over which the front travels (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014)). The fact that singularities can arise within solutions of pressure-driven growth is unsurprising because a formal equivalence has been proven (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) between pressure-driven growth and another equation that admits singularities, namely the eikonal equation, a well-known hyperbolic partial differential equation originally formulated in the context of optics (Arnold Reference Arnold2004).

Returning to consider the case of pressure-driven growth, when a Lagrangian scheme is used to evolve the front location, any concave corners that occur need to be propagated differently from surrounding front material points (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014; Mas-Hernández et al. Reference Mas-Hernández, Grassia and Shokri2015a ,Reference Mas-Hernández, Grassia and Shokri b ). In such numerical schemes, the consequences of not propagating the concave corners correctly are very serious indeed: segments of the front cross-over one another, forming topologically infeasible spurious loops in the representation of the front, and these spurious loops then grow indefinitely (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014).

These issues should not arise of course for a wholly convex front. If concavities do nevertheless appear in a computed foam front shape, we can draw one of two conclusions: either a numerical scheme has artificially introduced a concavity where one is not supposed to be, or else the foam front really does have a concavity according to the underlying model.

It is explained by Grassia et al. (Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) that it is relatively easy to introduce a concavity inadvertently in a numerical scheme designed to track Lagrangian material points on a foam front: particularly when one is adding new material points in the process of regridding a numerical representation of the front, such regridding needing to occur from time to time whenever material points are separating from one another. If the new material point is placed too far back, a concavity is created at the newly placed point itself. On the other hand, if the new material point is placed too far forward, a concavity is created above or below the point (or both). The problem is particularly acute on any parts of the front where the front curvature happens to be low: since then the set of locations within which new material points can be safely placed is very limited indeed.

If, even in spite of best efforts to place newly added numerical points in such a way as to avoid artificially introducing concavities, it still turns out that concavities manage to arise, the question then arises as to whether the concavities might actually be an inherent part of the front shape. This is the question we wish to explore here, employing asymptotic rather than numerical techniques (in order to be confident that any predictions we make are not simply numerical artefacts associated with Lagrangian schemes as described above).

3.1 Derivation of the Velde solution

Recall that in the situation under consideration here, gas is injected into surfactant solution initially along a vertical injection well, so that the foam front is initially vertical and its normal is initially horizontal. Although the direction of the front normal does not remain horizontal at all times, the Velde solution recognises that at sufficiently early times, the front normal will point nearly horizontally and the front motion is primarily horizontal with only a small vertical component superposed.

The $x$ component of (2.5) can be written

(3.1) $$\begin{eqnarray}\frac{\text{d}x}{\text{d}t}=\frac{y}{s}\cos \unicode[STIX]{x1D6FC},\end{eqnarray}$$

where we recall that $\text{d}/\text{d}t$ denotes a time derivative following a material point (a contrast from the notation $\text{d}/\text{d}y$ , which denotes a spatial derivative with respect to vertical location along the foam front from material point to material point). Since for nearly horizontal motion, $s$ is nearly the same as $x$ and $\cos \unicode[STIX]{x1D6FC}$ is close to unity, it follows from (3.1) that

(3.2) $$\begin{eqnarray}x\approx \sqrt{2yt},\end{eqnarray}$$

where the temporal variation of $y$ has been ignored with respect to that of $x$ . Equivalently

(3.3) $$\begin{eqnarray}y\approx \frac{x^{2}}{2t},\end{eqnarray}$$

showing that the predicted front shape is a parabola. This parabolic shape relies on the fact that gas is injected initially over the entire vertical depth of the front: if gas had not been injected over the entire vertical line but instead over a much more localised region as has been considered by Kovscek et al. (Reference Kovscek, Patzek and Radke1997), rather different front shapes then result.

The Velde solution (3.2) whilst undoubtedly useful has a number of issues. Firstly, since it implies

(3.4) $$\begin{eqnarray}\frac{\text{d}x}{\text{d}y}\approx \sqrt{\frac{t}{2y}}\end{eqnarray}$$

but assumes a nearly vertical front, i.e.  $\text{d}x/\text{d}y\ll 1$ , it requires $y\gg t/2$ . Under these conditions (combining (3.4) with (2.7)) we find

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}\approx \frac{\text{d}x}{\text{d}y}\approx \sqrt{\frac{t}{2y}}\ll 1.\end{eqnarray}$$

This is however invalid for small $y$ values of order $t/2$ or less, but this is actually not problematic for computing how the front shape evolves. We are dealing with $t\ll 1$ here, and hence $y$ must be exceedingly small if it is of order $t/2$ or less: there is little net front motion as $y\rightarrow 0$ anyway, because driving pressure and background hydrostatic pressure are nearly in balance.

Determining what is happening near the top of the front $y=1$ is more problematic. Although the Velde solution (3.2) gives the correct $x$ value for material points moving along the top boundary $x=\sqrt{2t}$ , it gives a value of $\text{d}x/\text{d}y$ at the top equal to $\sqrt{t/2}$ . We are assuming that $t\ll 1$ here, so $\sqrt{t/2}$ is a small parameter, but nonetheless it differs from zero, despite the fact that the boundary condition at the top requires that it should be zero. The explanation why the top boundary condition is not satisfied by the Velde solution is considered below.

3.2 Vertical motion predicted by the Velde solution

The reason why the Velde solution (3.2) is problematic near the top boundary can be seen by examining the vertical motion of material points. The $y$ component of (2.5) is

(3.6) $$\begin{eqnarray}\frac{\text{d}y}{\text{d}t}=-\frac{y}{s}\sin \unicode[STIX]{x1D6FC}.\end{eqnarray}$$

Since in the regime of interest, $s\approx x$ and $\sin \unicode[STIX]{x1D6FC}\approx \tan \unicode[STIX]{x1D6FC}\equiv \text{d}x/\text{d}y$ , it follows using (3.2) and (3.4) that

(3.7) $$\begin{eqnarray}\text{d}y/\text{d}t\approx -1/2\end{eqnarray}$$

uniformly for all $y$ . In other words, any material points to which the Velde solution applies also obey (3.7) and hence by time $t$ invariably find themselves at $y$ values below $1-t/2$ .

Thus the Velde solution is only valid in the lower region of the solution domain, namely for $y<1-t/2$ . In the upper region, i.e. in the domain $1-t/2\leqslant y\leqslant 1$ , a different solution must be found, and this solution must meet the boundary condition that $\unicode[STIX]{x1D6FC}=0$ on the top boundary. We will analyse the upper region of the solution domain in § 4, but before doing that, we wish to consider the consequences that (3.7) implies for the lower region.

3.3 Improving the Velde solution

Equation (3.7) makes it possible to improve upon the Velde solution (3.2), which was originally based on the assumption of constant rather than variable $y$ . In what follows, the improvement to the Velde solution is derived: those readers who wish to skip the derivation should turn directly to (3.13)–(3.14) and the results in § 3.3.1. The derivation that we follow is similar to one that has already been employed by Grassia (Reference Grassia2017) to describe pressure-driven growth within porous media with anisotropic permeability. Here the system is considered to be isotropic, and there are some subtle (albeit significant) differences in the equations obtained: see Grassia (Reference Grassia2017) for details.

The improvement to the Velde solution can be obtained starting from (2.5) by recognising that $s$ at any instant is slightly larger than $x$ (owing to the vertical component of motion being superposed upon the horizontal component), that $\cos \unicode[STIX]{x1D6FC}$ is well approximated by $1-\unicode[STIX]{x1D6FC}^{2}/2$ , and also that historical values of $y$ for any given material point were slightly larger than the current values.

Specifically

(3.8) $$\begin{eqnarray}s=\int _{0}^{x}\left(1+\frac{({\dot{y}})^{2}}{({\dot{x}})^{2}}\right)^{1/2}\,\text{d}x\approx \int _{0}^{x}\left(1+\frac{1}{2}\frac{({\dot{y}})^{2}}{({\dot{x}})^{2}}\right)\,\text{d}x,\end{eqnarray}$$

where ${\dot{x}}$ and ${\dot{y}}$ are shorthand for time derivatives. If we substitute ${\dot{x}}$ from the Velde solution (3.2), replace $\text{d}x$ by ${\dot{x}}\,\text{d}t$ , set ${\dot{y}}\approx -1/2$ and integrate, we deduce

(3.9) $$\begin{eqnarray}s\approx x(1+t/(12y)).\end{eqnarray}$$

It follows (for small $t$ )

(3.10) $$\begin{eqnarray}s^{-1}\approx x^{-1}(1-t/(12y)).\end{eqnarray}$$

Meanwhile

(3.11) $$\begin{eqnarray}\cos \unicode[STIX]{x1D6FC}\approx 1-\frac{\unicode[STIX]{x1D6FC}^{2}}{2}\approx 1-\frac{(\text{d}x/\text{d}y)^{2}}{2}\approx 1-\frac{t}{4y},\end{eqnarray}$$

where (3.4) has been used. Additionally, if a material point is at location $y$ at time $t$ , then, at some earlier time $t^{\prime }$ , its location must have been $y+(t-t^{\prime })/2$ or equivalently $y(1+(t-t^{\prime })/(2y))$ .

Substituting the above into (3.1) it follows

(3.12) $$\begin{eqnarray}x\approx \sqrt{2y\int _{0}^{t}(1-t^{\prime }/(12y))(1+(t-t^{\prime })/(2y))(1-t^{\prime }/(4y))\,\text{d}t^{\prime }}.\end{eqnarray}$$

Performing the integrals, retaining only terms through to order $t^{2}$ (discarding any higher powers of $t$ ) we deduce

(3.13) $$\begin{eqnarray}x\approx \sqrt{2yt+t^{2}/6}.\end{eqnarray}$$

Equation (3.13) can be viewed as a correction or improvement to the Velde solution incorporating contributions to $x$ at order $t^{3/2}$ over and above the leading-order term $\sqrt{2yt}$ . It is clear that (at any given $y$ ) the value of $x$ according to (3.13) is slightly larger than what is predicted by the Velde solution (3.2). In other words the front is displaced to the right, i.e. it travels slightly further than the Velde solution predicts, benefitting from the fact that historically material points were moving more rapidly having been higher up than their current $y$ location.

Equivalent to (3.13) we can write

(3.14) $$\begin{eqnarray}y\approx \frac{x^{2}}{2t}-\frac{t}{12},\end{eqnarray}$$

which predicts that the ‘improved’ Velde solution is a vertical distance $t/12$ below the original given by (3.3) regardless of the value of $x$ . As a result there is less unswept area underneath the front (i.e. less gravity override) with the ‘improved’ Velde solution (3.13) than with the original (3.2). Rearranging (3.2) to obtain (3.3), and integrating equation (3.3) from $x=0$ to the leading edge of the front at $x=\sqrt{2t}$ , we deduce the unswept area under the original Velde solution to be $\sqrt{2t}/3$ . The improvement to the Velde solution (i.e. (3.14) obtained from (3.13)) however reduces this unswept area by an amount $\sqrt{2}t^{3/2}/12$ , thereby helping to arrest the growth in unswept area and limit the override.

3.3.1 Comparison with simulation data

In figure 2 we have compared the prediction of (3.13) with a numerically simulated foam front, all results corresponding to a time $t=0.5$ . These numerical simulations were performed using the same algorithm for computing the trajectories of Lagrangian material points as described in detail in Grassia et al. (Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014). The simulations used 40 material points initially (albeit with additional material points being added over time as points separate from one another, there being 46 material points in total by time $t=0.5$ ) and a time step of $5\times 10^{-5}$ .

Figure 2. Velde solution (3.2) and ‘improved’ Velde solution (3.13) compared with numerical simulation data obtained from tracking Lagrangian material points on the front. Data correspond to time $t=0.5$ .

