КОЛЛОИДНЫЙ ЖУРНАЛ, 2013, том 75, № 6, с. 703-709
УДК 532.546
CELL MODEL FOR HYDROMAGNETIC AXIAL FLOW OVER A CYLINDER. PART I. TRANSVERSE MAGNETIC FIELD
© 2013 г. Sunil Datta1, Manju Agarwal1, Anatoly Filippov2, and Sergey Vasin2
department of Mathematics and Astronomy, Lucknow University Lucknow, 226007, India 2Department of Higher Mathematics, Gubkin Russian State University of Oil and Gas Leninsky Prospekt 65-1, Moscow, 119991, Russia Поступила в редакцию 18.04.2013 г.
Using cell model initiated by Happel, the filtration problem across a membrane composed of an aggregate of parallel circular cylinders subject to a uniform transverse magnetic field is studied. The system is simulated by a single cylinder enveloped by a concentric cylindrical enveloping surface with axial flow. The analysis leads to the evaluation of the permeability parameter. The results are then graphically presented and discussed. The effect of magnetic field is seen to increase the permeability.
DOI: 10.7868/S0023291213060049
1. INTRODUCTION
In the year of1959 Happel [1] considered the problem ofviscous flow relative to arrays of cylinders. Same year Kuwabara [2] investigated the forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow. Both employed Stokes equations for flow at small Reynolds number and laid down the foundation of the cell method which is now widely employed to tackle the difficult problem of slow viscous flow through a concentrated assemblage of particles. For our purpose it suffices to observe that in this method we isolate a particle and surround it by an enveloping cell and use appropriate boundary conditions at the cell surface to take care of the effect of other particles of the system; thus the problem is reduced to the flow past the particle enclosed in a cell. The method is admirably described in the book [3].
A porous medium consists of fluid flowing through a concentrated assemblage of a system of particles. Thus, cell model is an effective and simple way of evaluating permeability constant that is an important parameter in percolation and filtration problems. Using cell model, Vasin and Filippov [4] calculated the hy-drodynamic permeability for a system of solid spherical particles covered with a porous shell. Continuing their study, Vasin and Filippov [5] extended the results for the more general problem of determining the permeability of a complex porous media. In another paper [6] they investigated the flows in a concentrated media composed of rigid impenetrable cylinders covered with a porous layer; both transverse and longitudinal flows of filtering fluid were taken up. Amongst other works in this direction, we may mention the contributions of Kirsh [7, 8].
A crucial step in the use of cell method is the application of appropriate boundary conditions on the cell surface. Four variations of the boundary conditions have been advanced: Happel, Kuwabara, Kvashnin and Cunningham (or Mehta-Morse). For a discussion of these conditions the reader is referred to the book [3] or the cited papers [5, 6].
In 1937 Hartmann [9] studied the flow of a conducting fluid between parallel plates under transverse magnetic field. The flow is now known as the Hartmann flow. This led to the development on laboratory scale of branch of fluid dynamics viz. Magnetohydro-dynamics (MHD) that deals with the interaction between the flow of an electrically conducting fluid and magnetic field. One of the forces induced by magnetic field is the Lorentz force providing control over the fluid motion. Rossow paper [10] paved the way for using magnetic field to control the motion of electrically conducting fluids. Use of this has opened new directions in microfluidics for non-mechanical fluid pumping phenomena. Thus, there is growing use of hydromagnetic forces in application to systems involving small dimensions and our problem belongs to this category. Our aim here is to exploit the control offered by MHD forces in the filtration process for conducting fluids.
In this paper we propose to study the flow of an electrically conducting fluid through a membrane composed of an array of parallel circular non conducting cylinders. The flow is taken along the axes of the cylinders and the magnetic field in a fixed direction transverse to the flow. It may be remarked here that we have not taken the induced electric/magnetic field to be zero as is done by many authors; non-vanishing of the induced electric field is exhibited in the sequel. In
Fig. 1. Transverse view of the cell in non-dimensional form.
Hartmann's paper too this induced field is present. In Gold's [11] study of magnetohydrodynamic pipe flow, which has motivated us to take up this MHD problem, induced electromagnetic fields appear and are significant. Also, it is seen that the problem is not axial symmetric and quantities are functions of both radial distance r and polar angle. It should also be noted that in this problem there is only one component of velocity along the axes of the cylinders and then as in the concerned references [4] and [6] mentioned earlier the four cell flow boundary conditions are all identical. Electromagnetic boundary conditions are also needed. All the needed boundary conditions are specified in the Section 3. In another paper, Part II of this study of cell model for hydromagnetic axial flow over a cylinder, we shall present the corresponding problem wherein the impermeable inner cylinder has concentric covering of a porous shell and subject to a radial magnetic field.
