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)}

(1) \begin{equation} \frac{1}{\eta_f}\frac{\partial p}{\partial x}=\frac{\partial^2 v_x}{\partial z^2}, \end{equation}

and

(2) \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

(3) \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

(4) \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

(5) \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

(6) \begin{equation}\eta_f\frac{\partial v_x}{\partial z}\approx 0 \end{equation}

Finally, the discharge or volumetric flow is given by

(7) \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

(8) \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

(9) \begin{equation}v_i=\frac{3Q}{2l\delta}, \end{equation}

and likewise the mean velocity $\bar{v}_x$ is

(10) \begin{equation}\bar{v}_x=\frac{Q}{\delta l}, \end{equation}

which considering Equation (9) becomes

(11) \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)}

(12) \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

(13) \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

(14) \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

(15) \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

(16) \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}