At time $t=0.5$ , the error in vertical position between the ‘original’ Velde solution (3.3) and the ‘improved’ solution (3.14) is $t/12\approx 0.04$ . Considering that $y$ varies between 0 and 1 here, the difference between the ‘original’ and ‘improved’ Velde solutions is still small in relative terms, even for $t=0.5$ . Nonetheless it is clear that the numerical simulation data in figure 2 agree much better with the ‘improved’ Velde solution (3.13) than with the original Velde solution (3.2). Indeed, given that the ‘improved’ Velde solution is formally a $t\ll 1$ solution, it is remarkable that it agrees so well with $t=0.5$ numerical data. Nonetheless it is clear that the ‘improved’ solution still only applies in the lower region of the solution domain. Equation (3.13) is seen to deviate from the numerical simulation data in the upper region of the domain as $y\rightarrow 1$ . It is not so much a case of (3.13) ceasing to be a valid solution for pressure-driven growth per se, but instead a case of the upper region being influenced by a top boundary condition which (3.13) fails to respect. Not only does (3.13) fail to give $\text{d}x/\text{d}y\rightarrow 0$ in the limit as $y\rightarrow 1$ , it also gives the wrong value of $x$ , predicting $x\approx \sqrt{2t+t^{2}/6}$ in this limit, instead of the correct value $x=\sqrt{2t}$ .

3.4 Improved estimate of vertical motion

Having obtained an improved estimate of the displacement in the $x$ direction (via (3.13)) and having demonstrated in § 3.3.1 that it really does fit the front shape better, we can now proceed to obtain an improved estimate of the trajectory of $y$ versus $t$ . A derivation follows, but some readers may wish to skip to the final result in (3.23).

We start by considering (2.6), and note that on the right-hand side we can replace $y$ to a good approximation by the following

(3.15) $$\begin{eqnarray}y\approx y_{0}-t/2,\end{eqnarray}$$

where $y_{0}$ is the initial location of the material point currently at $y$ . Upon integrating (2.6) and Taylor expanding for small $t$

(3.16) $$\begin{eqnarray}s\approx \sqrt{2y_{0}t-\frac{t^{2}}{2}}\approx \sqrt{2y_{0}t}\left(1-\frac{t}{8y_{0}}\right)\end{eqnarray}$$

and hence

(3.17) $$\begin{eqnarray}\frac{1}{s}\approx \frac{1}{\sqrt{2y_{0}t}}\left(1+\frac{t}{8y_{0}}\right).\end{eqnarray}$$

Via (3.13)

(3.18) $$\begin{eqnarray}\frac{\text{d}x}{\text{d}y}\approx \frac{t}{\sqrt{2yt+t^{2}/6}}.\end{eqnarray}$$

Substituting from (3.15) and Taylor expanding for small $t$ it follows

(3.19) $$\begin{eqnarray}\frac{\text{d}x}{\text{d}y}\approx \sqrt{\frac{t}{2y_{0}}}\left(1+\frac{5t}{24y_{0}}\right).\end{eqnarray}$$

Moreover (again via a Taylor expansion)

(3.20) $$\begin{eqnarray}\sin \unicode[STIX]{x1D6FC}=\sin \left(\arctan \frac{\text{d}x}{\text{d}y}\right)\approx \frac{\text{d}x}{\text{d}y}-\frac{1}{2}\left(\frac{\text{d}x}{\text{d}y}\right)^{3}.\end{eqnarray}$$

Upon substituting from (3.19) and expanding for small $t$

(3.21) $$\begin{eqnarray}\sin \unicode[STIX]{x1D6FC}\approx \sqrt{\frac{t}{2y_{0}}}\left(1-\frac{t}{24y_{0}}\right).\end{eqnarray}$$

Substituting (3.15), (3.17) and (3.21) into (3.6)

(3.22) $$\begin{eqnarray}y-y_{0}\approx -\frac{1}{2y_{0}}\int _{0}^{t}(1+t^{\prime }/(8y_{0}))(y_{0}-t^{\prime }/2)(1-t^{\prime }/(24y_{0}))\,\text{d}t^{\prime }.\end{eqnarray}$$

Integrating, whilst discarding any terms higher than order $t^{2}$ , gives

(3.23) $$\begin{eqnarray}y\approx y_{0}-\frac{t}{2}+\frac{5t^{2}}{48y_{0}},\end{eqnarray}$$

where $y_{0}\leqslant 1$ is the material point’s initial location.

This equation indicates that material points actually migrate downward slightly slower than speed $1/2$ (a point we will return to much later on in § 8), the main contributing effect being that the net driving pressure (injection pressure less hydrostatic pressure) decays as points move downwards. Nonetheless material points still do migrate downwards, so the improvements to the Velde solution that we have presented here cannot apply all the way up to $y=1$ for any finite $t>0$ : that requires a different analysis as we now consider.

4.1 Rotation rate of material elements

In this section, an equation (namely (4.4)) for the rotation rate of material elements is derived. One of the key questions that we wish to address in this work is whether or not this equation for the rotation rate can be replaced by a slightly simpler equation, namely (4.5). The relevant derivations follow.

Material elements rotate due to non-uniformities in the speed of material points from place to place. It follows then that the rotation rate of a material element $\dot{\unicode[STIX]{x1D6FC}}$ depends on how the material point speed varies with arc length measured along the front, i.e.

(4.1) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FC}}=-\frac{\text{d}}{\text{d}{\mathcal{S}}}\left(\frac{y}{s}\right).\end{eqnarray}$$

Here $y/s$ is the material point speed according to (2.6) ( $y$ is vertical coordinate, $s$ is distance measured along the trajectory). Moreover ${\mathcal{S}}$ is distance measured along the front (in the downwards direction).

We can expand this as follows

(4.2) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FC}}=-\frac{1}{s}\frac{\text{d}y}{\text{d}{\mathcal{S}}}+\frac{y}{s^{2}}\frac{\text{d}s}{\text{d}y}\frac{\text{d}y}{\text{d}{\mathcal{S}}}.\end{eqnarray}$$

Since $\text{d}y/\text{d}{\mathcal{S}}=-\text{cos}\,\unicode[STIX]{x1D6FC}$ (by definition) we deduce

(4.3) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FC}}=\frac{\cos \unicode[STIX]{x1D6FC}}{s}\left(1-\frac{y}{s}\frac{\text{d}s}{\text{d}y}\right).\end{eqnarray}$$

At small times and near the top boundary, $\cos \unicode[STIX]{x1D6FC}\approx 1$ , $s\approx \sqrt{2t}$ , $y\approx 1$ and hence,

(4.4) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FC}}\approx \frac{1}{\sqrt{2t}}\left(1-\frac{1}{\sqrt{2t}}\frac{\text{d}s}{\text{d}y}\right).\end{eqnarray}$$

At early times we know that $s$ is well approximated by $x$ : being able to replace $s$ by $x$ is a significant simplification as $s$ is history dependent, but $x$ depends only on instantaneous location of a material point. It is valid to ask whether in the above formula for $\dot{\unicode[STIX]{x1D6FC}}$ we can then replace $\text{d}s/\text{d}y$ by $\text{d}x/\text{d}y$ (the latter being $\tan \unicode[STIX]{x1D6FC}$ , which is essentially the same as $\unicode[STIX]{x1D6FC}$ at small times when $\unicode[STIX]{x1D6FC}$ is invariably small). The difficulty is that near the top boundary $\text{d}x/\text{d}y$ vanishes, so (even though $x$ and $s$ are close), it is not a priori clear whether $\text{d}x/\text{d}y$ and $\text{d}s/\text{d}y$ are actually similar in magnitude or not.

Suppose that it is possible to make this replacement. Then

(4.5) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FC}}\sim \frac{1}{\sqrt{2t}}\left(1-\frac{1}{\sqrt{2t}}\,\frac{\text{d}x}{\text{d}y}\right)\approx \frac{1}{\sqrt{2t}}\left(1-\frac{\unicode[STIX]{x1D6FC}}{\sqrt{2t}}\right).\end{eqnarray}$$

At the top boundary, $\unicode[STIX]{x1D6FC}\rightarrow 0$ , so $\dot{\unicode[STIX]{x1D6FC}}\approx 1/\sqrt{2t}$ regardless of whether (4.4) or (4.5) is used. Below the top boundary however, the solution for $x$ versus $y$ might be expected to match onto the lower part of the solution domain, where the Velde solution $x\approx \sqrt{2yt}$ as given by (3.2) should be (approximately) valid. The matching point is expected to occur somewhere near $y\approx 1-t/2$ (with $t$ being chosen much smaller than unity here). This implies that $\unicode[STIX]{x1D6FC}\approx \text{d}x/\text{d}y\approx \sqrt{t/(2y)}$ according to (3.4), which for $y$ values close to unity (i.e. for $1-y\ll 1$ ) becomes $\unicode[STIX]{x1D6FC}\approx \sqrt{t/2}$ . Hence (according to (4.5) and assuming $\unicode[STIX]{x1D6FC}$ is a continuous function moving between the upper and lower regions) at the matching point, we must have $\dot{\unicode[STIX]{x1D6FC}}\approx 1/(2\sqrt{2t})$ , which is only half as big as $\dot{\unicode[STIX]{x1D6FC}}$ at the top boundary. It would appear that there are significant spatio-temporal non-uniformities in $\dot{\unicode[STIX]{x1D6FC}}$ across a comparatively small domain.

4.2 Vertical motion of material elements

According to the pressure-driven growth model, the vertical motion of material elements is as per (3.6). Since in the upper region we are interested in $y$ values near unity, $s$ values roughly $\sqrt{2t}$ , and $\unicode[STIX]{x1D6FC}$ much smaller than unity it follows

(4.6) $$\begin{eqnarray}{\dot{y}}\approx -\unicode[STIX]{x1D6FC}/\sqrt{2t}.\end{eqnarray}$$

Note this is apparently consistent with the foregoing analysis for the lower region discussed in § 3. The matching point with the Velde solution, should be close to $y=1$ when $t\ll 1$ , and is expected to satisfy $\unicode[STIX]{x1D6FC}\approx \sqrt{t/2}$ , at least under the assumption that the upper and lower regions are to join up with the same orientation (in which case (3.4) can be applied where the regions join). It then follows ${\dot{y}}\approx -1/2$ , and hence, on the basis of this assumption, the matching point is therefore predicted to be approximately a distance $t/2$ below the top boundary.

What we have demonstrated here is that, if the upper and lower regions match up with the same front orientation, then the vertical velocity component of the uppermost material point of the lower region is the same as the vertical velocity component of the lowermost material point of the upper region. Given these two material points are initially adjacent, if their subsequent vertical velocity components are also the same, then they must remain adjacent to one another as time proceeds. Under these circumstances, the matching point is (by definition) precisely where these two material points meet, and it too must move downwards with the same velocity component as the material points themselves do.

What we have not ruled out of course is a more complicated case where the upper and lower regions join up with different front orientations e.g. in a concave corner. It is known in the literature for instance that concave corners can consume material points (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014). Thus the material points either side of a concave corner at any given time are not necessarily the same as those either side of that concave corner at any other time. It follows then that the downward motion of the concave corner need not match that of material points either, since the identities of the material points immediately adjacent to either side of the corner are continually changing.

As we will see (in §§ 6–7), it is absolutely essential to consider this more complicated situation for a proper understanding of the shape of the foam front. For the present however, we wish to explore the consequences of assuming the simpler situation for which the upper and lower regions match with the same front orientation. Ultimately this assumption will lead us towards a contradiction (see § 5), but the analysis we perform here is nonetheless useful, as it is amenable to be generalised to the more complex situations in §§ 6–7.

4.3 Proposed similarity solution for front shape and its governing equation

In this section, we propose a similarity solution for the front shape, and derive a governing equation for that similarity solution. The derivation follows, but some readers may prefer to skip directly to the result given in (4.17).

We begin by defining a similarity variable

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D701}=(1-y)/(t/2),\end{eqnarray}$$

with $\unicode[STIX]{x1D701}=0$ at the top boundary ( $y=1$ ) and $\unicode[STIX]{x1D701}=1$ at the expected location of the matching point ( $y=1-t/2$ ). Clearly $\unicode[STIX]{x1D701}$ represents a ratio of distances, i.e. the ratio between the distance that a point $y$ is below the top surface, and the (assumed) distance between the top surface and the matching point itself.

