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HYDROMECHANICS
N. P. DZHORBENADZE and D. V. SHARIKADZE
ON THE FLOW OF A CONDUCTING VISCOUS FLUID BETWEEN TWO POROUS PLANES
(Presented by Academician N. N. Bogolyubov, 15 III 1960)
In the present paper we consider the problem of the flow of a viscous incompressible electrically conducting fluid in the space between two parallel porous planes, when a homogeneous constant external magnetic field \(H_0\) is applied perpendicular to the planes. The flow between porous planes without the influence of a magnetic field has been studied in works \((^{1-3})\).
Let the principal flow of the fluid take place parallel to the planes. The fluid simultaneously enters through one plane and leaves through the other, so that the amount of fluid entering and leaving through the pores remains unchanged along the flow.
We direct the axis \(Ox\) along the planes, and the axis \(Oz\) perpendicular to them. If the corresponding velocity components are denoted by \(v_x\) and \(v_z\), then in the case under consideration one may assume that they depend only on \(z\) and \(t\). Then from the continuity equation \(\operatorname{div}\mathbf v = 0\) it immediately follows that \(v_z = v_0(t)\), where \(v_0(t)\) is the value of the velocity at the boundary of the plane. The same also applies to the longitudinal field \(H_x\), arising as a result of the motion of the fluid.
The fundamental equations of magnetohydrodynamics \((^{4,6})\) in the case under consideration have exact solutions:
\[ v_x = v(z,t), \qquad v_z = v_0(t), \qquad H_x = H(z,t), \qquad H_z = H_0 = \mathrm{const}, \]
where \(v\) and \(H\) satisfy the equations
\[ \frac{\partial v}{\partial t} - \nu \frac{\partial^2 v}{\partial z^2} = \frac{H_0}{4\pi \rho}\frac{\partial H}{\partial z} - v_0(t)\frac{\partial v}{\partial z} + A(t), \]
\[ \frac{\partial H}{\partial t} - \lambda \frac{\partial^2 H}{\partial z^2} = H_0 \frac{\partial v}{\partial z} - v_0(t)\frac{\partial H}{\partial z} \tag{1} \]
where \(A(t)=\dfrac{1}{\rho}\dfrac{\partial p}{\partial x}\) is prescribed; \(\lambda\) is the coefficient of magnetic viscosity; \(\nu\) is the kinematic coefficient of viscosity; \(p\) is the ordinary pressure; \(\rho\) is the density. In this case the functions \(v\) and \(H\) satisfy the boundary conditions
\[ v(z,0)=v^0(z), \qquad H(z,0)=H^0(z), \]
\[ \left. v(z,t)\right|_{z=\pm a}=0, \qquad \left. H(z,t)\right|_{z=\pm a}=0, \tag{2} \]
where \(2a\) is the distance between the porous planes, and the plane \(z=0\) is located midway between them. The condition for \(H\) follows from the invariability of the external magnetic field \(H_0\), while the tangential component of \(\mathbf H\) at the boundary must be continuous.
The solution of this problem reduces to a system of integro-differential equations \((^{2,5})\)
\[ v(z,t)=\Phi(z,t)+\int_0^t d\tau\int_{-a}^{a} \left(\frac{H_0}{4\pi\rho}\frac{\partial H}{\partial \eta} -v_0(\tau)\frac{\partial v}{\partial \eta}\right) G(z,\eta,t-\tau)\,d\eta, \]
\[ H(z,t)=F(z,t)+\int_0^t d\tau\int_{-a}^{a} \left(H_0\frac{\partial v}{\partial \eta} -v_0(\tau)\frac{\partial H}{\partial \eta}\right) G(z,\eta,t-\tau)\,d\eta, \tag{3} \]
where \(\Phi(z,t)\) and \(F(z,t)\) satisfy the heat-conduction equations:
\[ \frac{\partial \Phi}{\partial t}-\nu\frac{\partial^2\Phi}{\partial z^2}=A(t), \qquad \frac{\partial F}{\partial t}-\lambda\frac{\partial^2F}{\partial z^2}=0 \tag{4} \]
and the boundary conditions (2), while \(G(z,\eta,t)\) is the Green’s function of the heat-conduction equation
\[ G(z,\eta,t)=-\frac{1}{2\sqrt{\pi nt}} \exp\left[-\frac{(z-\eta)^2}{4nt}\right]+g(z,\eta,t), \]
where \(g(z,\eta,t)\) is the solution of the heat-conduction equation
\[ \frac{\partial g}{\partial t}-n\frac{\partial^2 g}{\partial z^2}=0, \qquad n=\lambda,\nu, \]
which vanishes at the initial instant and satisfies the boundary conditions
\[ g(a,\eta,t)=\frac{1}{2\sqrt{\pi nt}} \exp\left[-\frac{(a-\eta)^2}{4nt}\right], \]
\[ g(-a,\eta,t)=\frac{1}{2\sqrt{\pi nt}} \exp\left[-\frac{(-a-\eta)^2}{4nt}\right]. \]
Hence it is clear that the determination of \(F,\Phi,g\) is reduced to a system of Volterra integral equations of the second kind with a regular kernel.
