NOMENCLATURE
- g
gravity
- H
magnetic field
- l
thickness of the permanent magnet or ferrofluid
- M
magnetisation
- p
pressure
- Q
volumetric flow
- S
surface area of the wing
- T
temperature
- u
velocity of air
- v
velocity of ferrofluid
- W
weight
- x
length coordinate
- z
normal coordinate
Greek symbol
$\delta$thickness of the ferrofluid film
- $\eta$
dynamic viscosity
- $\mu$
magnetic momentum
- $\rho$
density
- $\sigma$
surface tension
Subscripts
- a
air
- f
ferrofluid
- i
air–ferrofluid interface
- m
magnet
- o
interface
1.0 INTRODUCTION
Almost immediately after Prandtl’s boundary-layer theory was proposed, continuous research began to identify methods to mitigate its negative effects. Although a multitude of approaches have been suggested to tackle this problem, all of them in one way or another try to prevent, or at least delay, the detachment of the boundary layer from the wall (e.g. Refs. [Reference Modi1–Reference Gilarranz, Traub and Rediniotis10]). Suction and blowing, turbulence promoters, vortex generators, and moving walls are just some examples of the various techniques that can be found in literature. The aim of suction and blowing is to remove low-energy air, through suction slots or by blowing high-energy air through backward-directed slots, respectively. On the other hand, turbulence promoters and vortex generators attempt to control the flow separation on symmetric aerofoils by creating spots of high turbulence by using various elements such as baffles or wall roughness elements. Finally, one has moving solid walls, which are intended to remove the zero-slip condition as well as to inject momentum into the boundary layer.
The object of this work is to analyse a novel approach for lift enhancement. According to this concept, this goal is attained by preventing the growth of the air boundary layer through eliminating the zero-slip condition between the surface and the air stream, and more exactly by elimination of the air–wing direct contact. The concept would simulate all effects of a moving wall, leading to the appearance of a slip velocity at the gas–fluid interface, including the injection of momentum into the boundary layer, with one exception: there is no moving wall but instead a ferrofluid thin film pumped parallel and attached to the wall by a magnetic field. For this work, it suffices to know that a ferrofluid or ferromagnetic fluid is nothing more than a colloidal liquid that becomes strongly magnetised in the presence of a magnetic field due to the presence of suspended nanoscale ferromagnetic, or ferrimagnetic particles. For the sake of illustration, Fig. 1 shows a schematic of the concept investigated in this work. However, caution is called, Fig. 1 is not intended to give a definitive design, nor should it be misconstrued as an attempt to produce a definitive optimised application of the concept, as real applications could depart largely from this sketch. Nonetheless, it provides important guidance on the core idea proposed in this work.
Figure 1. Sketch of the core idea. (a): in normal conditions, the zero-slip condition leads to the formation of a boundary layer, limiting the lift. (b): because of the injection of a ferrofluid thin film attached to the wall by a magnetic field, the zero-slip condition disappears, which translates into lift enhancement.
Figure 2. Physical model of the region of the aerofoil covered by the ferrofluid film.
2.0 MATERIALS AND METHODS
2.1 The ferromagnetic thin film layer
To begin, consider Fig. 2, in which a ferrofluid thin film (with a thickness of a few millimetres or less) is attached to the wall of an aerofoil by a magnetic gradient field normal to the surface which is generated by, say, an array of permanent magnets attached below the wing. In addition, the ferrofluid is pumped tangentially to the surface of the wall. We choose the normal to the surface as the z-axis and the x-axis in the direction of motion of the fluid and fixing the origin of coordinates at the wall. Considering that the velocity depends only on the normal axis z, and that the considered film is a few millimetres thick or less, the convective term in the Navier–Stokes equations can be neglected in comparison with the viscous term, leading to (Reference Rosenweig11)
\begin{equation} \frac{1}{\eta_f}\frac{\partial p}{\partial x}=\frac{\partial^2 v_x}{\partial z^2}, \end{equation}
and
\begin{equation} \frac{\partial p}{\partial z}= \rho_fg+\mu_fM_f\frac{\partial H}{\partial z}, \end{equation}
where p is pressure; 
$v_x$ is the ferrofluid velocity parallel to the wall; 
$\eta_f$, 
$\rho_f$, 
$M_f$, and 
$\mu_f$ are the dynamic viscosity, density, magnetisation, and magnetic momentum of the ferrofluid, respectively; g is gravity; 
$\frac{\partial H}{\partial z}$ is the uniform normal magnetic gradient. After integrating Equation (2) one obtains
\begin{equation}p= p_1(x)+ \rho_fgz+\mu_fM_f\frac{\partial H}{\partial z}z \end{equation}