We now look for a similarity solution in the small-time limit assumed to take the form

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\sqrt{t/2}A(\unicode[STIX]{x1D701}),\end{eqnarray}$$

where $A(\unicode[STIX]{x1D701})$ is a function that we must determine. Matching onto the top boundary requires $A=0$ when $\unicode[STIX]{x1D701}=0$ and matching onto the lower region (assuming continuity of $\unicode[STIX]{x1D6FC}$ and hence of $A$ ) implies $A=1$ when $\unicode[STIX]{x1D701}=1$ . Note the geometrical interpretation of this: the larger the value of $A$ (and hence of $\unicode[STIX]{x1D6FC}$ for any given time $t$ ) the greater the extent to which a material element on the front has reoriented compared to the orientation at the top boundary, and hence the greater the curvature of the front.

Our aim here is to obtain a differential equation determining $A$ as a function of $\unicode[STIX]{x1D701}$ . The proposed similarity form given in (4.8) not only governs how $\unicode[STIX]{x1D6FC}$ varies with position on the front at fixed time $t$ , but also how $\unicode[STIX]{x1D6FC}$ and vertical position evolve with time following a specified material point. Taking a time derivative following a material point of the expression (4.7) for $\unicode[STIX]{x1D701}$ gives

(4.9) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D701}}=-\frac{{\dot{y}}}{t/2}-\unicode[STIX]{x1D701}/t.\end{eqnarray}$$

Since via (4.6) (applicable in the small-time limit) and (4.8)

(4.10) $$\begin{eqnarray}{\dot{y}}\approx -\unicode[STIX]{x1D6FC}/\sqrt{2t}\approx -A/2,\end{eqnarray}$$

it follows

(4.11) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D701}}\approx (A-\unicode[STIX]{x1D701})/t.\end{eqnarray}$$

We can also differentiate the expression (4.8) for $\unicode[STIX]{x1D6FC}$ with respect to time (at fixed  $y$ )

(4.12) $$\begin{eqnarray}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}t=1/(2t)\unicode[STIX]{x1D6FC}+\sqrt{t/2}(-\unicode[STIX]{x1D701}/t)A_{\unicode[STIX]{x1D701}}=\sqrt{t/2}A/(2t)+\sqrt{t/2}(-\unicode[STIX]{x1D701}/t)A_{\unicode[STIX]{x1D701}},\end{eqnarray}$$

where $A_{\unicode[STIX]{x1D701}}$ denotes the derivative of $A$ with respect to $\unicode[STIX]{x1D701}$ . Meanwhile

(4.13) $$\begin{eqnarray}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}y=-\sqrt{t/2}A_{\unicode[STIX]{x1D701}}/(t/2)\end{eqnarray}$$

and moreover (using (4.10))

(4.14) $$\begin{eqnarray}{\dot{y}}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}y\approx \sqrt{t/2}AA_{\unicode[STIX]{x1D701}}/t.\end{eqnarray}$$

Summing $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}t$ and ${\dot{y}}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}y$ we deduce for the rate of change of $\unicode[STIX]{x1D6FC}$ following a material point

(4.15) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FC}}\approx \sqrt{t/2}A/(2t)+\sqrt{t/2}(A-\unicode[STIX]{x1D701})A_{\unicode[STIX]{x1D701}}/t.\end{eqnarray}$$

We now equate this to $\dot{\unicode[STIX]{x1D6FC}}\sim (\sqrt{2t})^{-1}(1-\unicode[STIX]{x1D6FC}/\sqrt{2t})$ (via (4.5)) giving

(4.16) $$\begin{eqnarray}1-A/2\approx A/2+(A-\unicode[STIX]{x1D701})A_{\unicode[STIX]{x1D701}}\end{eqnarray}$$

and hence

(4.17) $$\begin{eqnarray}A_{\unicode[STIX]{x1D701}}\approx (1-A)/(A-\unicode[STIX]{x1D701}),\end{eqnarray}$$

which is the similarity equation (applicable in the small-time limit) we seek that governs the variation $A$ with respect to $\unicode[STIX]{x1D701}$ in the upper region of the front.

4.3.1 Implications for the similarity solution

Note the implication of (4.17) above: as $\unicode[STIX]{x1D701}$ grows, $A$ must be at least as large as $\unicode[STIX]{x1D701}$ , since if $\unicode[STIX]{x1D701}$ ever moves too close to $A$ , then $A$ necessarily suddenly increases. On the other hand, $A$ should not move too close to unity too quickly, otherwise $A$ stops increasing altogether. At the expected matching point then $\unicode[STIX]{x1D701}\rightarrow 1$ (or equivalently $y\rightarrow 1-t/2$ ), it is permitted only to have $A\rightarrow 1$ (corresponding to $A$ being continuous and hence $\unicode[STIX]{x1D6FC}$ being continuous at the expected matching point).

The above equation however assumes (via (4.5)) that the variation in the path length travelled from point to point on the front $\text{d}s/\text{d}y$ can be replaced by the variation in horizontal location $\text{d}x/\text{d}y$ . More generally however we have via (4.4) (still assuming a small-time limit)

(4.18) $$\begin{eqnarray}A_{\unicode[STIX]{x1D701}}\approx \left.\left(1-A/2-\frac{1}{\sqrt{2t}}\,\frac{\text{d}s}{\text{d}y}\right)\right/(A-\unicode[STIX]{x1D701}).\end{eqnarray}$$

If, as $y$ decreases moving down along the front starting from the top boundary, the value of $s$ happens to decrease more slowly than the value of $x$ , then the term in $\text{d}s/\text{d}y$ is smaller than $\text{d}x/\text{d}y$ , then this leads via (4.18) to a larger $A_{\unicode[STIX]{x1D701}}$ and hence a larger $A$ than previously, corresponding to a more strongly curved front. It is now possible for $A$ in the upper region of the front to exceed unity even for values of $\unicode[STIX]{x1D701}<1$ : this then allows for the appearance of a concave corner in the front shape, since the Velde solution implies that at the top of the lower region $\unicode[STIX]{x1D6FC}$ must jump immediately to $\sqrt{t/2}$ and hence $A\equiv (t/2)^{-1/2}\unicode[STIX]{x1D6FC}$ must jump immediately to unity. More subtly, it also allows for the location of the matching point between the upper and lower regions of the solution being not exactly a distance $t/2$ below the top boundary (although we will not demonstrate this fact until later on in § 6).

5.1 Implicit solution and limiting cases

Equation (4.17) has an (implicit) solution

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D701}=A-(1-A)\log (1/(1-A)).\end{eqnarray}$$

We deliberately use an equal sign here rather than an approximation sign to convey the notion that (5.1) is an exact solution of (4.17), even though (4.17) is itself only approximate, namely a small-time asymptotic approximation to the governing equation of the front. Later on we will have call to derive expressions which are themselves approximations to (5.1) and for these we will revert to using an approximation sign in order to distinguish them.

A graph of the solution (5.1) is plotted in figure 3. In the first instance we focus on solutions with $0\leqslant \unicode[STIX]{x1D701}\leqslant 1$ and $0\leqslant A\leqslant 1$ which is what (5.1) describes (although figure 3 includes a solution branch defined all the way up to $\unicode[STIX]{x1D701}=2$ and $A=2$ , as we will discuss shortly).

Figure 3. (a) Graph of $A$ versus $\unicode[STIX]{x1D701}$ (a rescaled version of angle through which the front turns $\unicode[STIX]{x1D6FC}$ versus distance from the top boundary $1-y$ ) applicable to the upper region of the solution domain. The curve labelled $c=1$ is based on assuming that $\text{d}s/\text{d}y$ (with $s$ being path length travelled) is the same (at leading order) as $\text{d}x/\text{d}y$ and corresponds to (5.1). Although the $c=1$ case does admit a branch of solutions with $A>1$ (equivalent to saying that $\unicode[STIX]{x1D6FC}$ can overshoot the value predicted by the Velde solution lower down in the solution domain) the $A$ versus $\unicode[STIX]{x1D701}$ curve plotted here exhibits an inflection at $A=1$ (which is not observed in simulation data (Torres-Ulloa Reference Torres-Ulloa2015)). Alternate solutions assuming $\text{d}s/\text{d}y$ is less than $\text{d}x/\text{d}y$ are also given, parameterised by a value $c$ (the ratio between $\text{d}s/\text{d}y$ and $\text{d}x/\text{d}y$ being equal to $2c-1$ ): these solutions are given by (6.2). The case $c=0.9$ is plotted (and is close to the original $c=1$ data set). The case of main interest is however $c=0.75$ . Data obtained from an integro-differential equation system (7.19) and (7.21) are also shown and are very close to the $c=0.75$ data set. (b) A zoomed view comparing the $c=0.75$ data set with the numerical solution of the integro-differential equation which is discretised into 1000 intervals. The curve labelled ‘less refined’ is for the integro-differential equation discretised into only 100 intervals. A discretisation into 10 000 intervals (albeit not plotted here) is indistinguishable on the scale of the graph from that for 1000 intervals. The vertical lines indicate the $\unicode[STIX]{x1D701}$ values at which the solutions generated are found to match onto the lower region of the solution domain $\unicode[STIX]{x1D701}\approx 0.94$ (for the $c=0.75$ solution) and $\unicode[STIX]{x1D701}\approx 0.954$ (for the integro-differential equation).

Note that for small $A\ll 1$ , the right-hand side of (5.1) Taylor expands to $A^{2}/2$ and hence for $\unicode[STIX]{x1D701}\ll 1$ , it follows $A\sim \sqrt{2\unicode[STIX]{x1D701}}$ . In that case, $\unicode[STIX]{x1D6FC}\equiv \sqrt{t/2}A$ becomes approximately $\sqrt{t\unicode[STIX]{x1D701}}\equiv \sqrt{2(1-y)}$ . This is the same behaviour (i.e. a mild singularity) near the top boundary as has been found for the late-time asymptotic solution (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) (and is applicable here because the material points in question have spent the overwhelming majority of their history on the top boundary, only having left the top boundary comparatively recently). It is interesting to note that this finding justifies the numerical procedure proposed by Grassia et al. (Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) for placement of material points that were newly introduced near the top boundary which assumed a mild singularity was present. Formally Grassia et al. (Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) only proved the validity of this procedure in the long-time limit $t\gg 1$ , but it evidently has broader applicability. Moreover this same behaviour near the top boundary can actually also be obtained from (4.18) as both (4.17) and (4.18) reduce to $A_{\unicode[STIX]{x1D701}}\approx 1/A$ in this limit (with in addition $A\gg \unicode[STIX]{x1D701}$ in this same limit). This predicted behaviour immediately adjacent to the top boundary does not then require an assumption that $\text{d}s/\text{d}y\approx \text{d}x/\text{d}y$ since (4.18) does not make that assumption.

Moving away from the top boundary, equation (5.1) given above rearranges to

(5.2) $$\begin{eqnarray}1-\unicode[STIX]{x1D701}=(1-A)(1+\log (1/(1-A))).\end{eqnarray}$$

When $\unicode[STIX]{x1D701}$ and $A$ are both close to unity (which we expect to happen moving downwards through the upper region towards the lower region), we have an approximation

(5.3) $$\begin{eqnarray}1-\unicode[STIX]{x1D701}\approx (1-A)\log (1/(1-A))\end{eqnarray}$$

and hence

(5.4) $$\begin{eqnarray}\log (1-\unicode[STIX]{x1D701})\approx \log (1-A)+\log \log (1/(1-A)).\end{eqnarray}$$

To a very crude approximation it follows from (5.4) that $\log (1-A)\sim \log (1-\unicode[STIX]{x1D701})$ and hence substituting this result into (5.2)

(5.5) $$\begin{eqnarray}1-A\approx (1-\unicode[STIX]{x1D701})(1+\log (1/(1-\unicode[STIX]{x1D701})))^{-1}.\end{eqnarray}$$

The implication is that, in the limit as $\unicode[STIX]{x1D701}\rightarrow 1$ and $A\rightarrow 1$ , $A$ varies more slowly than $\unicode[STIX]{x1D701}$ , i.e. rescaled front orientation angle varies more slowly than rescaled distance. If indeed the matching point between the upper and lower regions occurs at $\unicode[STIX]{x1D701}=A=1$ , this is consistent with the curvature of the front being small near the matching point. We have already commented (see § 2.5) that numerical schemes for simulating pressure-driven growth become particularly difficult to implement when curvature is low, as even small perturbations can then convert convex shapes into a potentially problematic concave ones: it is unsurprising therefore that previous work (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014) has encountered numerical challenges here.

