2. The generalized reciprocal theorem

The generalized reciprocal theorem relates the velocity and stress fields of the problem of interest, $({\bf\sigma}^{(1)},\boldsymbol{u}^{(1)})$ , and a simpler test problem, $({\bf\sigma}^{(2)},\boldsymbol{u}^{(2)})$ , for which the solution is known. Both $({\bf\sigma}^{(1)},\boldsymbol{u}^{(1)})$ and $({\bf\sigma}^{(2)},\boldsymbol{u}^{(2)})$ are solutions of the flow past the same body (a spheroid in the present case), but with different boundary conditions and possibly governed by different dynamical equations (Leal Reference Leal1979; Subramanian & Koch Reference Subramanian and Koch2005, Reference Subramanian and Koch2006). Since the quantities of interest here are both the inertial and viscoelastic torques on a sedimenting spheroid, $({\bf\sigma}^{(1)},\boldsymbol{u}^{(1)})$ is taken to correspond to a spheroid translating in an otherwise quiescent non-Newtonian fluid, with the inertial acceleration being taken into account. The test problem, $({\bf\sigma}^{(2)},\boldsymbol{u}^{(2)})$ corresponds to the inertialess rotation of a spheroid (with the same instantaneous orientation) in a Newtonian fluid at rest and about an axis transverse to its axis of symmetry. The reciprocal theorem then yields the following identity:

where $\boldsymbol{n}$ is the unit normal pointing out of the fluid domain $V$ . As will be seen below, the disturbance velocity fields decay sufficiently rapidly away from the particle, so the surface integrals at infinity may be neglected, and the bounding surface $S$ in the integrals in (2.1) is that of the spheroid ( $S_{p}$ ).

The non-dimensional equations of motion and the continuity equation for the problem of interest, $({\bf\sigma}^{(1)},\boldsymbol{u}^{(1)})$ , are given by:

with where $\hat{\boldsymbol{U}}$ is a unit vector along the ambient flow. In (2.2), $\boldsymbol{u}^{(1)}$ is the velocity field in a reference frame that translates with the spheroid, $\text{D}\boldsymbol{u}^{(1)}/\text{D}t=\boldsymbol{u}^{(1)}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)}$ is the convective acceleration, and ${\bf\sigma}^{(1)}$ is the total stress tensor of the non-Newtonian fluid. The time derivative in the inertial terms is neglected because the unsteadiness occurs due to the rotation of the spheroid on a time scale that is $O(\mathit{Re}^{-1})$ larger than the nominal time scale of $O(L/U)$ . In obtaining (2.2)–(2.5), the length scale chosen is the semi-major axis of the spheroid (in spheroidal coordinates, $L=d{\it\xi}_{0}$ , where ${\it\xi}$ is the analogue of the radial distance in spheroidal coordinates, with ${\it\xi}_{0}$ denoting the surface of the particle and $d$ denoting half the interfocal distance; see § 3), while the velocity scale $U$ may be taken as the nominal scale for the settling velocity obtained from balancing buoyancy with the $O({\it\mu}UL)$ hydrodynamic drag. The choice of the major axis ensures that the particle length is the relevant scale in the limit of a slender fibre, as is the case for inertial forces, where the dominant contribution to the torque comes from a volume of $O(L^{3})$ around the fibre (Subramanian & Koch Reference Subramanian and Koch2005). A transformation to oblate spheroidal coordinates leads to the diameter being the relevant length scale for the limiting case of a flat disk. With this choice, the Reynolds number in (2.2) is $\mathit{Re}=({\it\rho}UL)/{\it\mu}$ . The dimensionless stress, ${\bf\sigma}^{(1)}$ , in (2.2) may be written in the form

where the second term is the deviatoric Newtonian stress with $\boldsymbol{e}^{(1)}=(\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)}+\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)^{\dagger }})/2$  being the rate-of-strain tensor where $\boldsymbol{{\rm\nabla}}u^{\dagger }$ refers to the transpose of the velocity gradient tensor and ${\bf\sigma}_{NN}^{(1)}$ denotes the viscoelastic stress. The Deborah number is a dimensionless measure of viscoelasticity, being defined as $\mathit{De}=U{\it\lambda}/L$ , and may be regarded as the ratio of the intrinsic relaxation time of the microstructure to the imposed flow time scale. The present analysis is valid in the limit $\mathit{De}\ll 1$ , corresponding to slow, slowly varying flows. In this regime, the fluid rheology is governed by the retarded-motion expansion (see Bird et al. Reference Bird, Armstrong and Hassager1987; Larson Reference Larson1988). To $O(\mathit{De})$ , this expansion reduces to (2.6), the constitutive relation for a second-order fluid, which rigorously accounts for the first effects of viscoelasticity. With ${\bf\sigma}_{NN}^{(1)}$ defined as the stress in a second-order fluid (see (2.8) below), (2.6) predicts a constant shear viscosity and a quadratic shear-rate dependence of the first and second normal stress differences ( $N_{1}$ and $N_{2}$ ) in a viscometric flow. The intrinsic relaxation time can be expressed as the ratio of the sum of normal stress coefficients to the shear viscosity ( ${\it\lambda}=({\it\psi}_{1}+{\it\psi}_{2})/{\it\mu}$ , ${\it\psi}_{i}=N_{i}/\dot{{\it\gamma}}^{2}$ ; $i=1,2$ ). In the original expansion, shear-thinning appears at $O(\mathit{De}^{2})$ , shear-rate dependence of the normal stress coefficients appears at $O(\mathit{De}^{3})$ , and so on. However, the orientational degeneracy in the Stokes limit, arising from dependence on initial conditions, is already resolved at $O(\mathit{De})$ , and these higher-order effects will not qualitatively alter the predicted behaviour for small $\mathit{De}$ .

Using (2.6), the equations of motion, (2.2), may now be written as:

where the second-order fluid stress may be written as the sum of a co-rotational $({\bf\sigma}_{NNC}^{(1)})$ and a quadratic $({\bf\sigma}_{NNQ}^{(1)})$ contribution:

with (Koch & Subramanian Reference Koch and Subramanian2006; Subramanian & Koch Reference Subramanian and Koch2007)

Here, ${\bf\omega}^{(1)}=(\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)}-\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)\dagger })/2$ is the vorticity tensor, ${\it\epsilon}$ being the second-order fluid parameter. Note that ${\it\epsilon}=-1$ with ${\bf\sigma}_{NNQ}^{(1)}=0$ corresponds to the co-rotational limit. An alternate convention exists in the literature (see; Joseph Reference Joseph1990; Feng et al. Reference Feng, Joseph, Glowinski and Pan1995; Wang & Joseph Reference Wang and Joseph2004) where the second-order stress is written as $\mathit{De}_{d}{\bf\sigma}_{NN}^{(1)}$ , with $\mathit{De}_{d}={\it\psi}_{1}U/(2{\it\mu}L)$ , and ${\bf\sigma}_{NN}^{(1)}$ defined in terms of the first and second Rivlin–Ericksen tensors as:

where the first term within brackets is proportional to the second-order Rivlin–Ericksen tensor, and ${\it\epsilon}_{d}=-2({\it\psi}_{1}+{\it\psi}_{2})/{\it\psi}_{1}$ . The relation between the above expression and one used in the present work may be seen from substituting $\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)\dagger }=\boldsymbol{e}^{(1)}-{\bf\omega}^{(1)}$ and $\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)}=\boldsymbol{e}^{(1)}+{\bf\omega}^{(1)}$ , whence one finds ${\it\epsilon}_{d}=1/{\it\epsilon}$ .