Now, we define the boundary conditions for Equation (1). First, on the solid boundary, the ferrofluid velocity vanishes, which give the first boundary condition as
\begin{equation}v_x(z=0)=0 \end{equation}
Second, at the air–ferrofluid interface (
$z=\delta$), the components of the viscous stress tensor are continuous(Reference Landau and Lifshitz12), thus
\begin{equation}\eta_f\frac{\partial v_x}{\partial z}\bigg|_{z=\delta}+\eta_a\frac{\partial u_x}{\partial z}\bigg|_{z=\delta}=0, \end{equation}
where u and 
$\eta_a$ are the air velocity and dynamic viscosity, respectively. For a very thin film (
$\delta\rightarrow 0$) and considering that 
$\eta_f\gg \eta_a$, one can assume that 
$\eta_f\frac{\partial v_x}{\partial z}\gg \eta_a\frac{\partial u_x}{\partial z} $, in which case Equation (5) simplifies to
\begin{equation}\eta_f\frac{\partial v_x}{\partial z}\approx 0 \end{equation}
Finally, the discharge or volumetric flow is given by
\begin{equation}Q= l\int_0^{\delta}v_x(z)\mathrm{d}z \end{equation}
Taking into account the set of boundary conditions, the solution of Equation (1) yields
\begin{equation}v_x=\frac{3Q}{2l\delta^2}\left[2z-\frac{z^2}{\delta} \right] \end{equation}
The above equation is familiar from Couette flow with a pressure gradient, as expected(Reference Bateman13). On the other hand, the interfacial velocity 
$v_x(z=\delta)=v_i$ is
\begin{equation}v_i=\frac{3Q}{2l\delta}, \end{equation}
and likewise the mean velocity 
$\bar{v}_x$ is
\begin{equation}\bar{v}_x=\frac{Q}{\delta l}, \end{equation}
which considering Equation (9) becomes
\begin{equation}\bar{v}_x=\frac{2v_i}{3} \end{equation}
2.2 Film stability
From Equation (8), one may be tempted to think that, by increasing the volumetric flow indefinitely, i.e. the pumping power, or by decreasing the thickness of the film, it could be possible to increase the interface velocity as desired. However, this is not the case. Actually, the maximum interface velocity is limited by Kelvin–Helmholtz instabilities which arise from the relative motion between the ferrofluid and the air stream. The criterion for instability in the magnetic Kelvin–Helmholtz problem when the ferrofluid film is under the action of a magnetic field is given by,(Reference Rosenweig11)
\begin{equation}(\bar{v}_x-u_o)^2> \frac{\rho_f+\rho_a}{\rho_f\rho_a} \left[2\left[g(\rho_f-\rho_a)\sigma \right]^{\frac{1}{2}}+\frac{(\mu_a-\mu_f)^2H_x^2}{\mu_a+\mu_f} \right]\!, \end{equation}
where 
$\bar{v}_x$ is the mean ferrofluid velocity, 
$u_o$ is the air free stream velocity; 
$\rho_f$ and 
$\rho_a$ are the density of the ferrofluid and the air, respectively; g is gravity; 
$\sigma$ is the surface tension, 
$\mu_a$ and 
$\mu_f$ are the magnetic permeability of the air and the ferrofluid, respectively. In the above equation, the uniform magnetic field 
$H_x$ is the magnetic field collinear with the direction of wave propagation (the stream direction), where it is known that a tangential applied magnetic field in the direction normal to the direction of wave propagation offers no stabilisation(Reference Rosenweig11).
Because 
$\rho_a\ll \rho_f$ and 
$\mu_a\ll\mu_f$, and taking into account Equations (11) and (12) becomes
\begin{equation}\frac{v_i}{u_o}> \frac{3}{2}\left[1+\frac{1}{u_o}\left[\frac{2(g\rho_f\sigma)^{\frac{1}{2}}+\mu_fH_x^2}{\rho_a} \right]^{\frac{1}{2}} \right] \end{equation}
However, although a uniform magnetic field normal to the direction of wave propagation provides null stabilisation, a magnetic gradient in this direction causes a normal body force per unit volume of
\begin{equation}F_m= \mu_0M_f\nabla_zH, \end{equation}
where 
$\mu_0$ is the permeability of free space, 
$M_f$ is the magnetisation of the ferrofluid and 
$\nabla_z H$ is the normal magnetic field gradient. Thus, because both gravity and the magnetic force are body forces, an effective acceleration 
$g_e$ may be defined as
\begin{equation}g_e = g+ \frac{\mu_0M\nabla_z H}{\rho_f} \end{equation}
Therefore, if only a gradient magnetic field is acting on the ferrofluid plus gravity and surface tension, the stability criterion in Equation (13) becomes
\begin{equation}\frac{v_i}{u_o}> \frac{3}{2}\left[1+\frac{1}{u_o}\left[ \frac{2\sigma^{\frac{1}{2}}(g\rho_f+\mu_0M_f\nabla_z H)^{\frac{1}{2}}}{\rho_a} \right]^{\frac{1}{2}} \right] \end{equation}
3.0 DISCUSSION
To obtain some idea of the shape of the curves predicted by Equations (13) and (16), we assume some typical values of the parameters for a ferrofluid: 
$\sigma= 70\times 10^{-3}\text{N/m}$; 
$g=9.8\text{m/s}^2$; 
$\rho_f=1.2\times 10^3\text{kg/m}^3$; 
$\rho_a= 1.0\text{kg/m}^3$; 
$M_f= 4.5\times 10^5\text{A/m}$, which corresponds to a realisable magnetic field of around 0.5T obtained from a typical hand-held permanent magnet; 
$\mu_0=4\pi\times 10^{-7}\text{H/m}$; 
$\mu_f=8\mu_0$. The resulting curves are shown in Figs. 3 and 4 for Equations (13) and (16), respectively, and considering practical achievable values for the magnetic field as well as the magnetic gradient.