Interestingly (4.17) for the upper region of the foam front actually admits solutions for $\unicode[STIX]{x1D701}$ and $A$ both greater that unity: if one replaces $\unicode[STIX]{x1D701}$ by $2-\unicode[STIX]{x1D701}$ and $A$ by $2-A$ within (5.1) then a different solution branch is obtained. This branch is also plotted on figure 3 covering the domain $1\leqslant \unicode[STIX]{x1D701}\leqslant 2$ and $1\leqslant A\leqslant 2$ . The existence of this solution branch raises the possibility that the value of $A$ , even within the upper region, might rise above unity, in which case in order to match onto the top of the lower region (which has $A=1$ ) a concave corner would be required. We will see later on (§ 6) that concave corners can and do indeed occur, albeit not in the fashion that (4.17) predicts. For the present however we continue to explore the consequences of assuming that the upper and lower regions join up without a concave corner.

5.2 Re-expressing solution in terms of injection time

It is possible to re-express the solution (5.1) above in terms of an ‘injection time’, defined as a time at which material points were injected from the top boundary onto the foam front, a concept that has already been described by Grassia et al. (Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014).

The injection time $t_{inj}$ can take any value between zero and the current time $t$ . If $t_{inj}\rightarrow t$ then material points are extremely close to the top boundary (i.e.  $\unicode[STIX]{x1D701}\ll 1$ and hence $A\ll 1$ ). On the other hand if $t_{inj}\ll t$ then material points are close to the matching point at which material which has been injected (constituting the upper region) joins up with material which has been continuously on the front since time zero (constituting the lower region). In that case we anticipate $\unicode[STIX]{x1D701}\approx 1$ and also (at least according to (5.1)) $A\approx 1$ .

It turns out that the solution (5.1) given above follows from a particularly simple functional form

(5.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=(t-t_{inj})/\sqrt{2t}.\end{eqnarray}$$

This equation is ‘exact’ in the same sense that (5.1) is, i.e. it manages to solve (4.17) exactly, notwithstanding the fact that (4.17) is itself a small-time approximation. Here we proceed by assuming that $\unicode[STIX]{x1D6FC}$ follows (5.6) and then demonstrating that (5.1) follows as a consequence. If (5.6) holds, then (using the definition $\unicode[STIX]{x1D6FC}\equiv \sqrt{t/2}A$ )

(5.7) $$\begin{eqnarray}A=1-t_{inj}/t,\end{eqnarray}$$

a very simple relation indicating that the higher the fraction of total time a material point has spent on the front since being injected, the more the material element upon which it finds itself has reoriented. On the other hand, if (5.6) holds, then we can solve (4.6) for $y$ versus $t$ . The solution is

(5.8) $$\begin{eqnarray}y=1-t/2+t_{inj}/2+(1/2)t_{inj}\log (t/t_{inj}).\end{eqnarray}$$

Since $\unicode[STIX]{x1D701}\equiv (1-y)/(t/2)$ , it follows

(5.9) $$\begin{eqnarray}\unicode[STIX]{x1D701}=1-\frac{t_{inj}}{t}+\frac{t_{inj}}{t}\log \frac{t}{t_{inj}}.\end{eqnarray}$$

Substituting from (5.7) then gives (5.1).

Note that differentiating $A$ and $\unicode[STIX]{x1D701}$ (or equivalently $\unicode[STIX]{x1D6FC}$ and $y$ ) with respect to $t_{inj}$ at constant $t$ , corresponds to moving from material point to material point for a given instantaneous curve shape. Notice that

(5.10) $$\begin{eqnarray}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x2202}y}=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}t_{inj}}{\unicode[STIX]{x2202}y/\unicode[STIX]{x2202}t_{inj}}=\frac{1/\sqrt{2t}}{(1/2)\log (t/t_{inj})},\end{eqnarray}$$

where the denominator diverges (albeit exceedingly slowly) as $t_{inj}\rightarrow 0$ . This is actually a demonstration that the predicted curvature falls to zero near the expected matching point, since (recalling that ${\mathcal{S}}$ is distance measured along the front in a downwards direction) the curvature becomes $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}{\mathcal{S}}$ which is equivalent to $-\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}y$ for a front that is nearly vertical, such as would happen when $t\ll 1$ .

This situation near the matching point with $t_{inj}\ll t$ contrasts with what is happening at the top boundary where $t_{inj}\rightarrow t$ . Near the top $\unicode[STIX]{x1D6FC}\sim \sqrt{2(1-y)}$ as we saw in § 5.1 and hence $-\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}y\sim 1/\sqrt{2(1-y)}$ , so curvature is infinite at the top boundary.

5.3 Horizontal mismatch of front shape

One elegant feature of the solution (5.1) for the upper region of the foam front is that at $y=1-t/2$ (equivalently at $\unicode[STIX]{x1D701}=1$ ) it has $A=1$ (equivalently $\unicode[STIX]{x1D6FC}=\sqrt{t/2}$ ) which is the same $\unicode[STIX]{x1D6FC}$ value as would be predicted by the Velde solution for the lower region of the front as discussed in § 3 (see e.g. (3.5) assuming $t\ll 1$ and hence with $y=1-t/2$ at the top of the lower region being close to unity). Thus we have two solutions for the distinct regions of the front which have the same orientation at the same vertical location, suggesting that the solutions for the upper and lower regions might join up with the same orientation angle, as we have assumed to date. Unfortunately however it turns out that there is a horizontal mismatch between these solutions. The detailed explanation is given below, but some readers may prefer to skip directly to view the horizontal mismatch plotted in figure 4.

Figure 4. (a) Graph of $-\unicode[STIX]{x1D701}$ versus $-\unicode[STIX]{x1D6EF}$ (a rescaled version of $y$ versus $x$ ) for the foam front. Solutions in the upper region of the domain are generated in the first instance assuming $\text{d}s/\text{d}y\approx \text{d}x/\text{d}y$ (curve labelled $c=1$ ; with angle $A$ obtained from (5.1), and $\unicode[STIX]{x1D6EF}$ obtained from integrating $A$ via (5.16)) and subsequently with $\text{d}s/\text{d}y<\text{d}x/\text{d}y$ (curve labelled $c=0.75$ ; $A$ obtained from (6.2)) and also shown is a curve where solutions were obtained via an integro-differential equation (7.19). These solutions for the upper region are required to intersect solutions that apply in the lower part of the domain (curve labelled lower region, given by (5.15)). For $c=1$ , no intersection is seen (at least not in the neighbourhood of $\unicode[STIX]{x1D701}=1$ ). (b) In this zoomed view, intersections are seen in the case $c=0.75$ and in the case of the integro-differential equation, respectively for $\unicode[STIX]{x1D701}\approx 0.94$ and $\unicode[STIX]{x1D701}\approx 0.954$ (horizontal lines).

We know via (3.13) that in the lower region of the domain, to a good approximation $x\approx \sqrt{2yt+t^{2}/6}$ which can be obtained via a perturbation expansion that improves upon the original Velde solution (3.2). We know moreover that the leading edge at the top of the front is at location $x=\sqrt{2t}$ with $y=1$ . Moving away from $y=1$ , if we define $\unicode[STIX]{x1D709}$ to be the horizontal distance behind the leading edge ( $\unicode[STIX]{x1D709}\equiv \sqrt{2t}-x$ ), it follows via a difference of squares (remembering that $x$ is itself close to $\sqrt{2t}$ )

(5.11) $$\begin{eqnarray}\unicode[STIX]{x1D709}\approx \frac{(\sqrt{2t}-x)(\sqrt{2t}+x)}{2\sqrt{2t}}\approx \frac{(2t-x^{2})}{2\sqrt{2t}}\approx \frac{2t-2yt-t^{2}/6}{2\sqrt{2t}}.\end{eqnarray}$$

If we put $y=1-t/2$ (the expected matching point), we deduce $\unicode[STIX]{x1D709}\approx 5t^{3/2}/(12\sqrt{2})$ via this ‘improved’ Velde solution (3.13).

There is however an alternative way of computing $\unicode[STIX]{x1D709}$ , as a function of $y$ , namely integrating $\text{d}x/\text{d}y$ to measure a change in $x$ for a given change in $y$ . For $t\ll 1$ , we can approximate $\text{d}x/\text{d}y$ by the angle $\unicode[STIX]{x1D6FC}$ , and therefore

(5.12) $$\begin{eqnarray}\unicode[STIX]{x1D709}\approx \int _{y}^{1}\unicode[STIX]{x1D6FC}\,\text{d}y\end{eqnarray}$$

and hence at $y=1-t/2$ ,

(5.13) $$\begin{eqnarray}\unicode[STIX]{x1D709}|_{y=1-t/2}\approx \int _{1-t/2}^{1}\unicode[STIX]{x1D6FC}\,\text{d}y,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ is based on the shape of the upper region. Using (5.8) to change the integration variable from $y$ to $t_{inj}$ , this can also be written

(5.14) $$\begin{eqnarray}\unicode[STIX]{x1D709}|_{y=1-t/2}\approx \int _{0}^{t}\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}y}{\unicode[STIX]{x2202}t_{inj}}\,\text{d}t_{inj}=\int _{0}^{t}\frac{(t-t_{inj})}{\sqrt{2t}}\frac{1}{2}\log \frac{t}{t_{inj}}\,\text{d}t_{inj}=3t^{3/2}/(8\sqrt{2}),\end{eqnarray}$$

where (5.6) has also been used in addition to (5.8). This $\unicode[STIX]{x1D709}$ value in (5.14) is smaller than the result obtained earlier by a factor $3/8\times 12/5=9/10$ .

In other words, moving downwards from the top boundary, the solution $\unicode[STIX]{x1D6FC}$ versus $y$ predicted by (5.1) does not curve sufficiently as to join up with the $x$ versus $y$ function in the lower region. The predicted horizontal location of the bottom of the upper region (5.14) is not the same as that of the top of the lower region ((5.11) with $y=1-t/2$ ). We can see this graphically in figure 4. In this figure we have defined a variable $\unicode[STIX]{x1D6EF}=(\sqrt{2t}-x)/t^{3/2}\equiv \unicode[STIX]{x1D709}/t^{3/2}$ along with $\unicode[STIX]{x1D701}=(1-y)/(t/2)$ , and have plotted $-\unicode[STIX]{x1D701}$ against $-\unicode[STIX]{x1D6EF}$ , which has the same orientation as a graph of $y$ versus $x$ .

In terms of $\unicode[STIX]{x1D6EF}$ , equation (5.11) describing the lower region of the front in the neighbourhood of the matching point becomes

(5.15) $$\begin{eqnarray}\unicode[STIX]{x1D6EF}\approx \frac{\unicode[STIX]{x1D701}}{2\sqrt{2}}-\frac{1}{12\sqrt{2}}\end{eqnarray}$$

whilst in the upper region we have (via (5.12))

(5.16) $$\begin{eqnarray}\unicode[STIX]{x1D6EF}\approx \frac{1}{2\sqrt{2}}\int _{0}^{\unicode[STIX]{x1D701}}A\,\text{d}\unicode[STIX]{x1D701},\end{eqnarray}$$

where $A$ is obtained in the first instance from (5.1).

Clearly in figure 4 there is no intersection between the upper and lower region curves near $\unicode[STIX]{x1D701}=1$ , at least when (5.1) is used to describe the upper region. In order to make the curves intersect, what we require in the upper region is a solution for $\unicode[STIX]{x1D6FC}$ versus $y$ that curves slightly more than that computed above, giving a slightly larger $\unicode[STIX]{x1D6FC}$ at any given $y$ (or equivalently a slightly larger $A$ at any given $\unicode[STIX]{x1D701}$ ). As was mentioned in § 4.3.1, the solution to (4.18), in lieu of (4.17) which is what we have been solving to date, should have that property.