The test problem, $({\bf\sigma}^{(2)},\boldsymbol{u}^{(2)})$ in (2.1), is the inertialess rotation of a spheroid, with the same instantaneous orientation as in the actual problem, in an otherwise quiescent Newtonian fluid. Since the translating spheroid in the actual problem does not have a spin about its symmetry axis, the test angular velocity, ${\it\bf\Omega}^{(2)}$ , may be chosen as being orthogonal to this axis. The equations of motion and continuity in this case are given by:

with where we have used that ${\bf\sigma}^{(2)}=-p^{(2)}\boldsymbol{I}+2\boldsymbol{e}^{(2)}$ , with $p^{(2)}$ and $\boldsymbol{e}^{(2)}$ being the pressure and rate-of-strain fields in the test problem.

Returning to the reciprocal theorem identity, equation (2.1), the second surface integral on the left-hand side in (2.1) is identically zero since the spheroid in the test problem exerts only a torque but no force. Using the boundary condition, (2.14), the first surface integral in (2.1) reduces to – $\boldsymbol{L}_{sed}\boldsymbol{\cdot }{\it\bf\Omega}^{(2)}$ , $\boldsymbol{L}_{sed}$ being the torque acting on the translating spheroid. Thus, (2.1) takes the form:

where $\boldsymbol{L}_{sed}$ is the non-dimensional torque measured in units of ${\it\mu}UL^{2}$ . It is convenient to write the inertial acceleration in terms of the disturbance velocity field, which, as will be seen in § 3, is readily expressed in terms of decaying spheroidal harmonics. Writing $\boldsymbol{u}^{(1)}=\hat{\boldsymbol{U}}+\boldsymbol{u}^{(1)^{\prime }}$ , with $\boldsymbol{u}^{(1)^{\prime }}\rightarrow 0$ for $\boldsymbol{r}\rightarrow \infty$ , we have:

Noting that the velocity field in the test problem is linear in ${\it\bf\Omega}^{(2)}$ , one may write $\boldsymbol{u}^{(2)}=\unicode[STIX]{x1D650}^{(2)}\boldsymbol{\cdot }{\it\bf\Omega}^{(2)}$ , where $\unicode[STIX]{x1D650}^{(2)}$ is a second-order tensor dependent only on the geometry of the spheroidal particle. Apart from the constraint of being orthogonal to the spheroidal axis of symmetry, ${\it\bf\Omega}^{(2)}$ is arbitrary, and since there can be no torque along the symmetry axis, one obtains from (2.17):

The Stokesian limit leads to a zero torque, and expectedly, $\boldsymbol{L}_{sed}$ involves terms proportional to $\mathit{Re}$ and $\mathit{De}$ . Apart from the restriction to second-order-fluid rheology, (2.18) does not yet exploit the simplifications that accrue in the limit $\mathit{De},\mathit{Re}\ll 1$ , since the integrands in (2.18) involve $\boldsymbol{u}^{(1)^{\prime }}$ , which requires the solution of the nonlinear equations of motion for finite $\mathit{Re}$ . If one only needs the $O(\mathit{Re})$ and $O(\mathit{De})$ contributions, however, one may replace $\boldsymbol{u}^{(1)^{\prime }}$ by $\boldsymbol{u}_{s}^{(1)}$ , where $\boldsymbol{u}_{s}^{(1)}$ is the disturbance velocity field due to a translating spheroid at $\mathit{Re}=\mathit{De}=0$ . To see that the $\mathit{Re}$ - and $\mathit{De}$ -dependent terms in $\boldsymbol{u}^{(1)^{\prime }}$ lead only to higher-order corrections to $\boldsymbol{L}_{sed}$ , it is sufficient to verify that the two volume integrals in (2.18), with $\boldsymbol{u}^{(1)^{\prime }}$ replaced by $\boldsymbol{u}_{s}^{(1)}$ , remain convergent, and this convergence is readily illustrated using the far-field behaviour of the Stokes velocity fields. In the integrand for the inertial torque, the dominant contribution for large distances ( $r\gg O(L)$ ) comes from the linearized inertial term, $\hat{\boldsymbol{U}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)^{\prime }}$ , corresponding to convection by the ambient flow. Since $\boldsymbol{u}_{s}^{(1)}\sim O(1/r)$ , this term decays as $O(1/r^{2})$ , and since the disturbance velocity field due to a rotating spheroidal particle decays as $O(1/r^{2})$ , the integrand for the inertial torque is $O(1/r^{4})$ , implying convergence for large $r$ . Considering the viscoelastic torque integral, the non-Newtonian stress ${\bf\sigma}_{NN}^{(1)}\sim (\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{(1)})^{2}\sim O(1/r^{4})$ ; with $\unicode[STIX]{x1D650}^{(2)}\sim O(1/r^{2})$ , $\boldsymbol{{\rm\nabla}}\unicode[STIX]{x1D650}^{(2)}\sim O(1/r^{3})$ , the integrand is $O(1/r^{7})$ and decays faster than that for the inertial torque contribution, and the resulting integral is again convergent. The convergence of the volume integrals indicates the regular nature of both the inertial and viscoelastic torque contributions. Although the elastic corrections for small $\mathit{De}$ , based on the retarded-motion expansion, typically lead to convergent integrals, this is not the case with inertia. For instance, the well-known $O(\mathit{Re})$ correction to the Stokes drag on a sphere (Leal Reference Leal1992), and more generally for any fore–aft symmetric body (including the spheroid considered here in appendix A), has a singular nature. Thus, a reciprocal theorem formulation for the drag leads to a volume integral that would be divergent on using the Stokes estimates as above, and convergence requires use of the Oseen velocity field at leading order. Physically, the inertial correction to the torque on a spheroid is dominated by the contribution from a volume of $O(L^{3})$ around it, while the corresponding correction to the force originates at distances of the order of $O(L\mathit{Re}^{-1})$ , the inertial screening length. In light of the above scaling estimates, one may write (2.18) in the form:

where the subscript ‘ $s$ ’ denotes a Stokesian field.

3. The spheroidal harmonics formalism

In this section, we discuss a novel analytical framework, the vector spheroidal harmonics formalism, introduced originally by Kushch and co-workers in the context of determining the averaged properties of a composite consisting of an elastic matrix with spheroidal inclusions (Kushch Reference Kushch1997, Reference Kushch1998). The formalism applies to the case where both the matrix and the inclusions are governed by the Lame equations of linear elasticity given by:

where $\tilde{\boldsymbol{u}}$ is the displacement field and ${\it\nu}$ is Poisson’s ratio. On identifying $\tilde{\boldsymbol{u}}$ in the matrix with the velocity field, the combination $(2(1-{\it\nu})/(1-2{\it\nu}))\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\tilde{\boldsymbol{u}}$ , which tends to a finite value at any point even as ${\it\nu}\rightarrow 1/2,\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\tilde{\boldsymbol{u}}\rightarrow 0$ , with the pressure field enforcing the incompressibility constraint, and taking the limit where the moduli of the spheroidal inclusions tend to infinity, the same formalism becomes applicable to the case of hydrodynamically interacting rigid spheroidal particles immersed in a Newtonian fluid that satisfies the Stokes equations. Such an identification has already been made by Kushch and Sangani, who have used the formalism to determine the effective viscosity of a periodic array of such particles, but this has unfortunately not been published yet. In what follows, we therefore give some of the details of this formalism, mainly those relevant to the motion of a single particle, before using it to obtain the relevant disturbance velocity and stress fields.