If instead of (3) we consider the system with parameter \(\delta\) and differentiate it once with respect to \(z\), then for determining \(\partial v/\partial z\) and \(\partial H/\partial z\) we obtain
\[ \frac{\partial v}{\partial z} = \frac{\partial \Phi}{\partial z} +\delta\int_0^t d\tau\int_{-a}^{a} \left[ \frac{H_0}{4\pi\rho}\frac{\partial H}{\partial \eta} -v_0(\tau)\frac{\partial v}{\partial \eta} \right] \frac{\partial G}{\partial z}\,d\eta, \]
\[ \frac{\partial H}{\partial z} = \frac{\partial F}{\partial z} +\delta\int_0^t d\tau\int_{-a}^{a} \left[ H_0\frac{\partial v}{\partial \eta} -v_0(\tau)\frac{\partial H}{\partial \eta} \right] \frac{\partial G}{\partial z}\,d\eta. \tag{5} \]
We shall seek the functions \(\partial v/\partial z\) and \(\partial H/\partial z\) in the form of a series
\[ \frac{\partial v}{\partial z}=\sum_m^\infty \delta^m W_m, \qquad \frac{\partial H}{\partial z}=\sum_m^\infty \delta^m U_m. \tag{6} \]
To determine the terms of the series we obtain the recurrence formulas
\[ W_0=\frac{\partial\Phi}{\partial z}, \qquad U_0=\frac{\partial F}{\partial z}, \]
\[ W_m=\int_0^t d\tau\int_{-a}^{a} \left[ \frac{H_0}{4\pi\rho}U_{m-1} -v_0(\tau)W_{m-1} \right] \frac{\partial G}{\partial z}\,d\eta, \]
\[ U_m=\int_0^t d\tau\int_{-a}^{a} \left[ H_0 W_{m-1} -v_0(\tau)U_{m-1} \right] \frac{\partial G}{\partial z}\,d\eta. \]
It is easy to show that the inequalities
\[ |U_m|,\ |W_m|<M(2NK)^m t^{m/2}\frac{\Gamma^m(1/2)}{\Gamma(m/2+1)}, \tag{7} \]
hold, where \(M, N, K\) are constants satisfying the conditions
\[ \left|\frac{\partial F}{\partial z}\right|,\ \left|\frac{\partial \Phi}{\partial z}\right|<M,\qquad |H_0|,\ |\upsilon(t)|<N,\qquad \int_{-a}^{a}\sqrt{t-\tau}\left|\frac{\partial G}{\partial z}\right|\,d\eta<K, \]
and \(\Gamma\) is Euler’s gamma-function. From (7) it is clear that the series (6) converge absolutely and uniformly when \(t<\infty\). For \(\delta=1\), the series (6) gives the solution of our problem.
If \(\upsilon_0(t)=0\), then from (3) we obtain the solution of the nonstationary problem of flow in the space between two plane-parallel solid planes
\[ \upsilon(z,t)=\Phi(z,t)+\int_0^t d\tau\int_{-a}^{a} \frac{H_0}{4\pi\rho}\frac{\partial H}{\partial \eta} G(z,\eta,t-\tau)\,d\eta, \]
\[ H(z,t)=F(z,t)+\int_0^t d\tau\int_{-a}^{a} H_0\frac{\partial \upsilon}{\partial \eta} G(z,\eta,t-\tau)\,d\eta. \]
The solution of the stationary problem could have been obtained from the corresponding nonstationary problem considered above by passing to the limit \(t\to\infty\), but this limiting transition involves considerable mathematical difficulties; therefore we shall consider the stationary problem directly.