Figure 3. Stability curve predicted by Equation (13) as a function of the air free stream.
Figure 4. Stability curve predicted by Equation (16) as a function of the air free stream.
3.1 Experimental measurements
To obtain a preliminary idea of the lift enhancement when using ferrofluid thin films, full experiments on specific aerofoil designs are not required. Instead, because from an aerodynamic point of view there is no substantial difference if the interfacial velocity is generated by a thin film or, say, a moving solid surface, the thin film can be regarded to some extent as a moving solid surface (both being introduced into the differential Navier–Stokes equations as a simple boundary condition), thus it is possible to take advantage of experimental measurements available in literature on the lift enhancement and flow separation from moving solid surfaces for several aerofoil designs. Hence, after obtaining experimental measurements of the interfacial velocity from the ferrofluid film, the lift and angle-of-attack can be inferred from the available literature on moving surfaces with the same interfacial velocity. With this goal, a set of experiments were performed to find the attainable interfacial velocity for the concept of the ferrofluid film.
The set-up consisted in a square polycarbonate cavity with 
$\delta=l$ and 170mm long, filled with ferrofluid. Below the cavity, a train of hand-held neodymium permanent magnets were located. The ferrofluid was pumped through the cavity by using a peristaltic pump, electronically regulated by the number of revolutions per minute. Several cavities were used, i.e. different 
$\delta$ values, but always keeping the same length. The ferrofluid employed was 
$\text{Mn}_{0.5}\text{Zn}{0.5}\text{Fe}_2\textit{O}_4$ in water at room temperature 
$T=298\text{K}$. The air stream was propelled by a blower parallel to the cavity. The interfacial velocity 
$v_i$ was measured by using a Fluke 922 airflow meter. The cavity was positioned on a simple laboratory adjustable lifting-rotating platform to allow measurements at different angles of inclination. The magnetic field from the array of magnets was measured as 0.12T at 3mm from its surface using a FW BELL 5170 Gauss/Tesla meter. Figures 5 and 6 show a sketch and the real experimental set-up, respectively.
Figure 5. Sketch of the experimental set-up.
Figure 6. Close-up of experimental set-up.
4.0 RESULTS AND CONCLUSIONS
The resulting experimental curves are shown in Figs. 7, 8, and 9. Figure 7 shows the ratio 
$\frac{v_i}{u_o}$ as a function of the volumetric flow and for several thicknesses of the cavity. Figure 8 shows the ratio 
$\frac{v_i}{u_o}$ as a function of the free stream air, and Fig. 9 shows the effect of the angle of inclination (by taking measurements at different inclination angles of the platform from totally horizontal at 
$0^\circ$ to totally vertical 
$90^\circ$; see Fig. 5) of the cavity on the ratio 
$\frac{v_i}{u_o}$ and for several thicknesses of the cavity.
Figure 7. 
$\frac{v_i}{u_o}$ as a function of the volumetric flow with u o
$=\text{0.5}\text{m/s}$.
Figure 8. 
$\frac{v_i}{u_o}$ as a function of the free stream air.
Figure 9. 
$\frac{v_i}{u_o}$ as a function of the angle of inclination and for several thicknesses of the cavity.
Finally, Figs. 10 and 11 show shows the predicted lift enhancement and the increase of the angle-of-attack for a NACA0015 derived from using the experimental interfacial velocity obtained before and applied to the plots of the experimental data on moving walls reported by Ref. [Reference Boukenkoul, Li, Chen and Zhang14], i.e. by replacing the velocity of the wall by the interfacial velocity of the film.
Figure 10. Lift coefficient as a function of 
$\frac{v_i}{u_o}$.
Figure 11. Angle-of-attack as a function of 
$\frac{v_i}{u_o}$.