The formalism expresses the general solution of the Stokes equations, around an arbitrary number of spheroidal particles, as a superposition of growing and decaying partial vectorial solutions centred at each particle, and being defined in terms of a local spheroidal coordinate system aligned with the axis of symmetry of the particle under consideration. Thus, one writes the general expression for the velocity field around $N$ particles in the form:

where $\boldsymbol{r}_{p}=\boldsymbol{x}-\boldsymbol{x}_{p}$ is the position vector relative to the centre ( $\boldsymbol{x}_{p}$ ) of the $p$ th spheroid. The growing (regular) solutions are given by:

with $\boldsymbol{s}_{ts}^{(i)}\rightarrow \mathbf{0}$ for $\boldsymbol{r}_{p}\rightarrow 0$ , and the decaying (singular) solutions are given by: with $\boldsymbol{S}_{ts}^{(i)}\rightarrow \mathbf{0}$ for $\boldsymbol{r}_{p}\rightarrow \infty$ . Here, ${\it\beta}_{t}=(t+3)/((t+1)(2t+3))$ , $C_{ts}=(t+s+1)$ $(t-s+1){\it\beta}_{t}$ , with $t=0,1,\dots ;~|s|\leqslant t$ ; further, $\boldsymbol{e}_{1}=(\mathbf{1}_{x}+\text{i}\mathbf{1}_{y})/2$ , $\boldsymbol{e}_{2}=(\mathbf{1}_{x}-\text{i}\mathbf{1}_{y})/2$ , with $\mathbf{1}_{z}$ being directed along the axis of symmetry of the spheroidal coordinate system. The $\text{D}_{i}$ denote differential operators with $\text{D}_{1}=(\partial /\partial x-\text{i}\partial /\partial y)$ , $\text{D}_{2}=(\partial /\partial x+\text{i}\partial /\partial y)$ and $\text{D}_{3}=(\partial /\partial z)$ . In (3.3)–(3.5) and (3.6)–(3.8), the functions $F_{t}^{s}\equiv F_{t}^{s}(\boldsymbol{r}_{p},d_{p})$ and $f_{t}^{s}\equiv f_{t}^{s}(\boldsymbol{r}_{p},d_{p})$ are, respectively, the singular and regular solid spheroidal harmonics of the form $F_{t}^{s}=Q_{t}^{s}({\it\xi})Y_{t}^{s}({\it\eta},{\it\phi})$ and $f_{t}^{s}=P_{t}^{s}({\it\xi})Y_{t}^{s}({\it\eta},{\it\phi})$ , with $Y_{t}^{s}({\it\eta},{\it\phi})=P_{t}^{s}({\it\eta})\exp (\text{i}s{\it\phi})$ being the familiar scalar surface harmonics, and $P_{t}^{s}$ and $Q_{t}^{s}$ being the associated Legendre functions of the first and second kind, respectively (Morse & Feshbach Reference Morse and Feshbach1953).

In all of the above, $({\it\xi},{\it\eta},{\it\phi})$ denote the spheroidal coordinates. The above expressions for the partial vectorial solutions are valid for the prolate case, in which case $({\it\xi},{\it\eta},{\it\phi})$ are related to the Cartesian coordinates $(x,y,z)$ as $x+\text{i}y=d\sqrt{{\it\xi}^{2}-1}\sqrt{1-{\it\eta}^{2}}\exp (\text{i}{\it\phi}),z=d{\it\xi}{\it\eta}$ , with $1\leqslant {\it\xi}\leqslant \infty ,|{\it\eta}|\leqslant 1,0\leqslant {\it\phi}<2{\rm\pi}$ . The constant- ${\it\xi}$ surfaces denote a family of confocal prolate spheroids, with the interfoci distance being equal to $2d$ , and ${\it\xi}_{0}$ denotes the surface of the particle. ${\it\xi}=1$ reduces to the line segment connecting the foci, and the limit ${\it\xi}_{0}\rightarrow 1$ therefore corresponds to the limit of an infinitely slender particle. The constant- ${\it\eta}$ surfaces represent a family of confocal two-sheeted hyperboloids, while the constant- ${\it\phi}$ surfaces are planes passing through the axis of symmetry. The oblate spheroidal coordinate system may be obtained by replacing ${\it\xi}$ by $\text{i}\sqrt{{\it\xi}^{2}-1}$ and $d$ by $-\text{i}d$ in the above expressions for the prolate case. With this substitution, the constant- ${\it\xi}$ surfaces are confocal oblate spheroids with ${\it\xi}=1$ degenerating to the focal circle with diameter $2d$ , and the limit ${\it\xi}_{0}\rightarrow 1$ now denotes the limit of a flat disk. The constant- ${\it\eta}$ surfaces are confocal single-sheeted hyperboloids, while the constant- ${\it\phi}$ surfaces remain planes through the symmetry axis as in the prolate case. With increasing ${\it\xi}$ , or decreasing $d$ , the constant ${\it\xi}$ -surfaces, in both the prolate and oblate cases, approach a spherical shape and the constant- ${\it\eta}$ surfaces approach a conical geometry. The important limiting case of a spherical particle of radius $a$ may therefore be obtained by taking ${\it\xi},{\it\xi}_{0}\rightarrow \infty$ , $d\rightarrow 0$ , with $d{\it\xi}=r$ being fixed and equal to the radial distance to the point of interest, and $d{\it\xi}_{0}=a$ being the radius of the spherical particle.

The general superposition, (3.2), together with re-expansion formulae (addition theorems) for the partial solutions due to an arbitrary translation and rotation of the coordinate system, reduce the satisfaction of the boundary conditions at the surfaces of the rigid particles, of the form $\boldsymbol{u}(\boldsymbol{r}_{p})=\boldsymbol{U}_{p}+{\it\bf\Omega}_{p}\wedge \boldsymbol{r}_{p}$ , to an infinite system of linear equations in the unknown amplitudes $A_{ts}^{(i)(p)}$ and $a_{ts}^{(i)(p)}$ . Such an approach has been successfully applied to the case of spherical particles (Sangani & Mo Reference Sangani and Mo1994), and Kushch, in a series of earlier articles, has extended the formulation to spheroidal particles (Kushch Reference Kushch1995, Reference Kushch1997, Reference Kushch1998). The aforementioned re-expansion formulae come into play only when there are multiple interacting particles, and are therefore beyond the scope of the present paper, which deals with the orientation of a single sedimenting particle. The exterior velocity disturbance field in the domain ${\it\xi}\geqslant {\it\xi}_{0}$ around a single spheroidal particle, due to an imposed ambient flow, may be expressed in terms of the decaying partial solutions alone as:

where we note from (3.6)–(3.8) that the $\boldsymbol{S}_{ts}^{(1)}$ and $\boldsymbol{S}_{ts}^{(2)}$ are vector harmonic functions ( ${\rm\nabla}^{2}\boldsymbol{S}_{ts}^{(i)}=0$ for $i=1,2$ ). The $\boldsymbol{S}_{ts}^{(3)}$ , which are vector biharmonic functions ( ${\rm\nabla}^{4}\boldsymbol{S}_{ts}^{(3)}=0$ ), lead to a non-trivial pressure field. We have

so the associated pressure field is $(2/d)D_{t}^{s}$ , with $D_{t}^{s}$ being a scalar harmonic given by $D_{t}^{s}=\text{d}\text{D}_{3}F_{t-1}^{s}$ , where $\text{D}_{3}F_{t-1}^{s}=\text{D}_{1}F_{t-1}^{s+1}=-\text{D}_{2}F_{t-1}^{s-1}$ . The singular solutions in (3.9) satisfy the additional properties:

with $\widetilde{\boldsymbol{S}}_{ts}^{(1)}$ and $\widetilde{\boldsymbol{S}}_{ts}^{(2)}$ having the same form as those given by (3.6) and (3.7), but with $F_{t}^{s}$ replaced by $D_{t}^{s}$ . The associated Legendre functions of the second kind, $Q_{t}^{s}({\it\xi})$ , that appear in (3.6)–(3.8) approach zero for ${\it\xi}\rightarrow \infty$ and, thereby, account for the decaying nature of the velocity field in (3.9) due to the singular partial solutions. Specifically, the index $t$ in $\boldsymbol{S}_{ts}^{(i)}$ is a measure of the rapidity of decay of the velocity disturbance field, with $\lim _{r\rightarrow \infty }\boldsymbol{u}(\boldsymbol{x})\propto r^{-t}$ , this arising from the large- ${\it\xi}$ behaviour of the $Q_{t}^{s}$ . Since it is the $\boldsymbol{S}_{ts}^{(3)}$ that include the fundamental singularities of the Stokes equations, in light of the above large- $r$ behaviour of $\boldsymbol{u}$ , one expects the $\boldsymbol{S}_{1s}^{(3)}$ to be relevant to the translation problem ( $\boldsymbol{u}(\boldsymbol{x})\propto 1/r$ ), the $\boldsymbol{S}_{2s}^{(3)}$ to come into play in the presence of an ambient linear flow ( $\boldsymbol{u}(\boldsymbol{x})\propto 1/r^{2}$ ), the $\boldsymbol{S}_{3s}^{(3)}$ to be relevant to an ambient quadratic flow ( $\boldsymbol{u}(\boldsymbol{x})\propto 1/r^{3}$ ), and so on. The second index $s$ in all these cases denotes the variation of the velocity field as a function of the azimuthal angle in the plane transverse to the symmetry axis, with $s=0$ corresponding to an axisymmetric exterior field.