In this case the magnetohydrodynamic parameters \(\upsilon, H, p\) do not depend explicitly on time, and (1), (2) take the form:
\[ \nu\frac{d^2\upsilon}{dz^2}+\frac{H_0}{4\pi\rho}\frac{dH}{dz} -\upsilon_0\frac{d\upsilon}{dz}=0, \]
\[ \lambda\frac{d^2H}{dz^2}-\upsilon_0\frac{dH}{dz} +H_0\frac{d\upsilon}{dz}=0; \tag{8} \]
\[ \upsilon(z)\big|_{z=\pm a}=0,\qquad H(z)\big|_{z=\pm a}=0. \tag{9} \]
The solution of equation (8) under the conditions (9) can be written in the form
\[ \upsilon=\frac{A\upsilon_0}{\beta}z+ \frac{A}{\nu(k_2-k_1)} \left\{ B_2\left(\operatorname{cth} k_1a- \frac{\operatorname{ch} k_1z+\operatorname{sh} k_1z}{\operatorname{sh} k_1a}\right) -\right. \]
\[ \left. -\,B_1\left(\operatorname{cth} k_2a- \frac{\operatorname{ch} k_2z+\operatorname{sh} k_2z}{\operatorname{sh} k_2a}\right) \right\}, \]
\[ H=-\frac{4\pi A}{H_0} \left\{ \left(\frac{\upsilon_0^2}{\beta}+1\right)z+ \frac{a}{\nu(k_2-k_1)} \left[ B_2C_1\left(\operatorname{cth} k_1a- \frac{\operatorname{ch} k_1z+\operatorname{sh} k_1z}{\operatorname{sh} k_1a}\right) -\right.\right. \]
\[ \left.\left. -\,B_1C_2\left(\operatorname{cth} k_2a- \frac{\operatorname{ch} k_2z+\operatorname{sh} k_2z}{\operatorname{sh} k_2a}\right) \right] \right\}, \]
where
\[ C_1=(\upsilon_0-\nu k_1),\qquad C_2=(\upsilon_0-\nu k_2),\qquad \beta=\frac{H_0^2}{4\pi\rho}-\upsilon_0^2, \]
\[ B_1=-\frac{\upsilon_0a}{\beta}C_1+ \left(\frac{\upsilon_0^2}{\beta}+1\right),\qquad B_2=-\frac{\upsilon_0a}{\beta}C_2+ \left(\frac{\upsilon_0^2}{\beta}+1\right), \]
\[ k_{1,2}= \frac{\upsilon_0(\lambda+\nu)\mp \sqrt{\upsilon_0^2(\lambda-\nu)^2+\frac{\lambda\nu}{\pi\rho}H_0^2}} {2\lambda\nu}. \]
If one considers steady flow between two solid planes, then \(v_0=0\), and we obtain the known solution found by Hartmann \((^{6,7})\):
\[ v=\bar v_0\,\frac{\operatorname{ch} ka-\operatorname{ch} kz}{\operatorname{ch} ka-1}, \qquad H=-\,\frac{\bar v_0\,4\pi\sqrt{\sigma\eta}}{c}\, \frac{\dfrac{z}{a}\operatorname{sh} ka-\operatorname{sh} kz}{\operatorname{ch} ka-1}, \]
where
\[ k=k_1=-k_2,\qquad \bar v_0=\frac{A}{\nu k}\,\frac{\operatorname{sh} ka}{\operatorname{ch} ka-1}. \]
For \(v_0=0\) and
\[ ak=\frac{aH_0}{c}\sqrt{\frac{\sigma}{\eta}}\ll 1 \]
one obtains the value
\[ v=\bar v_0\left(1-\frac{z^2}{a^2}\right), \]
which coincides with the result of ordinary hydrodynamics \((^8)\).
In conclusion, we express our deep gratitude to Prof. K. P. Stanyukovich and Prof. D. E. Dolidze for discussion of the results and valuable suggestions.
Tbilisi Mathematical Institute
named after A. M. Razmadze
Academy of Sciences of the Georgian SSR
Tbilisi State University
named after I. V. Stalin
Received
14 III 1960
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