5.4 Predicted difference between $\text{d}s/\text{d}y$ and $\text{d}x/\text{d}y$

In addition to the horizontal mismatch discussed above, it is possible to demonstrate that (5.1) contradicts the assumptions used to derive it. It turns out to predict that $\text{d}s/\text{d}y$ (with $s$ being a history-dependent path length variable) is significantly less than $\text{d}x/\text{d}y$ , even though (4.17) (of which (5.1) is the solution) was derived assuming $\text{d}s/\text{d}y\approx \text{d}x/\text{d}y$ . Some readers may wish to turn directly to (5.25)–(5.26) which demonstrate that this assumption is invalid. Nonetheless we will see that (5.1) still manages to make useful predictions at least on parts of the front, because there are parts of the front for which the local shape is unaffected in spite of the aforementioned contradiction.

The analysis proceeds as follows. To a good approximation

(5.17) $$\begin{eqnarray}s\approx \int _{0}^{x}\left(1+\frac{1}{2}\frac{({\dot{y}})^{2}}{({\dot{x}})^{2}}\right)\,\text{d}x\approx \int _{0}^{x}\left(1+\frac{1}{2}\unicode[STIX]{x1D6FC}^{2}\right)\,\text{d}x,\end{eqnarray}$$

where we have used the fact that at early time vertical velocity is much weaker than horizontal velocity $|{\dot{y}}|\ll |{\dot{x}}|$ , and also at leading order near the top boundary, ${\dot{y}}\approx -\unicode[STIX]{x1D6FC}/s$ and ${\dot{x}}\approx 1/s$ via e.g. (3.6) and (3.1).

Evaluating the integral in (5.17) is not possible without a priori information about  $\unicode[STIX]{x1D6FC}$ .

We know however via (5.6) that a material point injected onto the front at time $t_{inj}$ has an $\unicode[STIX]{x1D6FC}$ value $\unicode[STIX]{x1D6FC}=(t-t_{inj})/\sqrt{2t}$ . Substituting this into (5.17) we deduce

(5.18) $$\begin{eqnarray}s\approx x+\int _{t_{inj}}^{t}\frac{(t-t_{inj})^{2}}{4t}\,\frac{\text{d}t}{\sqrt{2t}}=x+\frac{(t-t_{inj})^{3}}{12\sqrt{2}t^{3/2}},\end{eqnarray}$$

where we have used the fact that $\text{d}x={\dot{x}}\,\text{d}t\approx \text{d}t/s\approx \text{d}t/\sqrt{2t}$ .

We know moreover that

(5.19) $$\begin{eqnarray}x=\sqrt{2t}-\unicode[STIX]{x1D709}\approx \sqrt{2t}-\int _{y}^{1}\unicode[STIX]{x1D6FC}\,\text{d}y.\end{eqnarray}$$

Changing the integration variable from $y$ to $t_{inj}$ using (5.8) which implies $\unicode[STIX]{x2202}y/\unicode[STIX]{x2202}t_{inj}=(1/2)\log (t/t_{inj})$ , we deduce

(5.20) $$\begin{eqnarray}x\approx \sqrt{2t}-\int _{t_{inj}}^{t}\frac{(t-t_{inj})}{\sqrt{2t}}\frac{1}{2}\log (t/t_{inj})\,\text{d}t_{inj}\end{eqnarray}$$

and hence

(5.21) $$\begin{eqnarray}x\approx \sqrt{2t}-\frac{t^{3/2}}{2\sqrt{2}}\left(-\frac{t_{inj}}{t}\left(1-\frac{t_{inj}}{2t}\right)\log (t/t_{inj})-\frac{t_{inj}}{t}\left(1-\frac{t_{inj}}{4t}\right)+\frac{3}{4}\right).\end{eqnarray}$$

Combining (5.18) and (5.21) we deduce

(5.22) $$\begin{eqnarray}s\approx \sqrt{2t}-\frac{t^{3/2}}{2\sqrt{2}}\left(-\frac{t_{inj}}{t}\left(1-\frac{t_{inj}}{2t}\right)\log (t/t_{inj})-\frac{t_{inj}}{t}\left(1-\frac{t_{inj}}{4t}\right)+\frac{3}{4}\right)+\frac{t^{3/2}}{12\sqrt{2}}(1-t_{inj}/t)^{3}.\end{eqnarray}$$

The physical content of both (5.21) and (5.22) is that as $t_{inj}$ falls below $t$ (corresponding to choosing material points further and further below the top boundary) both $x$ and $s$ fall short of the value $\sqrt{2t}$ (which is the displacement of the front on the top boundary), however they fall short of $\sqrt{2t}$ by differing amounts. The predicted ratio between the $s$ -shortfall and the $x$ -shortfall is sensitive to $t_{inj}/t$ (and hence is sensitive to the precise location within the upper region of the front which depends on $t_{inj}/t$ ). In what follows the limiting case $t_{inj}\rightarrow t$ (i.e. the behaviour at the very top of the upper region) is considered.

5.4.1 Limiting case $t_{inj}\rightarrow t$

A Taylor expansion of (5.21) for $t_{inj}$ close to $t$ (or in other words for vertical locations exceedingly close to the top boundary) reveals

(5.23) $$\begin{eqnarray}x\approx \sqrt{2t}-\frac{t^{3/2}}{6\sqrt{2}}(1-t_{inj}/t)^{3}\end{eqnarray}$$

from which it then follows

(5.24) $$\begin{eqnarray}s\approx \sqrt{2t}-\frac{t^{3/2}}{12\sqrt{2}}(1-t_{inj}/t)^{3},\end{eqnarray}$$

which can also be expressed in the form (using results from Grassia et al. (Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014))

(5.25) $$\begin{eqnarray}\displaystyle & \displaystyle x\approx \sqrt{2t}-\frac{2\sqrt{2}}{3}(1-y)^{3/2} & \displaystyle\end{eqnarray}$$

(5.26) $$\begin{eqnarray}\displaystyle & \displaystyle s\approx \sqrt{2t}-\frac{\sqrt{2}}{3}(1-y)^{3/2}. & \displaystyle\end{eqnarray}$$

Hence very close to the top boundary of the front at least, the variation of $s$ with respect to position is exactly half the variation of $x$ with respect to position, i.e.  $\text{d}s/\text{d}y$ is half of $\text{d}x/\text{d}y$ .

Some comments are pertinent. The predictions of (5.21) and (5.22) for the extent that $x$ and $s$ vary potentially contain errors/artefacts as those equations were derived based on an approximate front shape that assumed equality between $\text{d}x/\text{d}y$ and $\text{d}s/\text{d}y$ , but subsequently discovered that assumption is contradicted. Equations (5.23)–(5.24) or equivalently (5.25)–(5.26), although they only apply to $t_{inj}$ values very close to $t$ , and need not apply all the way to the matching point for which $t_{inj}\ll t$ , do not suffer from such artefacts. In the present limit $t_{inj}/t\rightarrow 1$ (which also implies $(1-y)\ll t/2$ and hence similarity variable $\unicode[STIX]{x1D701}\equiv (1-y)/(t/2)\ll 1$ ), we obtain the same results regardless of whether we use (4.4) or (4.5) to obtain $\dot{\unicode[STIX]{x1D6FC}}$ , or equivalently regardless of whether we use (4.17) or (4.18) to obtain $A_{\unicode[STIX]{x1D701}}$ . In the neighbourhood of the top of the front, the numerators of both (4.17) and (4.18) involve subtracting an exceedingly small term from unity, implying insensitivity to what the exact value of that exceedingly small term is. Hence the difference that is predicted between $\text{d}x/\text{d}y$ and $\text{d}s/\text{d}y$ neighbouring the top of the front is not merely an artefact of using (4.17) instead of (4.18). This then gives a strong indication that it might be possible for $\text{d}s/\text{d}y$ to remain smaller than $\text{d}x/\text{d}y$ throughout the entire upper region of the front, all the way down to the matching point, in which case (4.18) predicts a different $A$ versus $\unicode[STIX]{x1D701}$ relation from that given in (5.1) which relied upon (4.17). This alternative $A$ versus $\unicode[STIX]{x1D701}$ relation for the upper region should join up with the solution in the lower region without exhibiting the horizontal mismatch that was observed previously (in § 5.3) although the cost of eliminating the horizontal mismatch, could be that the front orientation angles might cease to be aligned where the upper and lower regions meet (i.e. a concave corner might be present). In the following section, we present a technique for determining the alternative $A$ versus $\unicode[STIX]{x1D701}$ relation.

6.1 Matching or ‘cross-over’ point

Assuming that (6.1) with $c=3/4$ is a good representation of the shape of the front over the entire upper region of the domain, it is possible to identify a matching or ‘cross-over’ point at which the front shapes computed in the upper and lower region of the domain intersect or ‘cross-over’ one another. The cross-over point (still assuming $c=3/4$ ) is found to be around $\unicode[STIX]{x1D701}_{cross}\approx 0.94$ , corresponding to an $A$ value $A_{cross}\approx 1.18$ : these values can be easily read off from figures 3(b) and 4(b) respectively.

The fact that $A_{cross}$ exceeds unity is encouraging: it suggests that the orientation angle of the upper region of the front overshoots that on the lower region, so the front has a concave corner there. The value of $A$ jumps from $A_{cross}\approx 1.18$ immediately above the concavity to unity immediately below it, or in other words, the front orientation angle $\unicode[STIX]{x1D6FC}$ , which is related to $A$ via $\unicode[STIX]{x1D6FC}\equiv \sqrt{t/2}A$ , jumps from around $0.834\sqrt{t}$ to around $0.707\sqrt{t}$ , implying a jump of 0.09 radians or approximately $5^{\circ }$ at the instant $t=0.5$ (which happens to be the specific time at which the distance through which the top of the front has advanced, namely $\sqrt{2t}$ , is the same as the front’s depth, i.e. unity).

The prediction that $\unicode[STIX]{x1D701}_{cross}$ is smaller than unity is slightly curious: it implies that the depth $1-y$ below the top boundary at which the concave corner is located is slightly less than the depth $t/2$ to which the material point initially at the top of the lower region of the front is expected to migrate. This in turn suggests that, rather than the concave corner consuming material points on the front on either side of it (as we expect in conventional pressure-driven growth (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014)), material points are actually being extracted from the concavity on the side below the concave corner (a behaviour which has only previously been observed in pressure-driven growth systems for which the reservoir properties are taken to be highly anisotropic (Grassia et al. Reference Grassia, Torres-Ulloa, Berres, Mas-Hernández and Shokri2016), and not in isotropic systems as are considered here). Indeed recognising that the location $\unicode[STIX]{x1D701}=1$ on the lower part of the curve (which is already a small but finite distance below the concavity) corresponds to a material point which was initially adjacent to the top surface, any points between $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{cross}$ and $\unicode[STIX]{x1D701}=1$ must have been extracted from the corner itself.

The reason the points at the top of the lower region of the front manage to ‘outrun’ those at the bottom of the upper region (despite the fact that points on the upper region immediately adjacent to the corner are instantaneously moving downward faster owing to their greater $A$ value remembering that ${\dot{y}}\approx -A/2$ via (4.10)) is that the upper region of the curve exhibits strong spatio-temporal non-uniformities in the vertical velocity component. Thus knowing the instantaneous vertical velocity alone (which is $-A_{cross}/2$ immediately above the corner) is inadequate for determining the vertical distance that points in the upper region have propagated over a given time interval up to time $t$ , even in the case of arbitrarily small values of $t$ . A conventional pressure-driven growth picture of a concavity that consumes material points on both sides of it (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014), would apply to a system for which velocities of neighbouring material points are locally nearly uniform and nearly constant in time, but that is not the case here.

To reiterate, the material point which is instantaneously at location $\unicode[STIX]{x1D701}_{cross}$ at time $t$ cannot have spent its entire history at or near that particular $\unicode[STIX]{x1D701}$ value. This follows because points at location $\unicode[STIX]{x1D701}_{cross}$ have an instantaneous vertical velocity ${\dot{y}}$ equal to $-A_{cross}/2$ . Were the point to move with that velocity at all times between zero and $t$ , it would attain a $y$ value $y=1-A_{cross}t/2$ and hence a $\unicode[STIX]{x1D701}$ value of $\unicode[STIX]{x1D701}=A_{cross}$ instead of its actual location $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{cross}$ . Spatio-temporal non-uniformity in the vertical trajectories of material points is an essential feature of the upper region.