The structure of the spheroidal harmonics formalism developed above is similar to the well-known solution, in terms of vector spherical harmonics, given originally by Lamb (see chapter 4 of Kim & Karrila Reference Kim and Karrila1991), and that has often been used for determining the effective properties of composites with spherical inclusions (Sangani & Mo Reference Sangani and Mo1994). For the case of a single particle, not acting as a source of mass, and whose induced disturbance velocity field is required to decay for large distances, this solution is of the form:

where $\boldsymbol{x}$ is now the position vector relative to the sphere centre with $c_{n}=(n+3)/(2(n+1)(2n+3))$ and $b_{n}=-n/((n+1)(2n+3))$ . Here, $p_{n}$ , ${\it\chi}_{n}$ and ${\it\phi}_{n}$ are (scalar) solid spherical harmonics of the $n$ th order. Since, for instance, $p_{-n-1}=r^{-n-1}\!\sum _{m=-n}^{n}A_{mn}Y_{n}^{m}({\it\theta},{\it\phi})=r^{-n-1}\!\sum _{m=-n}^{n}A_{mn}P_{n}^{m}(\cos {\it\theta})\text{e}^{\text{i}m{\it\phi}}$ , one may write (3.15) in the form:

where each of $p_{n,m}$ , ${\it\chi}_{n,m}$ and ${\it\phi}_{n,m}$ are proportional to $r^{n}P_{n}^{m}(\cos {\it\theta})\text{e}^{\text{i}m{\it\phi}}$ . From comparing (3.16) with (3.9) and (3.6)–(3.8), it may be seen that the terms in (3.15) involving ${\it\chi}_{n}$ and ${\it\phi}_{n}$ are harmonic, and are therefore similar to $\boldsymbol{S}_{ts}^{(1)}$ and $\boldsymbol{S}_{ts}^{(2)}$ , with $\boldsymbol{S}_{ts}^{(1)}$ being the analogue of $\boldsymbol{{\rm\nabla}}{\it\phi}_{-n-1,m}$ , since both describe decaying irrotational flow contributions. Importantly, $\boldsymbol{S}_{ts}^{(3)}$ is the analogue of the term involving $p_{n,m}$ . The latter term also satisfies the biharmonic equation and has an associated pressure field. The Kushch formalism is, in fact, a slightly more economical reformulation of the Lamb series in that the velocity disturbance of canonical flow problems are described by fewer terms in the general series given by (3.9). For example, the disturbance velocity field produced by a translating sphere has an $O(1/r)$ component corresponding to the action of the point force at the sphere centre, and a more rapidly decaying $O(1/r^{3})$ irrotational component that arises from the finite size of the sphere, being equivalent to an additional (degenerate) quadrupole at the centre. For the Lamb series, this velocity field requires the term involving $p_{-2}$ that generates the Stokeslet part and the potential term proportional to $\boldsymbol{{\rm\nabla}}{\it\phi}_{-2}$ that generates the $O(1/r^{3})$ terms (Kim & Karrila Reference Kim and Karrila1991). On the other hand, as will be seen below, the entire velocity field due to a translating spheroid is contained in $\boldsymbol{S}_{1s}^{(3)}$ for $s=1,0,-1$ (the different values of $s$ correspond to translations along and transverse to the axis of symmetry). This additional simplification is possible owing to the addition of terms proportional to $\bar{{\it\xi}}_{0}^{2}$ in (3.8), and these terms taken together are in the form of the gradient of a scalar. This, in effect, includes the additional irrotational component of the disturbance velocity field that is needed in order to satisfy the no-slip boundary condition on the surface of the particle ( ${\it\xi}={\it\xi}_{0}$ ). In contrast, the Lamb series solution evidently does not involve the particle size $a$ . Thus, while each of the terms in (3.15) is a homogeneous function of $r$ , the biharmonic component in $\boldsymbol{S}_{ts}^{(3)}$ in Kushch’s formalism is not a homogeneous function of ${\it\xi}$ , at leading order, in the limit of large ${\it\xi}$ (with $d{\it\xi}$ finite). As will be shown in a companion paper (Dabade et al. Reference Dabade, Marath and Subramanian2015), the above remains true even for the case of an ambient linear flow. In this case, the disturbance velocity field due to a force-free torque-free spherical particle has an $O(1/r^{2})$ part that corresponds to the stresslet singularity and an irrotational $O(1/r^{4})$ contribution, and these arise from the terms proportional to $p_{-3}$ and ${\it\phi}_{-3}$ , respectively, in the Lamb series. In contrast, the entire disturbance velocity field due to a spheroidal particle in an axisymmetric extensional flow aligned with the particle axis of symmetry is generated by $\boldsymbol{S}_{2,0}^{(3)}$ . Similarly, the entire disturbance field in a planar extension, transverse to the symmetry axis, is given by a linear combination of $\boldsymbol{S}_{2,2}^{(3)}$ and $\boldsymbol{S}_{2,-2}^{(3)}$ .

Having clarified the general structure of the spheroidal harmonics formalism, we construct the disturbance velocity fields that enter the reciprocal theorem formulation given in § 2. The requirement of far-field decay is already satisfied by (3.9). The satisfaction of the no-slip boundary condition on the particle surface requires expressions for the three decaying partial vectorial solutions for ${\it\xi}={\it\xi}_{0}$ , which, for the prolate spheroid, are given by:

with the corresponding expressions for an oblate spheroid resulting from the transformation mentioned above.

The simplest velocity field is that due to a spheroid rotating about its axis of symmetry. As for a rotating sphere, this field is harmonic without an associated pressure field, being proportional to $\boldsymbol{S}_{2,0}^{(2)}$ ; the Lamb series for a rotating sphere gives $\boldsymbol{{\rm\nabla}}\wedge (\boldsymbol{x}{\it\chi}_{-2})$ . The inertial acceleration and the non-Newtonian stress field that appear in the torque integrals in (2.19) require the disturbance velocity field due to a spheroid translating in an arbitrary orientation. The linearity of the Stokes equations imply that $\boldsymbol{u}_{s}^{(1)}$ may be written as the following superposition of the axial and transverse translation fields:

and $\boldsymbol{U}=U(\sin {\it\alpha}\mathbf{1}_{x}+\cos {\it\alpha}\mathbf{1}_{z})$ is the velocity of translation of the spheroid with ${\it\alpha}$ being the angle between $\boldsymbol{U}$ and the spheroidal axis. Thus, $\boldsymbol{S}_{1,0}^{(3)}$ in (3.20) gives the disturbance velocity field for axial translation, and the particular combination $\boldsymbol{S}_{1,1}^{(3)}-\boldsymbol{S}_{1,-1}^{(3)}$ corresponds to transverse translation along the $x$ -direction ( $\boldsymbol{S}_{1,1}^{(3)}+\boldsymbol{S}_{1,-1}^{(3)}$ gives the velocity field for $y$ -translation). Here, we have chosen the $xz$ plane as the one containing the vector $\hat{\boldsymbol{U}}$ and the unit vector along the axis of symmetry, and the torque vector is therefore perpendicular to this plane. Accordingly, the test velocity field, $\boldsymbol{u}^{(2)}$ in (2.1) corresponds to a rotation about the $y$ -axis, and for an angular velocity with unit magnitude, $\boldsymbol{u}_{s}^{(2)}$ is given by:

and $\unicode[STIX]{x1D650}^{(2)}$ in (2.19) is given by $\boldsymbol{u}_{s}^{(2)}\mathbf{1}_{y}$ . Note that while the axial rotation above only involved an $\boldsymbol{S}_{ts}^{(2)}$ , the transverse rotation field also involves the $\boldsymbol{S}_{ts}^{(3)}$ since the rotation of a non-spherical particle leads to a non-trivial pressure field. A detailed comparison that shows the equivalence of (3.20) and (3.21) to the corresponding expressions obtained from the method of singularities (Chwang & Wu Reference Chwang and Wu1974, Reference Chwang and Wu1975) is given in appendix B.

