This paper analyses the effects of small injection/suction Reynolds number, Hartmann parameter, permeability parameter and wave number on a viscous incompressible electrically conducting fluid flow in a parallel porous plates forming a channel. The plates of the channel are parallel with the same constant temperature and subjected to a small injection/suction. The upper plate is allowed to move in flow direction and the lower plate is kept at rest. A uniform magnetic field is applied perpendicularly to the plates. The main objective of the paper is to study the effect of the above parameters on temporal linear stability analysis of the flow with a new approach based on modified Orr-Sommerfeld equation. It is obtained that the permeability parameter, the Hartmann parameter and the wave number contribute to the linear temporal stability while the small injection/suction Reynolds number has a negligible effect on the stability.
The study of Couette flow in a rectangular channel of an electrically conducting viscous fluid under the action of a transversely applied magnetic field has immediate applications in many devices such as magnetohydrodynamic (MHD) power generators, MHD pumps, accelerators, aerodynamics heating, electrostatic precipitation, polymer technology, petroleum industry, purification of crude oil and fluid droplets spray. Channel flows of a Newtonian fluid with or without heat transfer were studied with or without Hall currents by many authors [
Indeed, the heat source and the Soret effect on hydromagnetic oscillatory flow through a porous medium bounded by two vertical parallel porous plates have been analyzed by Chand [
In the present paper, we studied the effects of above parameters on the linear temporal stability of the fluid, in a Couette horizontal porous channel flow with the presence of a uniform transverse magnetic field fixed relative to the fluid. We used a new approach based on a derived equation named modified Orr- Sommerfeld equation. The corresponding eigenvalue problem is resolved in order to study the linear stability of the flow. The plates of the channel are considered porous and flow within the channel is due to the uniform motion of the upper plate. Such linear temporal stability analysis through the so-called modified Orr-Sommerfeld equation has been made earlier [
The paper is organized as follows: Sect. 2 addresses the so-called modified Orr-Sommerfeld equation governing the stability analysis in the hydromagnetic Couette horizontal porous plates flow. Section 3 deals with analysis of the effects of small injection/suction Reynolds number R e ω , Hartmann parameter M , wave number k and permeability parameter K p on the flow. The conclusion is presented in the last section.
We considere a Couette viscous incompressilbe, electrically conducting fluid flow between two porous parallel plates of infinite lengh, distant h apart in the presence of uniform transverse constant magnetic field B 0 applied parallel to y * axis which is normal to the planes of the plates. We considered the simple case where, B 0 is fixed relative to the fluid. We work at constant temperature, the heat transfer aspect of the flow is not studied. We applied a small constant injection V ω , at the lower plate and a same small constant suction V ω , at the upper plate. The upper plate is allowed to move with non-zero uniform velocity U = U 0 in flow direction and the lower plate is kept at rest. We choose the origine on the plane ( x * ,0, z * ) such as − h ≤ y * ≤ h and x * parallel to the direction of the motion of the upper plate. We assumed the magnetic Reynolds number very small for metallic liquids and neglected the induced magnetic field in comparison with the applied one [
∇ ⋅ V = 0 , (1)
∂ V ∂ t * + ( V ⋅ ∇ ) V = − 1 ρ ∇ p + ν ∇ 2 V + 1 ρ J ∧ B − μ V ρ k * , (2)
∇ ∧ B = μ e ( J + ε e E ) , (3)
∇ ∧ E = − ∂ B ∂ t * , (4)
∇ ⋅ B = 0 , (5)
∇ ⋅ E = 0 , (6)
∇ ⋅ J = 0 , (7)
where (1)-(7) are continuity, Newton’s second law, Ampere’s law, Faraday’s law, Maxwell’s law and Gauss law equations respectively, with
J = σ ( E + V ∧ B ) , (8)
and V ( u * , v * , w * ) , B , E , J , σ , μ e , ε e are the velocity, the magnetic field, the electric field, the current density vector, the fluid electrical conductivity, the magnetic permeability and absolute permittivity of the fluid respectively and t * denotes the time. The physical model of the problem is illustrated in
B = ( 0 , B 0 , 0 ) , (9)
E = ( E x , E y , E z ) , (10)
J = ( J x , 0 , J z ) ; (11)
where B 0 is a constant. We assumed that no applied polarization voltage exists (i.e., E = 0 ). Then Equation (8) and Equation (11) give
J = σ B 0 ( − w , 0 , u ) (12)
and Equation (7) yields
σ B 0 ( ∂ u ∂ z − ∂ w ∂ x ) = 0 (13)
We introduce the following non-dimensional quantities x ˜ = x * h , y ˜ = y * h , z ˜ = z * h , t ˜ = U t * h , u ˜ = u * U , v ˜ = v * V ω , w ˜ = w * U , p ˜ = p * ρ U 2 , R e = U h ν (hydrodynamic Reynolds number), R e ω = V ω h ν (injection/suction Reynolds number), M = B 0 h σ μ (Hartmann parameter), K p = k * h 2 (permeability parameter).