6.2 Minimum permitted injection time

As was mentioned above, at any given time $t$ , the material point in the upper region of the solution domain immediately adjacent to the concave corner $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{cross}$ , cannot have always been at that location, but instead must have migrated there from smaller $\unicode[STIX]{x1D701}$ values (which must also have a lesser downward velocity component) earlier on. Points in the upper region of the front continue to be consumed by the concave corner, so there must exist a well defined injection time $t_{inj}$ (itself dependent on the current time $t$ ) at which the point instantaneously at the concave corner was injected onto the front from the top boundary. This is the minimum permitted injection time (in the sense that, at time $t$ , out of all the currently surviving points on the upper region of the front, the point that is currently adjacent to the concave corner was injected earlier than any of the others): we denote this minimum permitted injection time $t_{inj(min)}$ . Any material points which were injected onto the front earlier than time $t_{inj(min)}$ have already been consumed by the concave corner. Consequently, whereas points in the lower region of the solution domain have been moving downward over the entire interval of time from zero to $t$ , those in the upper region have only been moving downward for an interval no longer than $t-t_{inj(min)}$ (with $t_{inj(min)}$ depending on $t$ ).

We expect the following relations (which serve to fix the value of $t_{inj(min)}$ ) to be valid

(6.3) $$\begin{eqnarray}\int _{t_{inj(min)}}^{t}\frac{A}{2}\,\text{d}t^{\prime }=\unicode[STIX]{x1D701}_{cross}\frac{t}{2},\end{eqnarray}$$

(which uses the fact that the downward velocity component ${\dot{y}}$ of material points is $A/2$ , with $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{cross}$ being a distance $1-y=\unicode[STIX]{x1D701}_{cross}t/2$ below the top boundary) and also

(6.4) $$\begin{eqnarray}\int _{t_{inj(min)}}^{t}\frac{(A-\unicode[STIX]{x1D701})}{t^{\prime }}\,\text{d}t^{\prime }=\unicode[STIX]{x1D701}_{cross}\end{eqnarray}$$

(using the expression (4.11) for $\dot{\unicode[STIX]{x1D701}}$ of a material point). Nonetheless $t_{inj(min)}$ (or more precisely the ratio between $t_{inj(min)}$ and $t$ ) can be obtained most straightforwardly by manipulating the expression (4.11) for $\dot{\unicode[STIX]{x1D701}}$ as follows

(6.5) $$\begin{eqnarray}\text{d}\log t/\text{d}\unicode[STIX]{x1D701}=1/(A-\unicode[STIX]{x1D701})\end{eqnarray}$$

and hence (integrating from $t_{inj(min)}$ to $t$ , or equivalently from 0 to $\unicode[STIX]{x1D701}_{cross}$ )

(6.6) $$\begin{eqnarray}\log (t/t_{inj(min)})=\int _{0}^{\unicode[STIX]{x1D701}_{cross}}\text{d}\unicode[STIX]{x1D701}/(A-\unicode[STIX]{x1D701}).\end{eqnarray}$$

If $A$ versus $\unicode[STIX]{x1D701}$ happened to be known numerically at any given time (e.g. from a numerical simulation technique such as that described later on in § 8) the above equation gives a way of determining $t_{inj(min)}/t$ at the matching point $\unicode[STIX]{x1D701}_{cross}$ without even making a priori assumptions about the existence of a similarity solution. Moreover to the extent that (6.1) applies

(6.7) $$\begin{eqnarray}\log (t/t_{inj(min)})=\int _{0}^{A_{cross}}\frac{\unicode[STIX]{x1D701}_{A}\,\text{d}A}{(A-\unicode[STIX]{x1D701})}=\int _{0}^{A_{cross}}\frac{\text{d}A}{(1-cA)}=\frac{1}{c}\log (1/(1-cA_{cross}))\end{eqnarray}$$

and hence

(6.8) $$\begin{eqnarray}t_{inj(min)}/t=(1-cA_{cross})^{1/c}.\end{eqnarray}$$

In other words, the ratio $t_{inj(min)}/t$ for the longest surviving point on the upper region of the front (i.e. the longest surviving point which has not yet been consumed by the lower region of the front), can be obtained solely in terms of the parameters $c$ and $A_{cross}$ (with $c$ being estimated as $3/4$ and $A_{cross}$ being estimated as the ratio of the $\unicode[STIX]{x1D6FC}$ values either side of the concave corner at which the upper and lower regions of the front join together). Taking the value $A_{cross}\approx 1.18$ quoted previously, still with $c=3/4$ , we estimate $t_{inj(min)}\approx 0.056t$ .

6.3 Distances that material points travel to the matching point

One of the issues we have not yet discussed concerning the concave corner at the matching point is that material points immediately adjacent to the corner on either side could potentially have followed different paths to reach their current position. Hence there could potentially be two adjacent points on the front which have different $s$ values, i.e.  $s$ is not necessarily continuous at a concave corner.

Adjacent to a concave corner, the direction of motion of material points is certainly discontinuous, since motion is along the front normal, and the front normal undergoes a discrete change in direction. However any discontinuity in $s$ would also imply that the speed of material points (which is just $y/s$ according to (2.6)) is necessarily discontinuous there. Nonetheless significant discontinuities are unlikely to manifest themselves at early times, since motion is primarily horizontal at early times, so that $s$ at leading order is almost the same as $x$ , regardless of the path followed to arrive at any $(x,y)$ location.

Specifically the relationship giving $s$ for the point initially at the top of the lower region is (following § 3.3)

(6.9) $$\begin{eqnarray}s\approx \int _{0}^{x}\left(1+\frac{1}{2}\frac{({\dot{y}})^{2}}{({\dot{x}})^{2}}\right)\,\text{d}x=x+\int _{0}^{t}\frac{1}{2}\frac{1/4}{1/(2t^{\prime })}\,\frac{\text{d}t^{\prime }}{\sqrt{2t^{\prime }}}=x+(2t)^{3/2}/24\approx x(1+t/12),\end{eqnarray}$$

whereas that giving $s$ within the upper region is by analogy

(6.10) $$\begin{eqnarray}s\approx x+\int _{t_{inj}}^{t}\frac{1}{2}\frac{A^{2}/4}{1/(2t^{\prime })}\,\frac{\text{d}t^{\prime }}{\sqrt{2t^{\prime }}}=x+t_{inj}^{3/2}\int _{0}^{\unicode[STIX]{x1D713}}\frac{\sqrt{2}A^{2}}{8}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })^{5/2}},\end{eqnarray}$$

where we have defined a new variable $\unicode[STIX]{x1D713}$ to be

(6.11) $$\begin{eqnarray}\unicode[STIX]{x1D713}=1-t_{inj}/t,\end{eqnarray}$$

(with $\unicode[STIX]{x1D713}$ being the fraction of time relative to the current time $t$ that a given material point has been present on the front), and a corresponding dummy integration variable $\unicode[STIX]{x1D713}^{\prime }$ such that $\unicode[STIX]{x1D713}^{\prime }=1-t_{inj}/t^{\prime }$ , from which it follows $t^{\prime }=t_{inj}/(1-\unicode[STIX]{x1D713}^{\prime })$ and hence $\text{d}t^{\prime }=t_{inj}\,\text{d}\unicode[STIX]{x1D713}^{\prime }/(1-\unicode[STIX]{x1D713}^{\prime })^{2}$ . We can also consider $A$ to be a function of $\unicode[STIX]{x1D713}$ instead of being a function of either $\unicode[STIX]{x1D701}$ or $t_{inj}/t$ . The value of $\unicode[STIX]{x1D713}$ at the matching or cross-over point, which we denote $\unicode[STIX]{x1D713}_{cross}$ , is given by $\unicode[STIX]{x1D713}_{cross}=1-t_{inj(min)}/t$ . For the material point within the upper region immediately adjacent to the concave corner, the above (6.10) becomes

(6.12) $$\begin{eqnarray}\displaystyle s & {\approx} & \displaystyle x+\sqrt{2}t^{3/2}(1-\unicode[STIX]{x1D713}_{cross})^{3/2}\int _{0}^{\unicode[STIX]{x1D713}_{cross}}\frac{A^{2}}{8}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })^{5/2}}\nonumber\\ \displaystyle & {\approx} & \displaystyle x\left(1+t(1-\unicode[STIX]{x1D713}_{cross})^{3/2}\int _{0}^{\unicode[STIX]{x1D713}_{cross}}\frac{A^{2}}{8}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })^{5/2}}\right).\end{eqnarray}$$

Comparing (6.9) and (6.12), at any given time $t$ , and depending on the form of the function $A$ versus $\unicode[STIX]{x1D713}$ , it appears at first sight as though a discontinuity in the value of $s$ of up to order $t$ times $s$ itself could potentially occur.

Remember from § 6.1 however an unusual property for the concave corner: instead of consuming material points on both sides, immediately above the corner the system is consuming material points but immediately below the corner the system is creating them. As a result (6.9) gives the $s$ value for the material point within the lower region that was initially at the top of the front, but not the $s$ value of newly created points that lie between it and the concave corner. Indeed those newly created material points need to be assigned an $s$ value at the time they are created. If points created immediately below the corner are given the same $s$ value as those that are being destroyed immediately above the corner, then continuity of $s$ through the corner can be ensured. Interestingly a recent simulation study (Torres-Ulloa Reference Torres-Ulloa2015) on pressure-driven growth using a completely different numerical simulation technique (i.e. an Eulerian technique which does not make reference to material points) did indeed indicate continuity of $s$ through the concave corner at least to the extent that the numerical technique could resolve. More discussion of this Eulerian technique will be given in § 8.

7.1 Integral expression for $\text{d}s/\text{d}y$

Here we derive an integral expression for $\text{d}s/\text{d}y$ . In order to do this we re-express derivatives along the front with respect to $y$ in terms of derivatives along the front with respect to the injection time $t_{inj}$ . The derivation is quite detailed, and some readers may prefer to skip directly to the final result which is given by (7.13).

The derivation proceeds as follows. We know via (2.6) (with in addition $y\equiv 1$ for any time $t$ prior to the injection time, i.e. for $0\leqslant t\leqslant t_{inj}$ ) that

(7.1) $$\begin{eqnarray}s^{2}=2t+\int _{t_{inj}}^{t}2(y-1)\,\text{d}t^{\prime },\end{eqnarray}$$

where (at leading order for small $t$ ) via (4.10)

(7.2) $$\begin{eqnarray}y-1\approx -\int _{t_{inj}}^{t}A/2\,\text{d}t^{\prime \prime }.\end{eqnarray}$$

Alternatively we can write

(7.3) $$\begin{eqnarray}s^{2}\approx 2t+\int _{t_{inj}}^{t}F(t^{\prime },t_{inj})\,\text{d}t^{\prime },\end{eqnarray}$$

where $F$ is a function defined as

(7.4) $$\begin{eqnarray}F(t,t_{inj})=-\int _{t_{inj}}^{t}A\,\text{d}t^{\prime \prime }.\end{eqnarray}$$

Hence if we differentiate $s^{2}$ with respect to $t_{inj}$ at fixed $t$

(7.5) $$\begin{eqnarray}\unicode[STIX]{x2202}s^{2}/\unicode[STIX]{x2202}t_{inj}\approx -F(t_{inj},t_{inj})+\int _{t_{inj}}^{t}\unicode[STIX]{x2202}F/\unicode[STIX]{x2202}t_{inj}\,\text{d}t^{\prime }=\int _{t_{inj}}^{t}\unicode[STIX]{x2202}F/\unicode[STIX]{x2202}t_{inj}\,\text{d}t^{\prime }\end{eqnarray}$$

since $F(t_{inj},t_{inj})$ vanishes by construction.