The velocity fields given by (3.20) and (3.21) may now be used in (2.19) and, as will be seen in § 4, an integration in spheroidal coordinates yields analytical expressions for the torque on a prolate spheroid of an arbitrary aspect ratio as a function of $d$ and ${\it\xi}_{0}$ . The torques on an oblate spheroid may simply be obtained using ${\it\xi}_{0}\leftrightarrow \text{i}\sqrt{{\it\xi}_{0}^{2}-1},d\leftrightarrow -\text{i}d$ in the corresponding prolate forms. Before moving on to the results, it is worth spending a little time discussing earlier efforts based on a different approach, the method of singularities (Leal Reference Leal1992), that also analyse the motion of spheroidal particles and the associated disturbance velocity fields for various canonical scenarios. By distributing appropriate combinations of the fundamental Stokes singularities (Stokeslets, stresslets, potential dipoles and quadrupoles) along the axis of symmetry between the foci, Chwang and Wu, in a series of articles, obtained the velocity disturbance fields due to a prolate spheroid in a variety of flows, including ambient linear and quadratic flows (Chwang & Wu Reference Chwang and Wu1974, Reference Chwang and Wu1975; Chwang Reference Chwang1975). These representations of the disturbance velocity fields, together with similar representations for the Faxen relations, have been subsequently used by Kim and co-workers (Kim Reference Kim1985a ,Reference Kim b , Reference Kim1986b ) to analyse the interactions between a pair of sedimenting spheroids using the method of reflections (except near contact). Kim and co-workers (Kim & Arunachalam Reference Kim and Arunachalam1987; Kim & Karrila Reference Kim and Karrila1991) generalized the singularity representations to a tri-axial ellipsoid, there being an areal distribution of fundamental singularities on the surface of a focal ellipse in this case. In the limiting case of an oblate spheroid, there is an areal distribution on a focal disk in a plane transverse to the symmetry axis. The Faxen relations for a sphere were originally generalized to the case of a tri-axial ellipsoid much earlier (Brenner Reference Brenner1966), but the resulting representations were in the form of an infinite series in terms of singularities located at the geometric centre. These have been found to be numerically intractable (Claeys & Brady Reference Claeys and Brady1993b ), and are subject to convergence problems at small separations (Kim & Arunachalam Reference Kim and Arunachalam1987). Later, Claeys and Brady have used the results of Kim and co-workers to analyse the hydrodynamic interactions between a finite number of spheroidal particles (Claeys & Brady Reference Claeys and Brady1993b ), and in an infinite suspension, both ordered (Claeys & Brady Reference Claeys and Brady1993c ) and disordered (Claeys & Brady Reference Claeys and Brady1993a ). The authors extended the idea of the Stokesian dynamics method, originally proposed for suspensions of spherical particles (Brady & Bossis Reference Brady and Bossis1988), to spheroidal particles, and the results of Kim and co-workers were used to construct the far-field mobility matrix. The aforementioned efforts of Chwang and Wu for single particles, and those of Kim and Brady for both finite and infinite systems of particles, are for prolate spheroids. Recently, Shatz (Reference Shatz2004) has shown that the Stokes disturbance fields for an oblate spheroid may still be obtained by a line distribution of the fundamental Stokes singularities as opposed to the areal distribution mentioned above. This is possible by placing the singularities along an imaginary focal length, aligned with the symmetry axis, a singularity at a point along this axis being equivalent to a singularity all along the perimeter of a (real) focal circle transverse to the axis of symmetry. The notion of an imaginary focal length mimics the transition relations between the prolate and oblate spheroidal coordinates ( $d\leftrightarrow -\text{i}d$ ) given earlier. From its ready applicability to prolate and oblate spheroids of an arbitrary aspect ratio, and from the ease of obtaining closed-form analytical results, the Kushch formalism presented here appears more advantageous. The singularity method yields the disturbance velocity fields in terms of integral representations which are cumbersome to handle analytically (as already noted, Kim (Reference Kim1986a ), despite using the singularity representation of the velocity field calculated the viscoelastic torque integral numerically). Although these integrals can be evaluated in closed form, the Kushch solution yields a elegant rearrangement of the resulting analytical forms in a manner that mimics the original Lamb series. For these reasons we have adopted this formalism here, and in a companion paper (Dabade et al. Reference Dabade, Marath and Subramanian2015), to characterize the orientation dynamics of spheroidal particles in sedimentation and shearing flows.

4. The torques on a sedimenting spheroid

The final expression, equation (2.19), from the generalized reciprocal theorem formulation, with the expressions (3.20) and (3.21) for the velocity fields, are used to obtain closed-form expressions for the inertial and viscoelastic torques as a function of the particle aspect ratio for both prolate and oblate spheroids. The regular nature of the torque contributions, together with symmetry arguments, imply that the inertial and viscoelastic torques, apart from being proportional to $\mathit{Re}$ and $\mathit{De}$ , respectively, must have the general form $F({\it\xi}_{0})\hat{\boldsymbol{U}}\boldsymbol{\cdot }\boldsymbol{p}(\hat{\boldsymbol{U}}\wedge \boldsymbol{p})$ , $\boldsymbol{p}$ being the unit vector along the symmetry axis. For the viscoelastic case, there is an additional dependence on ${\it\epsilon}$ , the second-order fluid parameter, so that $F({\it\xi}_{0})_{visco}\equiv F({\it\xi}_{0};{\it\epsilon})_{visco}$ . Importantly, both torque contributions have an angular dependence given by $\sin 2{\it\alpha}$ , ${\it\alpha}$ being the angle between $\hat{\boldsymbol{U}}$ and $\boldsymbol{p}$ . The dimensional inertial (elastic) torque is given by $\mathit{Re}(\mathit{De})F({\it\xi}_{0}){\it\mu}UL^{2}\sin {\it\alpha}\cos {\it\alpha}$ , so that the peak value is attained at ${\it\alpha}=45^{\circ }$ , and the torque vanishes for the longitudinal ( $\hat{\boldsymbol{U}}\Vert \boldsymbol{p}$ ) and transverse ( $\hat{\boldsymbol{U}}\bot \boldsymbol{p}$ ) orientations; only one of these equilibria is stable, however. The detailed integration in spheroidal coordinates determines $F({\it\xi}_{0})$ , which then characterizes the torque dependence on aspect ratio. From figure 1, it is seen that a torque driving a prolate spheroid towards a longside-on orientation implies a positive $F({\it\xi}_{0})$ , while an approach towards a broadside-on orientation implies a negative $F({\it\xi}_{0})$ ; the opposite is true for an oblate spheroid.

Figure 1. The sense of the torques, and the corresponding signs of $F({\it\xi}_{0})$ for sedimenting prolate (a) and oblate (b) spheroids.

