International Journal of Mathematical Engineering and Science ISSN : Volume 1 Issue 2

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1 ISSN : Volume Issue The Mathematical Structure and Analsis of an MHD Flow in Porous Media () () Naji Qatanani and Mai Musmar () Department of Mathematics, An-Najah National Universit Nablus-Palestine () Al-quds open universit Nablus-Palestine nqatanani@najah.edu mmismar@qou.edu Abstract. This article is concerned with the formulation and analtical solution of equations for modeling a stead two-dimensional MHD flow of an electricall conducting viscous incompressible fluid in porous media in the presence of a transverse magnetic field. The governing equations, namel, Navier-Stoes equations and the Darc-Lapwood-Brinman model are emploed for the flow through the porous media. The solutions obtained for the Riabouchins-tpe flows are then classified into different tpes. Kewords: Porous medium, MHD flow, Navier-Stoes equations, Riabouchins flow. Introduction One of the oldest chapters of engineering sciences, the magnetohdrodnamic fluid flow in porous media due to its rapid epanse and widespread industrial and environmental applications, is still a subject of current research interest. For eample, the roc that constitutes the earth's crust is essentiall a porous medium that deforms over geological timescales. The flow through, and erosion of, this medium b magma leads to such phenomena on laered magma chambers and volcanic eruptions. The flow of groundwater through soil and / or roc has important applications in agriculture and pollution control. Other topics of interest include compaction of sedimentar basins and the phenomenon of frost heave, which occurs when groundwater freezes. As well as damaging roads and pavements, frost heave is responsible for geological formations [], [5], [6]. One of the original motivations for studing porous medium flows was the etraction of oil from rocs. It was found that the suction of viscous fluid from a porous medium is often unstable, tending to leave behind a sizeable proportion of the oil in small isolated pacets and increasing the etracted fraction is a constant challenge. Another important issue for the oil industr is upscaling from locall measured properties (e.g. permeabilit) [4]. Further application of porous media is most of the tissues in the bod (e.g. bone, cartilage, muscle) are deformable porous medium. The proper functioning of such materials

2 ISSN : Volume Issue depends cruciall on the flow of blood, nutrients through them. Porous medium models are used to understand various medical conditions (such as tumour growth) and treatment (such as injections) [3]. There are four independent fields of fluid dnamics namel, general fluid flow, flow through porous media, magnetohdrodnamics flow and analtical approaches to flow problems. In the field of general fluid flow, an enormous amount of wor has been carried out over the past two centuries, and man aspects of fluid flow theor are well developed [0]. and in their stud of inviscid aligned flow, b comple technique. Various aspects of magnetohdrodnamic flow have been studied, and man models have been developed to stud the interactions between the magnetic field, the electric field, and how the interact with a flowing fluid. Analsis of this tpe of flow has been proved b [6]. Some rotational and irrotational viscous incompressible aligned plane flow were discussed in [], [7]. As per fluid flow through porous media, interest in the field dates as far bac as 856 b Darc and currentl one finds various models governing tpes of fluid flow in man porous structures. The main tpe of single phase flow models have been developed and reviewed b [7], []. Although man models of flow through porous media are available, and various others are continuall being developed, a model of particular importance to this wor is the Darc-Lapwood-Brinman model [8], [3]. This accounts for both viscous shear and inertial effects of flow through porous media. A modified version of model in our current analsis, with the eception that a magnetic field is imposed on the flow field will be used. In this wor, we consider a single-phase fluid flow through porous media in the presence of a magnetic field. The aim is to analze the nonlinear model equation in an attempt to find possible solutions corresponding to a particular form of the stream function. The choice of the stream function in this wor is one that is linear with respect to one of the independent variables. This tpe of flow is referred to as the Riabouchins flow. In this case, the two dimensional Navier-Stoes equations, written as a fourth order partial differential equations in terms of the stream function, ma be replaced b fourth order ordinar differential equations in two unnown functions of a single variable. Solutions to the coupled set are then obtained based on the nowledge of particular integrals of one of the equations. A different tpe of flow ma then be obtained with the nowledge of one of the functions. Riabouchins [4] assumed one of the functions to be zero and studied the resulting flow which represents a plane flow in which the flow is separated in the two smmetrical regions b a vertical or a horizontal plane. In addition to the stud of Navier-Stoes flows and their applications [5], Riabouchins flows have also received considerable attention in the stud of non-newtonian flows [9] and in magnetohdrodnamics []. In this wor we find analtical solutions to the two-dimensional viscous fluid flows through porous media in the presence of a magnetic field. The governing equations are based on the Darc-Lapwood-Brinman model of flow through porous media. The medium is assumed to be traversed and aligned b a magnetic field. Solutions are obtained for Riabouchins-tpe flows, with a modified solution algorithm that has