Moreover since $F=-\int _{t_{inj}}^{t}A\,\text{d}t^{\prime \prime }$ , with $A=A(t,t_{inj})$ taken to be a function of both variables $t$ and $t_{inj}$ it follows

(7.6) $$\begin{eqnarray}\unicode[STIX]{x2202}F/\unicode[STIX]{x2202}t_{inj}=A(t_{inj},t_{inj})-\int _{t_{inj}}^{t}\unicode[STIX]{x2202}A/\unicode[STIX]{x2202}t_{inj}\,\text{d}t^{\prime \prime }=-\int _{t_{inj}}^{t}\unicode[STIX]{x2202}A/\unicode[STIX]{x2202}t_{inj}\,\text{d}t^{\prime \prime }\end{eqnarray}$$

since $A(t_{inj},t_{inj})$ vanishes (owing to the fact that when $t=t_{inj}$ the material point has just been injected at the top boundary, and the top boundary condition on the front then demands that $A$ vanishes).

It now follows from (7.5) and (7.6)

(7.7) $$\begin{eqnarray}\unicode[STIX]{x2202}s^{2}/\unicode[STIX]{x2202}t_{inj}\approx -\int _{t_{inj}}^{t}\left(\int _{t_{inj}}^{t^{\prime }}\unicode[STIX]{x2202}A/\unicode[STIX]{x2202}t_{inj}\,\text{d}t^{\prime \prime }\right)\,\text{d}t^{\prime }\end{eqnarray}$$

and also since $y-1\approx F/2$ (based on (7.2) and (7.4))

(7.8) $$\begin{eqnarray}\unicode[STIX]{x2202}y/\unicode[STIX]{x2202}t_{inj}\approx -\frac{1}{2}\int _{t_{inj}}^{t}\unicode[STIX]{x2202}A/\unicode[STIX]{x2202}t_{inj}\,\text{d}t^{\prime }.\end{eqnarray}$$

As in § 6.3, we define a variable $\unicode[STIX]{x1D713}$ such that $\unicode[STIX]{x1D713}=1-t_{inj}/t$ , and analogously a dummy integration variable $\unicode[STIX]{x1D713}^{\prime }$ (in terms of a dummy time variable $t^{\prime }$ ) such that $\unicode[STIX]{x1D713}^{\prime }=1-t_{inj}/t^{\prime }$ . We assume moreover that $A$ can be expressed in similarity form, as a function of the single variable $\unicode[STIX]{x1D713}$ instead of in terms of two variables $t$ and $t_{inj}$ . It follows that

(7.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}A}{\unicode[STIX]{x2202}t_{inj}}=\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime }}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}^{\prime }}{\unicode[STIX]{x2202}t_{inj}}=\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime }}\left(-\frac{1}{t^{\prime }}\right)=\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime }}\left(-\frac{(1-\unicode[STIX]{x1D713}^{\prime })}{t_{inj}}\right).\end{eqnarray}$$

Moreover, on the grounds that $\text{d}t^{\prime }=t_{inj}\,\text{d}\unicode[STIX]{x1D713}^{\prime }/(1-\unicode[STIX]{x1D713}^{\prime })^{2}$ , we deduce (via substitution into (7.7) and (7.8))

(7.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}s^{2}/\unicode[STIX]{x2202}t_{inj} & {\approx} & \displaystyle t_{inj}\int _{0}^{\unicode[STIX]{x1D713}}\left(\int _{0}^{\unicode[STIX]{x1D713}^{\prime }}\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime \prime }}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime \prime }}{(1-\unicode[STIX]{x1D713}^{\prime \prime })}\right)\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })^{2}}\nonumber\\ \displaystyle & = & \displaystyle (1-\unicode[STIX]{x1D713})t\int _{0}^{\unicode[STIX]{x1D713}}\left(\int _{0}^{\unicode[STIX]{x1D713}^{\prime }}\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime \prime }}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime \prime }}{(1-\unicode[STIX]{x1D713}^{\prime \prime })}\right)\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })^{2}}\end{eqnarray}$$

and

(7.11) $$\begin{eqnarray}\unicode[STIX]{x2202}y/\unicode[STIX]{x2202}t_{inj}\approx \int _{0}^{\unicode[STIX]{x1D713}}\frac{1}{2}\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime }}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })}.\end{eqnarray}$$

Since $s\approx \sqrt{2t}$ at leading order for points close to the top boundary, we deduce

(7.12) $$\begin{eqnarray}\frac{1}{\sqrt{2t}}\,\text{d}s/\text{d}y\approx \frac{1}{s}\,\text{d}s/\text{d}y=\frac{2s\unicode[STIX]{x2202}s/\unicode[STIX]{x2202}t_{inj}}{2s^{2}\unicode[STIX]{x2202}y/\unicode[STIX]{x2202}t_{inj}}\approx \frac{\unicode[STIX]{x2202}s^{2}/\unicode[STIX]{x2202}t_{inj}}{4t\unicode[STIX]{x2202}y/\unicode[STIX]{x2202}t_{inj}}\end{eqnarray}$$

implying finally via (7.10)–(7.11) that

(7.13) $$\begin{eqnarray}\frac{1}{\sqrt{2t}}\,\text{d}s/\text{d}y\approx \frac{\displaystyle \frac{1}{2}(1-\unicode[STIX]{x1D713})\int _{0}^{\unicode[STIX]{x1D713}}\left(\int _{0}^{\unicode[STIX]{x1D713}^{\prime }}\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime \prime }}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime \prime }}{(1-\unicode[STIX]{x1D713}^{\prime \prime })}\right)\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })^{2}}}{\displaystyle \int _{0}^{\unicode[STIX]{x1D713}}\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime }}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })}},\end{eqnarray}$$

which is the expression we have been seeking to derive. As long as the function $A$ versus $\unicode[STIX]{x1D713}$ is known, it is possible to deduce $\text{d}s/\text{d}y$ at that same $\unicode[STIX]{x1D713}$ value. Note that this depends on the entire set of $A$ values up to and including $\unicode[STIX]{x1D713}$ : in other words it depends on the entire history that a material point has experienced from injection (when by definition $\unicode[STIX]{x1D713}=0$ ) up to its current $\unicode[STIX]{x1D713}$ value.

Note that the expression above for $(2t)^{-1/2}\,\text{d}s/\text{d}y$ becomes ill behaved as $\unicode[STIX]{x1D713}\rightarrow 1$ . This implies that it is not permitted to consider $t_{inj}$ values that are arbitrarily small compared to $t$ . This is however not problematic if material points are being consumed by the concave corner, as in that case no points with arbitrarily small $t_{inj}$ can ever survive. Nonetheless in the limit as $t_{inj}\rightarrow t$ (corresponding to material points that are extremely close to the top boundary) the equations are well behaved. Indeed near the top boundary (i.e. for $\unicode[STIX]{x1D713}\ll 1$ ) it is possible to show $A\approx \unicode[STIX]{x1D713}$ : this actually follows from (5.7), which we already know to be accurate very close to the top of the upper region of the front (see e.g. the explanation for this given in § 5.4.1). Under these circumstances, with $\unicode[STIX]{x1D713}\ll 1$ , the above expression (7.13) for $(2t)^{-1/2}\,\text{d}s/\text{d}y$ very clearly approximates to $\unicode[STIX]{x1D713}/4$ , which (as is again expected via § 5.4.1) is only half as big as $(2t)^{-1/2}\,\text{d}x/\text{d}y\sim (2t)^{-1/2}\unicode[STIX]{x1D6FC}\sim A/2\sim \unicode[STIX]{x1D713}/2$ .

7.2 Derivation of integro-differential equation

Having obtained an expression for $\text{d}s/\text{d}y$ , an integro-differential equation for $A_{\unicode[STIX]{x1D713}}$ (the derivative of $A$ with respect to $\unicode[STIX]{x1D713}$ ) can now be derived analogously to the derivation for the differential equation for $A_{\unicode[STIX]{x1D701}}$ obtained earlier (see e.g. § 4.3). It is however now far more convenient to parameterise in terms of $\unicode[STIX]{x1D713}$ instead of in terms of $\unicode[STIX]{x1D701}$ .

We assume a small-time similarity solution of the form

(7.14) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\sqrt{t/2}A(\unicode[STIX]{x1D713})\end{eqnarray}$$

in other words we assume $A$ depends solely on $\unicode[STIX]{x1D713}\equiv 1-t_{inj}/t$ rather than upon $t$ and $t_{inj}$ separately. This then implies

(7.15) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FC}}=\frac{1}{2t}\sqrt{\frac{t}{2}}A+\sqrt{\frac{t}{2}}A_{\unicode[STIX]{x1D713}}\dot{\unicode[STIX]{x1D713}},\end{eqnarray}$$

where $\dot{\unicode[STIX]{x1D713}}$ is the time derivative of $\unicode[STIX]{x1D713}$ following a fixed material point (corresponding to a fixed $t_{inj}$ ). Hence

(7.16) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D713}}=t_{inj}/t^{2}=(1-\unicode[STIX]{x1D713})/t\end{eqnarray}$$

and it follows

(7.17) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FC}}=\frac{1}{\sqrt{2t}}\frac{A}{2}+\frac{1}{\sqrt{2t}}(1-\unicode[STIX]{x1D713})A_{\unicode[STIX]{x1D713}}.\end{eqnarray}$$

Equating the two expressions for $\dot{\unicode[STIX]{x1D6FC}}$ that from (4.4) (which recall is an approximation in the limit of small times) and that from (7.17) then gives

(7.18) $$\begin{eqnarray}\frac{1}{\sqrt{2t}}\left(1-\frac{1}{\sqrt{2t}}\,\frac{\text{d}s}{\text{d}y}\right)\approx \frac{A}{2}+(1-\unicode[STIX]{x1D713})A_{\unicode[STIX]{x1D713}}\end{eqnarray}$$

and hence substituting from (7.13)

(7.19) $$\begin{eqnarray}A_{\unicode[STIX]{x1D713}}\approx (1-\unicode[STIX]{x1D713})^{-1}\left(1-\frac{A}{2}-\frac{\displaystyle \frac{1}{2}(1-\unicode[STIX]{x1D713})\int _{0}^{\unicode[STIX]{x1D713}}\left(\int _{0}^{\unicode[STIX]{x1D713}^{\prime }}\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime \prime }}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime \prime }}{(1-\unicode[STIX]{x1D713}^{\prime \prime })}\right)\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })^{2}}}{\displaystyle \int _{0}^{\unicode[STIX]{x1D713}}\frac{\text{d}A}{\text{d}\unicode[STIX]{x1D713}^{\prime }}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })}}\right).\end{eqnarray}$$

This is an integro-differential equation that we can solve for $A$ versus $\unicode[STIX]{x1D713}$ .

7.2.1 Obtaining $A$ versus $\unicode[STIX]{x1D701}$

So far we have only derived an equation for $A$ versus $\unicode[STIX]{x1D713}$ , not for $A$ versus $\unicode[STIX]{x1D701}$ . Such a relation is however simple to obtain. We know (via (4.11)) that $\dot{\unicode[STIX]{x1D701}}\approx (A-\unicode[STIX]{x1D701})/t$ in the small-time asymptotic limit. Combining this with an assumption that $\unicode[STIX]{x1D701}$ can be expressed solely in terms of $\unicode[STIX]{x1D713}$ , and hence

(7.20) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D701}}=\dot{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D713}}=\frac{t_{inj}}{t^{2}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D713}}=\frac{(1-\unicode[STIX]{x1D713})}{t}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D713}}\end{eqnarray}$$

we can then deduce

(7.21) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D713}}\approx (A-\unicode[STIX]{x1D701})/(1-\unicode[STIX]{x1D713}).\end{eqnarray}$$

Once $A$ versus $\unicode[STIX]{x1D713}$ is known, it is possible to obtain $\unicode[STIX]{x1D701}$ versus $\unicode[STIX]{x1D713}$ (and hence $A$ versus $\unicode[STIX]{x1D701}$ parametrically of $\unicode[STIX]{x1D713}$ ). Indeed the solution for $\unicode[STIX]{x1D701}$ versus $\unicode[STIX]{x1D713}$ is

(7.22) $$\begin{eqnarray}\unicode[STIX]{x1D701}=(1-\unicode[STIX]{x1D713})\int _{0}^{\unicode[STIX]{x1D713}}\frac{A}{(1-\unicode[STIX]{x1D713}^{\prime })^{2}}\,\text{d}\unicode[STIX]{x1D713}^{\prime },\end{eqnarray}$$

which is an ‘exact’ solution of (7.21), although (7.21) is of course only an approximate representation of the front applicable at small times, and furthermore the integral within (7.22) typically needs to be obtained by quadrature.