For the inertial case, it will be seen in § 4.1 that $F({\it\xi}_{0})$ has the same sign for all ${\it\xi}_{0}\,{\it\epsilon}(1,\infty )$ , although it changes sign in going from a prolate to an oblate spheroid. The inertial torque favours the broadside-on configuration for both prolate and oblate spheroids. A prolate spheroid in this orientation has its $\boldsymbol{p}\bot \hat{\boldsymbol{U}}$ , while an oblate spheroid in the same orientation has $\boldsymbol{p}\Vert \hat{\boldsymbol{U}}$ , this being consistent with the change in sign of $F({\it\xi}_{0})$ . For $\mathit{Re}\sim O(1)$ or larger, the broadside-on orientation arises since the wake of the front portion of the spheroid ‘shields’ the rear, which catches up with the front as a result. Although not literally true for $\mathit{Re}\ll 1$ , the wake now being asymptotically far away in the Oseen region, the $O(\mathit{Re})$ correction to the Stokes velocity field, in the inner region, nevertheless reflects the asymmetry of the outer Oseen field. The nature of the inertial torque is generic, and occurs even when the anisotropic body is a compound object formed, for instance, as a result of two touching spheres; the tendency of the two spheres, in this configuration, to rotate onto a broadside-on orientation is well known (Fortes, Joseph & Lundgren Reference Fortes, Joseph and Lundgren1987).

For the viscoelastic case, to begin with, we restrict our attention to spheroids sedimenting in polymer solutions, in which case ${\it\epsilon}$ is restricted to the narrow interval $(-0.7,-0.5)$ . For these ${\it\epsilon}$ values, as will be in § 4.2, viscoelasticity stabilizes the longside-on orientation ( $\boldsymbol{p}\Vert \hat{\boldsymbol{U}}$ for a prolate spheroid and $\boldsymbol{p}\bot \hat{\boldsymbol{U}}$ for an oblate spheroid), and this may be explained based on the well-known idea of tensioned streamlines. At the microstructural level, the first effect of a weak ambient flow (small $\mathit{De}$ ) on the suspended polymers is to stretch them, on average, along the local extensional axis, leading to an enhancement in the shear viscosity. The second-order fluid rheology denotes the first departure from such a Newtonian response and, at this order, the distribution of polymer configurations tilts away from the local extensional axis towards the flow direction on account of the rotation induced by the local vorticity. This effect is captured by the co-rotational contribution to the polymeric stress given by (2.9). The slight flow-alignment of the polymers leads to a tensile stress component along the local flow direction and, at $O(\mathit{De})$ , one may therefore regard the streamlines as being similar to stretched rubber bands. The tendency of the bent streamlines to snap straight leads to a torque on the spheroid acting to align it to a longside-on orientation. Note that the quadratic stress component depends only on the rate-of-strain tensor, and yields $N_{1}=0$ , implying the absence of a flow-aligned tension.

Thus, physical arguments suggest that either inertia or elasticity (when consistent with a tensioned-streamline configuration), acting in isolation on a prolate or an oblate spheroid, leads to the same equilibrium orientation for all aspect ratios. Although $F({\it\xi}_{0})$ changes sign in going from inertia to viscoelasticity, the identical angular dependence implies that inertia and elasticity, even when acting simultaneously, cannot lead to equilibrium orientations intermediate between those parallel and perpendicular to $\hat{\boldsymbol{U}}$ , as observed in some earlier experiments (Joseph & Liu Reference Joseph and Liu1993; Liu & Joseph Reference Liu and Joseph1993). Intermediate equilibria are likely when one includes higher-order contributions, but these would be restricted to an asymptotically small angular interval of $O(\mathit{Re})$ or $O(\mathit{De})$ in the limit $\mathit{Re},\mathit{De}\ll 1$ . To the order considered here, therefore, depending on the ratio $\mathit{De}/\mathit{Re}$ , the so-called elasticity number $\mathit{El}$ , either the inertial or viscoelastic torque is dominant for all orientations. The ratio of the two torques, at leading order, is given by $[\mathit{Re}\,F({\it\xi}_{0})_{inertia}]/[\mathit{De}\,F({\it\xi}_{0};{\it\epsilon})_{visco}]$ , and equating this to unity yields the critical curve separating the two equilibria. This neutral curve may be written in the form $\mathit{El}=\mathscr{G}({\it\xi}_{0};{\it\epsilon})$ , with $\mathscr{G}({\it\xi}_{0})=F({\it\xi}_{0})_{inertia}/F({\it\xi}_{0};{\it\epsilon})_{visco}$ , and would separate a region of longside-on orientation above from a broadside-on orientation below. The detailed results obtained in § 4.2.2 determine this curve as a function of the spheroid aspect ratio.

The argument above, of the viscoelastic torque leading to a longside-on configuration, relies on the notion of tensioned streamlines. This in turn depends on $N_{1}$ being dominant and positive, as is true for polymer solutions. Other complex fluids in near-equilibrium scenarios have $N_{2}$ being comparable to $N_{1}$ , although still smaller in magnitude. An alternate scenario is where ${\it\epsilon}=0$ , when only the quadratic component of the polymeric stress survives, and one has an $N_{2}$ -dominant second-order fluid. That the effects of a non-zero $N_{2}$ are more subtle is known from the case of polymer solutions (Tanner Reference Tanner2000). More recently, the consequence of a dominant $N_{2}$ on the nature of the free-surface perturbation has been examined in the context of suspensions of non-Brownian particles (Boyer, Pouliquen & Guazzelli Reference Boyer, Pouliquen and Guazzelli2011). The observations have been explained based on the notion of tensioned vortex lines in the base-state azimuthal flow (Hinch Reference Hinch2011), although we note that such suspensions, unlike the second-order fluids considered here, are far from thermal equilibrium. Nevertheless, we consider the dependence of the viscoelastic torque on both ${\it\xi}_{0}$ and ${\it\epsilon}$ in § 4.2.4, and show that, for ${\it\epsilon}\geqslant 0$ , the torque acting on a prolate spheroid does change sign for certain aspect ratios, thereby stabilizing the broadside-on orientation. Further, this change in sign appears explainable based on a tensioned vortex line configuration, although the geometry is significantly more complicated than those of the viscometric flows considered in earlier efforts (Keentok et al. Reference Keentok, Georgescu, Sherwood and Tanner1980; Hinch Reference Hinch2011).

5. Viscoelastic torque on a fibre for large $\mathit{De}$

The second-order fluid regime, which underlies the analysis in the earlier sections, restricts the Deborah number to very small values. This restriction becomes particularly severe for large-aspect-ratio fibres, where the velocity gradient ranges from being $O(U/L)(\ln {\it\kappa})^{-1}$ , on length scales of $O(L)$ , to being $O(U/b)(\ln {\it\kappa})^{-1}$ close to the fibre. For such particles, the above restriction applies to the Deborah number based on the radius, implying that $\mathit{De}$ , as defined in § 2, must be smaller than $O(1/{\it\kappa})$ . Experiments carried out on sedimenting particles in polymer solutions conform neither to the stringent requirement on the relaxation time nor to the diluteness constraint in general (Leal Reference Leal1979; Chiba, Song & Horikawa Reference Chiba, Song and Horikawa1986; Cho, Cho & Park Reference Cho, Cho and Park1992; Joseph & Liu Reference Joseph and Liu1993). The observations indicate that the sense of the viscoelastic torque remains unaltered even outside of the second-order fluid regime, although a quantitative analysis would now have to account for the finite rate of microstructural relaxation even in the dilute regime. In what follows, we discuss the physical picture, along with scaling arguments and some analysis, relevant to a slender fibre sedimenting at finite $\mathit{De}$ in a dilute polymer solution.