So, Equation (1) and Equation (2) become
∂ u ˜ ∂ x ˜ + R e ω R e ∂ v ˜ ∂ y ˜ + ∂ w ˜ ∂ z ˜ = 0 (14)
∂ u ˜ ∂ t ˜ + u ˜ ∂ u ˜ ∂ x ˜ + R e ω R e v ˜ ∂ u ˜ ∂ y ˜ + w ˜ ∂ u ˜ ∂ z ˜ = − ∂ p ˜ ∂ x ˜ + ∇ 2 u ˜ R e − M 2 R e u ˜ − u ˜ K p R e , (15)
∂ v ˜ ∂ t ˜ + u ˜ ∂ v ˜ ∂ x ˜ + R e ω R e v ˜ ∂ v ˜ ∂ y ˜ + w ˜ ∂ v ˜ ∂ z ˜ = − R e R e ω ∂ p ˜ ∂ y ˜ + ∇ 2 v ˜ R e − v ˜ K p R e , (16)
∂ w ˜ ∂ t ˜ + u ˜ ∂ w ˜ ∂ x ˜ + R e ω R e v ˜ ∂ w ˜ ∂ y ˜ + w ˜ ∂ w ˜ ∂ z ˜ = − ∂ p ˜ ∂ z ˜ + ∇ 2 w ˜ R e − M 2 R e w ˜ − w ˜ K p R e . (17)
For the stability analysis, the flow is decomposed into the mean flow and the disturbance according to
u ˜ i ( r , t ) = U i ( r ) + u i ( r , t ) , (18)
p ˜ ( r , t ) = P ( r ) + p ( r , t ) . (19)
We take the dimensional basic flow for small suction and injection [
U * ( y ) = U 2 ( y * h + 1 ) , (20)
V * = V ω , (21)
W * = 0. (22)
By scaling these velocities as above, we obtain with h = ± 1 ( − 1 ≤ y * ≤ 1 ) the no-dimensional base flow
U ( y ) = y + 1 2 , (23)
V = 1 , (24)
W = 0. (25)
To obtain the stability equations for the spatial evolution of three-dimensional, we take the dependent on time disturbances
( u ( x , y , z , t ) ; v ( x , y , z , t ) ; w ( x , y , z , t ) ; p ( x , y , z , t ) ) ; (26)
which are scaled in the same way as above.
Inserting Equations (18) (26) into Equations (15)-(17), we get
∂ u ∂ t + U ∂ u ∂ x + R e ω R e ∂ u ∂ y + R e ω R e v ∂ U ∂ y = − ∂ p ∂ x + ∇ 2 u R e − M 2 R e u − u K p R e , (27)
∂ v ∂ t + U ∂ v ∂ x + R e ω R e ∂ v ∂ y = − R e R e ω ∂ p ∂ y + ∇ 2 v R e − v K p R e , (28)
∂ w ∂ t + U ∂ w ∂ x + R e ω R e ∂ w ∂ y = − ∂ p ∂ z + ∇ 2 w R e − M 2 R e w − w K p R e . (29)
The pressure terms can be eliminated from Navier-Stokes equations. For such a mean profile (base flow), the divergence of Navier-Stokes equations and continuity, give
∇ 2 p = − 2 R e ω R e d U d y ∂ v ∂ x + M 2 R e ω R e 2 ∂ v ∂ y . (30)
Taking the laplacian of Equation (28), we get after linearization with Equation (30)
[ ∂ ∂ t + U ∂ ∂ x + R e ω R e ∂ ∂ y + 1 K p R e − ∇ 2 R e ] ∇ 2 v − d 2 U d y 2 ∂ v ∂ x + M 2 R e ∂ 2 v ∂ y 2 = 0. (31)
The disturbances are taken to be periodic in the streamwise, spanwise directions and time, which allow us to assume solutions of the form
f ( x , y , z , t ) = f ^ ( y ) e i ( α x + β z − ω t ) ; (32)
where f represents either one of the disturbances u , v , w or p and f ^ the amplitude function; α = k x = k cos θ and β = k z = k sin θ are the wave numbers respectively on x and z axis directions; ω = α c is the frequency of the wave; i 2 = − 1 , θ = ( k x , k ) , c = c r + i c i is the wave velocity which is taken to be complex, α and β are real because of temporal stability analysis consideration. Then Equation (31) becomes
i α ( U − c − i R e ω D α R e − i α K p R e + i D 2 − k 2 α R e ) ( D 2 − k 2 ) v ^ = − ( M 2 D 2 R e − i α U ″ ) v ^ ; (33)
where D = d d y ; with boundary conditions for all ( x , ± 1 , z , t > 0 )
{ v ^ ( ± 1 ) = 1 , v ^ ′ ( ± 1 ) = 0. (34)
Taking
v p ( x , y , z , t ) = v ^ ( y ) e i ( α x + β z − ω t ) − 1, (35)
Equation (33) and the boundary conditions Equation (34), take the forms
{ [ ( U − i R e ω D α R e − i α K p R e + i D 2 − k 2 α R e ) ( D 2 − k 2 ) − ( U ″ + i M 2 D 2 α R e ) ] v ^ p = c ( D 2 − k 2 ) v ^ p , v ^ p ( ± 1 ) = v ^ ′ p ( ± 1 ) = 0. (36)
The first equation of system Equation (36) is a flow equation modified by the small injection/suction Reynolds number R e ω , the Hartmann parameter
#Math_100#, and permeability parameter ( K p = k * h 2 ) whichis the so-called
modified Orr-Sommerfeld equation, rewritten as an eigenvalue problem, where c is the eigenvalue and v ^ p the eigenfunction.