7.2.2 Obtaining $\unicode[STIX]{x1D701}$ versus $\unicode[STIX]{x1D6EF}$

Recall that we are interested in finding an intersection point between the upper and lower regions of the front. This is most easily determined in terms of a Cartesian coordinate plot in $(x,y)$ coordinates, or analogously a plot of $\unicode[STIX]{x1D701}$ versus $\unicode[STIX]{x1D6EF}$ ( $\unicode[STIX]{x1D701}$ being a rescaled version of the vertical distance measured downwards from the top boundary, and $\unicode[STIX]{x1D6EF}$ being a rescaled version of the horizontal distance measured backwards from the leading edge of the front). The value of $\unicode[STIX]{x1D6EF}$ versus $\unicode[STIX]{x1D713}$ in the upper region is given by (see (5.16) and substitute from (7.21))

(7.23) $$\begin{eqnarray}\unicode[STIX]{x1D6EF}\approx \frac{1}{2\sqrt{2}}\int _{0}^{\unicode[STIX]{x1D701}}A\,\text{d}\unicode[STIX]{x1D701}\approx \frac{1}{2\sqrt{2}}\int _{0}^{\unicode[STIX]{x1D713}}\frac{A(A-\unicode[STIX]{x1D701})}{(1-\unicode[STIX]{x1D713}^{\prime })}\,\text{d}\unicode[STIX]{x1D713}^{\prime }.\end{eqnarray}$$

Clearly $\unicode[STIX]{x1D701}$ versus $\unicode[STIX]{x1D6EF}$ can now be obtained parametrically (in terms of $\unicode[STIX]{x1D713}$ ) for the upper region, with (5.15) continuing to give the relation between $\unicode[STIX]{x1D6EF}$ and $\unicode[STIX]{x1D701}$ in the lower region. Our aim is to find the matching or ‘cross-over’ point $\unicode[STIX]{x1D713}_{cross}$ and corresponding values $\unicode[STIX]{x1D701}_{cross}$ and $A_{cross}$ such that the $\unicode[STIX]{x1D6EF}$ values in the upper and lower regions intersect. Note that $\unicode[STIX]{x1D701}_{cross}$ and $A_{cross}$ will differ from the values found earlier in § 6.1 because the integro-differential equation (7.19) is not the same as the differential equation (6.1) which was obtained via an ad hoc approximation. We will see however that the differences turn out to be exceedingly small, indicating that the ad hoc approximation of § 6.1 was a remarkably good one.

7.3 Numerical method for integro-differential equation

If integro-differential equation (7.19) can be solved for $A$ versus $\unicode[STIX]{x1D713}$ , then $\unicode[STIX]{x1D701}$ versus $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D6EF}$ versus $\unicode[STIX]{x1D713}$ can be solved by quadrature on (7.22) and (7.23). Our solution technique for (7.19) involved dividing the domain $0\leqslant \unicode[STIX]{x1D713}<1$ into a large number of intervals (denoted $N$ , and chosen to be either 100 or 1000 or 10 000). The size of each interval, which we denote $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}$ , equals $1/N$ . Suppose $\unicode[STIX]{x1D713}_{i(l)}$ and $\unicode[STIX]{x1D713}_{i(r)}$ denote the $\unicode[STIX]{x1D713}$ values at the left- and right-hand boundaries of interval $i$ , so that $\unicode[STIX]{x1D713}_{i(l)}=(i-1)\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D713}_{i(r)}=i\,\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}$ . We assumed that on each interval $A$ versus $\unicode[STIX]{x1D713}$ could be represented by a quadratic function

(7.24) $$\begin{eqnarray}A\approx A_{i(0)}+A_{i(1)}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D713}_{i(l)})+A_{i(2)}(\unicode[STIX]{x1D713}-\unicode[STIX]{x1D713}_{i(l)})^{2}.\end{eqnarray}$$

Recall from § 7.1 that, for the very first interval, we know that $A\approx \unicode[STIX]{x1D713}$ which tells us immediately that $A_{1(0)}=0$ and $A_{1(1)}=1$ . We guessed a value of $A_{1(2)}$ (initially zero) and then, on the basis of that guess, computed the right-hand side of (7.19) up to $\unicode[STIX]{x1D713}_{1(r)}$ using the trapezoidal rule to determine any integrals. This however should equal the left-hand side, which at $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}_{1(r)}$ must be $A_{1(1)}+2\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}A_{1(2)}$ . We used this to update our guess for $A_{1(2)}$ , and then recomputed the right-hand side, iterating until the process converged, which took typically 3 or 4 iterations.

Subsequent intervals beyond the first one were treated similarly. Values of $A_{i(0)}$ and $A_{i(1)}$ are known immediately, by requiring that $A$ and $A_{\unicode[STIX]{x1D713}}$ are continuous from the right-hand side of interval $i-1$ to the left-hand side of interval $i$ . Initial guesses for $A_{i(2)}$ could be obtained by using the corresponding value from interval $i-1$ , and $A_{i(2)}$ values were then improved iteratively, typically converging within a couple of iterations. It turns out that the integro-differential equation becomes badly behaved as $\unicode[STIX]{x1D713}\rightarrow 1$ , so we only took the computations as far as $\unicode[STIX]{x1D713}=0.95$ but (as we shall see) this proved sufficient to locate the matching point $\unicode[STIX]{x1D713}_{cross}$ .

7.4 Numerical results from the integro-differential equation

This numerical results section is laid out as follows. Data for the (rescaled) front orientation angle $A$ across the upper region of the front, as predicted by the integro-differential equation, are presented first (§ 7.4.1). We then do a convergence check (§ 7.4.2) to demonstrate that differences observed from the results of the differential equation approach of § 6 cannot be attributed merely to truncation error. Next we present data for $\unicode[STIX]{x1D6EF}$ , the (rescaled) amount that the horizontal displacement of points on the front fall short of that of the leading edge (§ 7.4.3) and for the variation of the path length $s$ swept out by material points in the upper region of the front (§ 7.4.4). We also look at unswept area beyond the front (§ 7.4.5), which is associated with the amount of gravity override, and furthermore discuss how to track material point trajectories traversing the upper region of the front (§ 7.4.6).

7.4.1 Data for front orientation angle $A$

A graph of $A$ versus $\unicode[STIX]{x1D713}$ computed from the numerical scheme used to solve the integro-differential equation is shown in figure 5. The function does display some deviation away from a straight line, a contrast from (5.7) which would predict an exact straight line. Computed values of $\unicode[STIX]{x1D701}$ versus $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D6EF}$ versus $\unicode[STIX]{x1D713}$ are also shown in the figure. These increase only very slowly with $\unicode[STIX]{x1D713}$ at first, but show more significant increases as $\unicode[STIX]{x1D713}$ grows. Such behaviour can be expected: indeed for $\unicode[STIX]{x1D713}\ll 1$ it is possible to show (using the known asymptotic behaviour near the top boundary of the front (Grassia et al. Reference Grassia, Mas-Hernández, Shokri, Cox, Mishuris and Rossen2014)) that $\unicode[STIX]{x1D701}\approx \unicode[STIX]{x1D713}^{2}/2$ and $\unicode[STIX]{x1D6EF}\approx \unicode[STIX]{x1D713}^{3}/(6\sqrt{2})$ .

Figure 5. Solutions (corresponding to the upper region of the solution domain) obtained from the integro-differential equation for $A$ versus $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D701}$ versus $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D6EF}$ versus $\unicode[STIX]{x1D713}$ , with $A$ being rescaled angle through which the front turns, $\unicode[STIX]{x1D713}$ being a measure of time that a given material point has been on the front (relative to total time elapsed), $\unicode[STIX]{x1D701}$ being rescaled vertical distance, and $\unicode[STIX]{x1D6EF}$ being rescaled horizontal distance. The domain has been divided into 1000 intervals. The vertical line indicates where matching is achieved with the lower region of the solution domain (which occurs when $\unicode[STIX]{x1D713}$ equals 0.948, and the solution of the integro-differential is stopped at this point).

Integro-differential equation data re-expressed in the form $A$ versus $\unicode[STIX]{x1D701}$ (which recall are equivalent to plotting $\unicode[STIX]{x1D6FC}$ versus $1-y$ , because $\unicode[STIX]{x1D6FC}\equiv \sqrt{t/2}A$ and $\unicode[STIX]{x1D701}\equiv (1-y)/(t/2)$ ) are shown in figure 3. Clearly the integro-differential equation solutions are extremely close to those predicted by (6.2) with $c=3/4$ : on the scale of figure 3(a) they are virtually indistinguishable. By implication the assumption upon which the $c=3/4$ data were derived (namely that $\text{d}s/\text{d}y$ is uniformly half of $\text{d}x/\text{d}y$ ) must be an exceedingly good one (a point to which we will return shortly). It is only in the zoomed view in figure 3(b) that a difference can be detected between the integro-differential equation data and the $c=3/4$ data, with the former giving slightly smaller $A$ values than the latter.

7.4.4 Data for path length $s$ travelled by material points

Recall that $\unicode[STIX]{x1D6EF}\equiv (\sqrt{2t}-x)/t^{3/2}$ is a rescaled measure of the amount that $x$ falls short of the leading edge of the front at location $\sqrt{2t}$ . We also define a rescaled measure of the extent to which path length $s$ exceeds horizontal coordinate $x$ . We denote this by the symbol $\unicode[STIX]{x1D70E}$ and define it by $\unicode[STIX]{x1D70E}\equiv (s-x)/t^{3/2}$ . It is clear from (6.10)–(6.11) that

(7.25) $$\begin{eqnarray}\unicode[STIX]{x1D70E}=(1-\unicode[STIX]{x1D713})^{3/2}\int _{0}^{\unicode[STIX]{x1D713}}\frac{\sqrt{2}A^{2}}{8}\,\frac{\text{d}\unicode[STIX]{x1D713}^{\prime }}{(1-\unicode[STIX]{x1D713}^{\prime })^{5/2}}\end{eqnarray}$$

meaning that $\unicode[STIX]{x1D70E}$ can be readily obtained by quadrature once $A$ versus $\unicode[STIX]{x1D713}$ is known. The difference $\unicode[STIX]{x1D6EF}-\unicode[STIX]{x1D70E}$ is a rescaled measure of the amount that $s$ falls short of $\sqrt{2t}$ , and the ratio $(\unicode[STIX]{x1D6EF}-\unicode[STIX]{x1D70E})/\unicode[STIX]{x1D6EF}$ (which is identical to the ratio $(\sqrt{2t}-s)/(\sqrt{2t}-x)$ ) is then a measure of the amount that $s$ varies in the upper region of the front relative to the amount that $x$ varies.

We have evaluated this ratio $(\unicode[STIX]{x1D6EF}-\unicode[STIX]{x1D70E})/\unicode[STIX]{x1D6EF}$ based on the numerical data for $\unicode[STIX]{x1D6EF}$ versus $\unicode[STIX]{x1D713}$ and $A$ versus $\unicode[STIX]{x1D713}$ which have already been presented in figure 5. Although we do not show the data for $(\unicode[STIX]{x1D6EF}-\unicode[STIX]{x1D70E})/\unicode[STIX]{x1D6EF}$ here, remarkably in the domain from zero to $\unicode[STIX]{x1D713}_{cross}$ the value of this ratio always falls between 0.5 and 0.51. This reiterates that the ad hoc assumption that was made in § 6, namely that $\text{d}s/\text{d}y$ is uniformly exactly half of $\text{d}x/\text{d}y$ is an extremely good approximation, and hence explains why the results obtained in that section agree so closely with the integro-differential equation results presented here.