For finite $\mathit{De}$ , the stress can be modelled using the Oldroyd-B constitutive equation (Larson Reference Larson1988) and the deviatoric polymeric stress ( ${\bf\sigma}_{NN}$ ) is governed by:

where ${\it\delta}_{U}/({\it\delta}t)$ is the upper convected derivative and $G=nkT$ is the shear modulus ( $n$ here is the polymer number density, and the dilute regime implies that $nR_{g}^{3}\ll 1$ , with $R_{g}$ being the characteristic size of a polymer coil). The polymer stress at any point can be obtained by inverting (5.1) along a particular streamline, leading to the following expression for the stress made non-dimensional by $G$ :

where $\unicode[STIX]{x1D60A}_{ij}^{-1}(t,t^{\prime })=(\partial x_{i}(t)/\partial r_{q}(t^{\prime }))(\partial x_{j}(t)/\partial r_{q}(t^{\prime }))$ , is the Finger tensor relating the configuration at time $t$ , $\boldsymbol{x}(t)$ , to that at an earlier time $t^{\prime }$ , $\boldsymbol{r}(t^{\prime })$ , and $\text{D}t^{\prime }$ denotes integration along a streamline.

For a slender fibre, it is natural to consider three physically distinct regimes, depending on the ordering of the two flow time scales, $L/U$ and $b/U$ , relative to the relaxation time scale ${\it\tau}$ . The second-order fluid regime, as indicated above, corresponds to ${\it\tau}\ll b/U\ll L/U$ . The other extreme, where ${\it\tau}\gg L/U\gg b/U$ , corresponds to $\mathit{De}\gg 1$ , when the polymer molecules do not relax as they are convected past the sedimenting fibre. In this limit, the upper convected derivative in (5.1) is dominant, and the polymer molecules are affinely deformed by the flow. The stress given by (5.2) reduces to just the Finger tensor, the response of a neo-Hookean solid (Rallison & Hinch Reference Rallison and Hinch2004). Viscous slender-body theory may now be used to estimate the contributions to the viscoelastic torque from the outer ( $r\sim O(L)$ ) and inner ( $r\sim O(b)$ ) regions. In the outer region, ${\bf\sigma}_{NN}\sim O(1/\text{ln}\,{\it\kappa})$ , corresponding to the strain resulting from an $O(1/\text{ln}\,{\it\kappa})$ velocity field acting over an $O(1)$ time interval, $\boldsymbol{{\rm\nabla}}\unicode[STIX]{x1D650}^{(2)}\sim O(1/\text{ln}\,{\it\kappa})$ , $\text{d}V\sim O(1)$ , so the torque integral, $\int \!\!{\bf\sigma}_{NN}\,:\,\boldsymbol{{\rm\nabla}}\unicode[STIX]{x1D650}^{(2)}\,\text{d}V\sim O[1/(\ln {\it\kappa})^{2}]$ . The polymer molecules spend only an $O(1/{\it\kappa})$ time in the inner region, the velocity field in this region being $O(\ln {\it\rho}/\text{ln}\,{\it\kappa})$ ; thus ${\bf\sigma}_{NN}\sim O(1/({\it\kappa}{\it\rho}\ln {\it\kappa}))$ , $\boldsymbol{{\rm\nabla}}\unicode[STIX]{x1D650}^{(2)}\sim O(1/({\it\rho}\ln {\it\kappa}))$ , $\text{d}V\sim O({\it\rho}^{2})$ , with ${\it\rho}\sim O(b)$ . The inner region contribution to the torque integral scales as $O[1/({\it\kappa}(\ln {\it\kappa})^{2})]$ . In contrast to the second-order fluid regime discussed in § 4.2.3, the arguments above suggest the dominance of the outer region for $\mathit{De}\gg 1$ . For a slender fibre, the logarithmically weak disturbance velocity field in the outer region implies that the stress at any given point may be obtained by integrating along the (straight) streamlines of the ambient uniform flow at leading logarithmic order, and this allows for a simple calculation in Fourier space. Using the convolution theorem, the volume integral for torque can be rewritten in Fourier space as:

where the $\hat{{\bf\sigma}}_{NN}(\boldsymbol{k})$ and $\hat{\boldsymbol{U}}^{(2)}\!(\boldsymbol{k})$ are the Fourier-transformed non-Newtonian stress and test velocity fields, the Fourier transform being defined by $\hat{\boldsymbol{f}}(\boldsymbol{k})=\int \!\text{e}^{-\text{i}2{\rm\pi}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}}f(\boldsymbol{r})\,\text{d}\boldsymbol{r}$ . Retaining only the material derivative on the left-hand side in (5.1), at leading order, and transforming, one obtains:

where $\hat{\boldsymbol{u}}^{(1)}(\boldsymbol{k})$ is the Fourier-transformed disturbance velocity field due to a translating fibre given by (Subramanian & Koch Reference Subramanian and Koch2005):

Figure 14. (a) Leading order non-dimensional elastic torque on a sedimenting slender fibre, as a function of the angle of inclination, for finite $\mathit{De}\gg O({\it\kappa}^{-1/2})$ . (b) A shift in the location of the peak torque, back towards $45^{\circ }$ , is apparent as $\mathit{De}$ decreases.

The Fourier transform of $\unicode[STIX]{x1D650}^{(2)}$ , proportional to the disturbance due to a rotating fibre, is given by (Subramanian & Koch Reference Subramanian and Koch2005):

Here, $\text{j}_{0}$ and $\text{j}_{1}$ are the zeroth- and first-order spherical Bessel functions. For infinite $\mathit{De}$ , using (5.4)–(5.6), the Fourier-space integral in (5.3) may be interpreted in the sense of a Cauchy principal value, and its evaluation leads to the following dimensional elastic torque:

This $O[GL^{3}/(\ln {\it\kappa})^{2}]$ torque again leads to a longside-on orientation. Although the angular dependence in (5.7) is more complicated than the simple $\sin 2{\it\alpha}$ dependence found for a second-order fluid, the torque still peaks at an angle very close to 45°. The torque integral may be evaluated numerically for finite $\mathit{De}$ , and the calculation shows that, with decreasing $\mathit{De}$ , the peak torque shifts to smaller angles, as seen in figure 14. For small enough $\mathit{De}$ , it again reverts to 45°, consistent with symmetry arguments applicable to a second-order fluid approximation in the outer region. The above analysis, based on the outer region alone, fails when the dominant torque contribution shifts to the inner region. This happens when $\mathit{De}$ reduces to $O({\it\kappa}^{-1/2})$ , and relaxation effects can no longer be neglected in the inner region (note that, for $\mathit{De}\gg O({\it\kappa}^{-1/2})$ , the stress in the inner region is proportional to the finger tensor, as in the outer region; the second-order fluid assumption of dominant relaxation becomes uniformly valid in the entire fluid domain only at an asymptotically smaller $\mathit{De}$ of $O(1/{\it\kappa})$ , when the leading-order torque contribution continues to come from the inner region; see § 4.2.3).

Although the analysis sketched above, based on slender-body theory, gives a convergent expression for the large- $\mathit{De}$ elastic torque, that this is the total torque is not obvious. This is because the scaling arguments above, that establish the dominance of the outer region, do not account for the detailed kinematics of the flow in the vicinity of the stagnation points. For spheres, it is known that the extensional nature of the flow close to the rear stagnation point, together with the large residence time involved, leads to highly stretched polymers once $\mathit{De}$ exceeds a value of order unity. The resulting region of stretched polymers in the vicinity of the rear stagnation streamline, the so-called ‘birefringent strand’ (Rallision & Hinch Reference Rallision and Hinch1988; Harlen Reference Harlen1990; Harlen, Rallison & Chilcott Reference Harlen, Rallison and Chilcott1990), extends for a considerably large distance downstream of the sedimenting sphere for large $\mathit{De}$ . In turn, this leads to a pronounced upstream–downstream asymmetry, with the flow downstream of the sphere relaxing at a much slower rate to the ambient in an extended elastic wake (Arigo et al. Reference Arigo, Rajagopalan, Shapley and Mckinley1995). The birefringent strand affects the drag on the sedimenting sphere for large $\mathit{De}$ . The sphere drag initially decreases from its Stokes value with increasing $\mathit{De}$ (as mentioned in § 1, this is captured even within the limited framework of an ordered fluid expansion, and is related to the decrease in the polymer contribution to the shear viscosity in a Lagrangian unsteady flow), but starts to increase again beyond $\mathit{De}\sim O(1)$ to a value higher than the Stokes value on account of the stress boundary layer, close to the rear stagnation point, associated with the birefringent strand; although, the precise extent of drag increase depends on the polymer extensibility (McKinley Reference McKinley2002).