2. BASIC EQUATIONS
The basic hydromagnetic field quantities are velocity u', pressure p\ electric field E', magnetic field B' and current density vector J'. The basic governing equations are presented below. The MHD Stokes equations
0 = -Vp 2 u' + J' X B', V • u' = 0.
(1) (2)
Here ^ is fluid viscosity and J' x B' is the electromagnetic body force called Lorentz force. The Faraday law
the Ampere law
Vx E' = 0,
Vx B' = ^ J',
(3)
(4)
where is magnetic permeability. The above two basic laws are coupled with subsidiary equations
v- E' = p'e/k , (5)
where p'e is total charge density and k dielectric constant and
V- B' = 0, (6)
It may be observed that in the above, displacement current has been neglected; there is no applied electric field and no free charges. The Ohm law in a moving media assumes the form
J' = a(E' + u' x B'), (7)
where a is electric conductivity. Also, Ampere's law provides
V- J' = 0. (8)
It will be convenient to non-dimensionalize the quantities as follows
b 1 , TT , uU
r = ar, z = az, c = - = -, u = Uu, p = £— p,
a y a (9)
B' = B0B, E' = UB0E, J' = aUB0J,
where a is the radius of the impermeable cylinder and b is the radius of the cell (Fig. 1), Uthe external stream velocity and B0 the applied magnetic field. Then, the set of equations (1) to (8) in non-dimensional form may be expressed as
0 = -Vp + V2u + M2J x B, (10)
V- u = 0. (11)
Here M2 = ctB02/p is square of Hartmann number.
Vx E = 0, (12)
Vx B = ReJ, (13)
where Rem = ^maaU is magnetic Reynolds number.
V- E = Pe, (14)
pe = peal(K-UB0) representing a dimensionless form of the total charge density and
V- B = 0, (15)
J = (E + u x B), (16)
V- J = 0. (17)
Relationship of Eqs (1)—(8) and (10)—(17) is quite obvious.
As pointed by Resler and Sears [12] the term u x B can be taken to represent a tiny generator or source of e.m.f. at any point in the moving fluid. The vector E represents the total electric field arising out of internal causes such as separation of charges or polarization and external causes such as charged boundaries of the flow. Thus the electric field cannot be dissociated from the fluid motion; its value within the fluid element is directly affected by the motion of the element and is tak-
en to be of the order u x B. Hence, keeping under consideration the continuity equation (17) for current density vector, we conclude that for E to vanish, we must have
V- (u x B) = 0. (18)
3. FORMULATION OF THE PROBLEM
We have seen that in the cell method, the problem gets reduced to the investigation of a single particle enveloped in a cell. Here the single particle is an impenetrable cylinder of radius 1 with axis along the z axis and the cell is a concentric cylinder of dimensionless radius c (Fig. 1).
The external flow, induced by constant axial pressure gradient is ez along the axis of the cylinder. The applied magnetic field is uniform field ex along the transverse x-direction. As observed earlier the problem is not axial symmetric, hence we cannot take quantities to be function of r alone. These will be functions of both x and y in Cartesian coordinates or both r and 9 in polar coordinates.
Thus, we take
u = u(x, y)e z = u(r, 0)e z, (19)
B = e x + b(x, y)e ^ = e x + b(r, 0)e ^. (20)
Here, since all conditions are satisfied, we may assume that the induced magnetic field b is only in the z-direc-tion (Fig. 2) and then find that
u x B = u(x, y)ey. (21)
Thus, providing non-zero
V- (u x B) = du. (22)
dy
Therefore, we have induced electric field
E = Exe x + Eye y, (23)
where in terms of electric potential ^
E = -V^ (24)
Now, inserting Lorentz force 1
J x B =
Re,
-(Vx B) x B =
1 ab2
ev --
1 db 2
e y +■
1 db.
(25)
2Rem dx 2Rem dy Remdx in Stokes equation, we get the component equations as
0 = _dp _ M2 db2 dx 2Rem dx
0 = _dP _ M2 db2 dy 2Re, dy '
0 = jp + V2« + ML db.
dz Rem dx
(26)
(27)
(28)
ex
r = c
r = 1
Fig. 2. Longitudinal view of the cell in non-dimensional form.
Again, Ampere's law together with Ohm's law provides Vx B = Rem(E + u x B). (29)
Now, the two equations (26) and (27) provide on integration
giving
p(x,y,z) = b2 - Pz
2Re,
dl = -P.
dz
The equation (28) now reduces to
P + V 2u + M_ db = o.
Re, dx
(30)
(31)
(32)
Next, taking curl of equation (13), using equations (12) and (15) and the setting of this problem, we obtain the equation providing induced magnetic field as
-
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