We consider a three-dimensional disturbances. We use a temporal stability analysis as mentioned above. With c complex as defined above, when c i < 0 , a stability mode takes place, c i = 0 corresponds to neutral stability and elsewhere corresponds to instability. We employ Matlab 7.8.0.(R2009a) version in all our numerical computations to find the eigenvalues. The Couette horizontal porous plates flow with the basic velocity profile
U = ( y + 1 2 , 1 , 0 ) (37)
for R e ω small (i.e. small suction) is considered. The eigenvalue problem Equation (36) is solved numerically with the suitable boundary conditions. The solutions are found in a layer bounded at y = ± 1 with U ( ± 1 ) = ( 0,1,0 ) . The results of calculations are presented in the figures below.
The black, red, green and blue colors are respectively, curves I, II, III, IV and the yellow color figure corresponds to the neutral mode c i = 0 . Frame a , b , c , d correspond respectively to k = 1 k = 1.02 , k = 2 and k = 3 .
may be concluded that except the frame c case where, the increasing of K p doesn’t contribute to stability for R e < 12500 , the increasing of permeabilty parameter contributes to the flow stability.
Finally, Figures 6-8 ( M ≠ 0 , electrically conducting fluid) show that for k = 1 and k = 1.02 , the small injection/suction has no effect on the linear temporal stability of the flow. But for k = 2 and k = 3 , we remark a little influence of the
small injection/suction on the stability only in a small range of R e .
The critical Reynolds numbers R e c for which transition occurs are presented in
In this paper, we have investigated the effects of small injection/suction Reynolds number, Hartmann parameter, permeability parameter and wave number on a viscous incompressilbe electrically conducting fluid flow, in a porous parallel plates forming a channel. We have derived the appropriate equation named modified Orr-Sommerfeld equation in order to make the
0.00 | 1.00 | 0.130 | 00 | 0.00 | 5347 |
0.00 | 1.00 | 0.048 | 00 | 0.00 | 19370 |
0.00 | 1.02 | 0.130 | 00 | 0.00 | 5139 |
0.00 | 1.02 | 0.048 | 00 | 0.00 | 18640 |
0.00 | 1.00 | 0.045 | 50 | 0.00 | 18650 |
0.00 | 1.00 | 0.045 | 80 | 0.00 | 7178 |
0.00 | 1.02 | 0.045 | 50 | 0.00 | 18010 |
0.00 | 1.02 | 0.045 | 80 | 0.00 | 6845 |
0.00 | 1.00 | 0.045 | 50 | 0.10 | 19610 |
0.50 | 1.00 | 0.045 | 50 | 0.10 | 19610 |
0.75 | 1.00 | 0.045 | 50 | 0.10 | 19610 |
1.00 | 1.00 | 0.045 | 50 | 0.10 | 19610 |
0.00 | 1.02 | 0.045 | 50 | 0.10 | 18940 |
0.50 | 1.02 | 0.045 | 50 | 0.10 | 18940 |
0.75 | 1.02 | 0.045 | 50 | 0.10 | 18940 |
1.00 | 1.02 | 0.045 | 50 | 0.10 | 18940 |
0.00 | 1.00 | 0.048 | 50 | 0.10 | 17750 |
0.50 | 1.00 | 0.048 | 50 | 0.10 | 17750 |
0.75 | 1.00 | 0.048 | 50 | 0.10 | 17750 |
1.00 | 1.00 | 0.048 | 50 | 0.10 | 17750 |
0.00 | 1.02 | 0.048 | 50 | 0.10 | 17140 |
0.50 | 1.02 | 0.048 | 50 | 0.10 | 17140 |
0.75 | 1.02 | 0.048 | 50 | 0.10 | 17140 |
1.00 | 1.02 | 0.048 | 50 | 0.10 | 17140 |
stability analysis of the flow. Through this approach, we have found that the small injection/suction has a negligible effect on the linear temporal stability of hydromagnetic Couette flow. We noticed that the permeability parameter (Darcy number), the Lorentz force (the Hartmann parameter) and the wave number contribute to the temporal linear stability of hydromagnetic Couette flow. We remarked also that at low wave numbers, the phase angle θ doesn’t contribute to the stability of the fluid flow, but for k = 2 and k = 3 , the stabilizing effect appears.
The authors would like to thank very much the anonymous referees whose useful criticisms, comments and suggestions have helped strengthen the content and the quality of the paper.
Monwanou, A.V., Hinvi, A.L., Miwadinou, H.C. and Chabi Orou, J.B. (2017) A New Approach for the Stability Analysis in Hydromagnetic Couette Flow. Journal of Applied Mathematics and Physics, 5, 1503-1514. https://doi.org/10.4236/jamp.2017.57123