In analogy with the sphere, one therefore expects the presence of a birefringent strand behind a sedimenting slender spheroid at large $\mathit{De}$ . Unlike a sphere, the asymmetry of the stagnation point configuration for a spheroid, sedimenting in an inclined orientation, implies that there will be an additional torque contribution associated with the strand. The finite $\mathit{De}$ motion of a slender spheroid/fibre in a simple shear flow is simpler in this sense since, on one hand, the time-dependent orientation implies there is not enough time to build up a birefringent strand and, on the other hand, the flow relative to the fibre being an extensional flow, the stagnation points are located at the fibre midplane, implying the absence of a birefringent-strand-induced torque. Slender-body theory in this case does yield the dominant correction to the angular velocity (Harlen & Koch Reference Harlen and Koch1993). Now, the location of the stagnation point on a sedimenting spheroid must, in principle, be derived as part of the solution, but simulations of suspensions of finitely extensible nonlinear elastic (FENE) dumbbells past sedimenting particles indicate a largely unmodified velocity field in the dilute limit (Binous & Phillips Reference Binous and Phillips1993); thus, some insight may be gained by examining the stagnation points of the Stokes velocity field. For the sedimentation angle ${\it\alpha}$ bounded away from 0 and ${\rm\pi}/2$ , the symmetrically located stagnation points are given by ${\it\eta}_{s}=\pm [1-4({\it\xi}_{0}-1)\tan ^{2}{\it\alpha}]$ . For ${\it\xi}_{0}\rightarrow 1$ , the rear stagnation point is located at an asymptotically small distance of $O(b/{\it\kappa})$ from the edge ( ${\it\eta}=1$ ), and the flow close to this stagnation point on length scales smaller than $O(b/{\it\kappa})$ is a nonlinear extensional flow with $u_{{\it\xi}}\sim ({\it\xi}-{\it\xi}_{0})^{2}$ , similar to that behind a sphere on length scales smaller than its radius. Interestingly, this suggests that the largest accessible strain rate for a slender spheroid is $O(U{\it\kappa}/b)$ in the vicinity of the stagnation point (and not $O(U/b)$ ), and thus, even when $\mathit{De}\ll O(1/{\it\kappa})$ , the second-order fluid regime described in earlier sections, there would a localized region of extended polymers in the vicinity of the rear stagnation point.

If the location of the stagnation points above is not significantly changed when $\mathit{De}\sim O(1)$ , the birefringent-strand-induced torque may be estimated as the force exerted by the strand times a moment arm of $O(L)$ . This force is $O({\bf\sigma}_{str}{\it\delta}^{2})$ , where ${\it\delta}$ is the strand thickness and ${\bf\sigma}_{str}$ is the Newtonian stress within the strand of highly stretched polymers. Assuming the stress response of the strand to be analogous to a semi-dilute solution of aligned (inextensible) fibres of the same length as the fully stretched polymers (Rallision & Hinch Reference Rallision and Hinch1988), we have ${\it\sigma}\sim {\it\mu}_{p}E$ , with ${\it\mu}_{p}$ being the enhanced strand viscosity in the semi-dilute regime and $E$ the rate of extension in the vicinity of the stagnation point. Based on the kinematics of the nonlinear extensional flow behind a sphere (Harlen Reference Harlen1990), one finds ${\it\delta}\sim O(b/{\it\kappa}\sqrt{E_{x}})$ , $E_{x}$ being the ratio of the fully stretched length of the polymer to its equilibrium extension ( $R_{g}$ ), and $b/{\it\kappa}$ being the smallest significant geometrical length scale. The rate of extension $E$ may be written as $O(U{\it\kappa}/b)f({\it\lambda})$ , following Harlen et al. (Reference Harlen, Rallison and Chilcott1990), where the single parameter ${\it\lambda}=(({\it\mu}_{p}{\it\delta}^{2})/{\it\mu})\log [b/({\it\kappa}{\it\delta})]$ characterizes the modified velocity field outside the birefringent strand. It may be shown, based on the analysis of Harlen et al. (Reference Harlen, Rallison and Chilcott1990), that $E$ varies from $O(U{\it\kappa}/b)$ for ${\it\lambda}=0$ (no strand) to $O(U{\it\kappa}/b){\it\lambda}^{-1/2}$ for ${\it\lambda}\gg 1$ . The extreme thinness of the strand in the present case implies that $E\sim O(U{\it\kappa}/b)$ in the present context. Combining these estimates, the torque contribution may be written as $\boldsymbol{L}_{str}\sim FL\sim ({\it\mu}_{p}{\it\delta}^{2})(U{\it\kappa}/b)L\sim ({\it\mu}_{p}/E_{x})(Ub/{\it\kappa})L$ . This torque evidently stabilizes a longside-on orientation since the birefringent strand at the edge of the slender spheroid is expected to ‘pull’ it onto a vertical orientation.

It is important to note that the birefringent strand above may not be the only region of highly stretched polymers. Most polymer molecules sufficiently close to the surface of the slender spheroid (at distances much smaller than $O(b)$ ) would be highly stretched when they reach the rear, having travelled only a distance of $O(b)$ along the spheroid surface. These stretched polymers would, however, move away from the spheroid before they traverse a distance of $O(L)$ , along the surface of the spheroid, needed to reach the rear stagnation point at the edge. It is only for distances from the surface that are much smaller than $O(b/{\it\kappa})$ that the normal velocity is small enough for the molecules to reach the rear stagnation point before moving away from the surface. Thus, most polymer molecules close to the spheroid go on to form a birefringent ‘sheet’ behind it, with a thickness of $O(b/\sqrt{E_{x}})$ , having undergone stretching by the nonlinear planar extensional flow in the transverse plane of the spheroid axis. The small fraction of polymer molecules that pass close to the rear stagnation point and are stretched by the three-dimensional nonlinear extensional flow in this region go on to form the much thinner birefringent strand above, with a transverse dimension of $O(b/\sqrt{E_{x}})$ . The birefringent sheet, being present all along the rear of the spheroid, does not lead to a torque and, therefore, does not enter the scaling arguments for the torque contributions above.

We are now in a position to compare the relative magnitudes of the torque due to the strand and that from the outer region. This ratio may be written as $\boldsymbol{L}_{outer}/\boldsymbol{L}_{str}\sim GL^{3}/(\ln {\it\kappa})^{2}[({\it\mu}_{p}/E_{x})(Ub/{\it\kappa})L]^{-1}$ . Using the known scale for the fibre suspension viscosity in the semi-dilute regime ${\it\mu}_{p}\sim {\it\mu}(nR_{g}^{3})E_{x}^{3}/\text{ln}\,E_{x}$ (Batchelor Reference Batchelor1971), this ratio may be rewritten in terms of the polymer relaxation time, ${\it\tau}={\it\mu}R_{g}^{3}/kT$ , as $\mathit{De}^{-1}[\ln E_{x}/(\ln {\it\kappa})^{2}]~(L/b)^{2}E_{x}^{-2}$ ; not withstanding logarithmic factors, the condition for the outer torque to be dominant turns out to be $\mathit{De}\ll (L/b/E_{x})^{2}$ – the aspect ratio of the spheroid must be greater than the ‘aspect ratio’ of the polymer (the aspect ratio here being defined as the ratio of the full extended length to the coil dimension). Finally, it must be mentioned that the polymer stress field that contributes to the outer torque will be set up in a time of $O(L/U)$ , but the formation of the birefringent strand in the vicinity of the stagnation point might take a significantly longer time.