It is easy to see the attractiveness of the proposed concept by comparing the lift coefficient enhancement as a function of 
$\frac{v_i}{u_o}$ in Fig. 10, and the allowable 
$\frac{v_i}{u_o}$ as a function of the air stream velocity from the curves of ferrohydrodynamic stability in Figs. 3 and 4. Thus, as an illustrative example, an air stream velocity of around 100m/s will allow for a thin film 
$\frac{v_i}{u_o}$ ratio of around 
$\approx3\text{m/s}$ before the Kelvin–Helmholtz effect will detach the film. With this ratio, the lift coefficient enhancement could be around 1.8, which is a considerable figure of merit. Additional research and development is required to explore the possibilities of using such ferrofluid thin films.
4.1 The weight penalty
There is no doubt that one of the major factors to be considered in an aircraft project is the weight. In the application of the proposed method, one of the main concerns could be related to the weight of the magnets needed to attach the ferrofluid film. However, because the ferrofluid film is very thin (a few millimetres or less), it is expected that the extra weight from the train of magnets attached in the wing of the aircraft will not result in additional concerns. To assess the extra weight per surface area of the wing caused by the magnets, we proceed as follows: First, we determine the external magnetic field needed to attain the magnetic saturation of the ferrofluid. The measured magnetisation of the ferrofluid as a function of the external magnetic field is shown in Fig. 12. It is clear from this figure that, with an external magnetic field of around 100kA/m, we attain 10% of saturation (
${\approx}32\text{kA/m}$), which translates into 
${\approx}3.2\text{kA/m}$, which for a film of a few millimetres will result in gradients 
$> 10^5\text{A/m}^2$, being sufficient to guarantee strong film attachment at the wall (Fig. 3).
Figure 12. Variation of magnetisation with the external applied magnetic field.
Second, we need to assess the amount of magnets needed to generate this magnetic field at a certain distance from the wing. To obtain an estimate of this, let us assume a wing with a certain thickness and covered by a film of ferrofluid in which there are a train of magnets attached below. The minimum skin thickness of a wing depends on several factors, for example, the load distribution, etc., and we can find commercial aircraft types, such as the 727, with a minimum skin thickness as low as 0.1mm or as high as 1.27mm for DC-8 and DC-9 aircraft. Therefore, to be on the safe and most conservative side, let us assume a thickness of around 1.27mm. On the other hand, let us consider a ferrofluid with a thickness of, say, 2mm. Therefore, the magnets will be at a distance of around 1.27mm + 2mm = 3.27mm. Then, we want to measure the magnetic field from a flat arrangement of magnets with a certain total thickness l at a distance of 3.27mm away from the surface. Finally, if it is desired to cover a surface area of the wing 
$S_w$ with a ferrofluid film, then the volume of magnet 
$V_m$ needed will be 
$V_m=S_wl$, and if the magnet has a density of 
$\rho_m$, this will result in a weight of 
$W_m=\rho_mS_wl$, with a weight of magnet per unit surface area of the wing of 
$\frac{W_m}{S_w}=\rho_ml$. The magnetic field as a function of the thickness of neodymium magnets, 
$\rho_m=7,612\text{kg/m}^3$, was measured, and the resulting curve is shown in Fig. 13. It is seem that, to generate the magnetic field of around 32kA/m, a density of around 
$\frac{W_m}{S_w}=2.5\text{kg/m}^2$ is required. The penalty caused by the excess weight is small compared with the lift enhancement, and will become negligible as the total weight of the aircraft increases. To see this, consider, for example, a DC-8 aircraft with a typical weight of around 130,000kg. With a lift enhancement of around 1.5, this will translate into an additional force of around 65,000kgf. However, with a wingspan of around 44m and a width of around 1m, the total surface area of the wing will be around 
$100\text{m}^2$, which with the aforementioned calculated figure of 
$\frac{W_m}{S_w}=2.5\text{kg/m}^2$ will result in an extra weight of 2,500kg, i.e. less than 0.5%. Even assuming the most pessimistic figure, the lift gain will be much greater. It must also be noted that the above rough calculations assume covering the entire wings with a ferrofluid layer, which of course will not be the case. In fact, the ferrofluid film concept will be applied in specific regions, for example at the point where the detachment of the boundary layer occurs, in order to delay it.
Figure 13. Magnetic field generated by a flat magnet at a distance of 
$\text{3.2}\text{mm}$ from the surface and as a function of the thickness of the magnet (upper abscissa), and the weight of the magnet per unit surface area of the wing (bottom abscissa).
ACKNOWLEDGEMENTS
This research was supported by the Spanish Ministry of Economy and Competitiveness under Ramon y Cajal fellowship grant RYC-2013-13459.
DECLARATION OF INTERESTS
The authors report no conflicts of interest.