3 ISSN : Volume Issue been developed to handle the tpe of flow. Solutions obtained are then classified into different tpes.. Flow Equations. The Darc-Lapwood-Brinman (DLB) Model The stead flow of an incompressible viscous fluid through porous media is governed b the conservation of mass and conservation of linear momentum principles. In the absence of sources and sins, conservation of mass principle taes the form V 0, () Where V is the macroscopic velocit vector. The conservation of linear momentum is given b the Navier-Stoes equations of the form P () (V )V V, Where P is the pressure, is the viscosit coefficient and is the fluid densit. When the macroscopic inertia and viscous shear are important, fluid flow through porous medium ma be described b the DLB of the form P (3) (V )V V V, Where is the permeabilit. Equation (3) is postulated to govern the flow through porous medium of high permeabilit and the flow through a mush zone undergoing rapid freezing. Through the viscous shear term, equation (3) is capable of handling the presence of a macroscopic boundar on which a non-slip condition is imposed. Comparison of the structure of equation (3) with that of the Navier-Stoes equation () which govern the flow of a viscous fluid in free-space shows the presence in equation (3) of the viscous damping term ( V) that is postulated to the Darc resistance to motion eerted b the medium on the transversing fluid. However, the presence of this term does not alter the nonlinear structure ehibited in the Navier- Stoes equations.. Flow through porous media in the presence of a magnetic field The stead flow of a viscous, incompressible, electricall conducting fluid through porous media in the presence of a magnetic field is governed b the following sstem of partial differential equations 3

4 ISSN : Volume Issue V 0, (4) (V )V P + V V ( H) H, (5) (VH) ( H) 0 (6) H 0 (7) Where V is the velocit vector field, H is the magnetic vector field, P is the pressure, is the electrical conductivit, is the medium permeabilit to the fluid, is the magnetic permeabilit, is the fluid viscosit, is the porous medium viscosit and is the fluid densit. In virtue of the vector identit (V )V ( V ) V ( V) (8) then equation (5) taes the form V V ( V) P + ( ). V H H (9) The governing equations are thus (4), (6), (7) and (9). It is required to solve this sstem of equations for the unnowns V, H and P..3 The case of two-dimensional flow Considering plane-transverse flow in two space dimensions and, with a velocit vector field V ( uv,,0) and a magnetic field H ( 00,, H ) which acts in a constant direction 0 z, then equation (7) is automaticall satisfied and the governing equations (4), (6) and (9) tae the following components form: u v 0 (0) () q v ( v u ) P u u H () q u( v u ) P v v H 4

5 ISSN : Volume Issue u H v H H 0, (3) Where the subscript notation denotes partial differentiation, and q u v is the square of the speed. The sstem of equations (0) (3) represent four scalar equations in the unnowns u(, ), v (, ), P(, ) and H (, ). It should be noted that if the fluid is infinitel conducting, then 0 and equation (3) reduces to H H 0 (4) If we now define u v P(, ) P H P P H (, ) [ ] P P H, (, ) [ ] Then the equations () and () tae the following forms respectivel: [ q ] v ( v u ) P u u (5) [ q ] u ( v u ) P v v (6) The governing equations for the case of finite electrical conductivit are therefore (0), (3), (5) and (6), and equations (0), (4), (5) and (6) are for the case of infinite conducting fluid..4 Equations for vorticit function and stream function Introducing the vorticit function (, ) and the pressure function h(, ) defined respectivel b (, ) v u (7) (8) h(, ) P q h p [ q ] 5

6 ISSN : Volume Issue h p [ q ]. Then equations (5) and (6) tae the following forms respectivel h v u u (9),. (0) h u v v Hence the governing equations are thus (0), (4), (9) and (0) with and h are given in (7) and (8) respectivel. Let (, ) be the stream function defined in terms of the components as u and v Then we can see clearl that the conductivit equation (0) is identicall satisfied since 0 () u v and the vorticit equation (4) becomes () Equations (9), (0) and (3) are then become in terms of the stream function as follows : h ( ) (3) h ( ) (4) H H H 0. (5) Equations ()-(5) are now required to be solved for (, ), (, ), H (, ) and h(, ). The velocit components u and v will be obtained from (), P from equation (8) and then follows P. If the fluid is infinitel electricall conducting, then the diffusion equation (5) is reduced to H H 0. A compatibilit equation can be derived from equations (3) and (4) using the integrabilit condition h h. Thus, differentiating equation (3) with respect to 6

7 ISSN : Volume Issue and equation (4) with respect to and using the integrabilit condition we obtain the compabilit equation ( ) ( ) (7) or (, ) 4, (8) (, ) Where and ;. 4. Hence once equation (6) is solved for (, ), the vorticit function can be calculated from equation (7) and H (, ) from equation (5). The pressure function h(, ) can be obtained from equations (3) and (4), while P(, ) is obtained from equation (8) and hence P(, ) follows. The velocit components can then be obtained from equation (). 3. An Overview of the Solution Methodolog 3. The case of Navier-Stoes Equation The two-dimensional stead flow of a viscous incompressible fluid in free space is governed b the continuit equation () and the Navier-Stoes equation (). In this form, it is required to solve () and () for V and P. These equations ma be written in vorticit-stream function form using definitions (7) and () to obtain stream function equation (9) Vorticit equation. (30) Integrabilit equation for the Navier-Stoes equation can be obtained b substituting equation (9) into equation (30) to ield (33)

8 ISSN : Volume Issue Let h(, ) P q (34) be the generalized pressure function for the Navier-Stoes equation. Then the momentum equation () ma be written in component form as h ( ) (35) h ( ). (36) Equation (33) can then be solved for (, ). Using equation (9) we can obtain (, ), u(, ) and v (, ) can then be obtained from (). The pressure function h(, ) ma then be obtained from equations (35) and (36) and the pressure function P(, ) is then follows from (34). In order to provide a solution for equation (33) we assume that the stream function (, ) is linear with respect to the independent variable and has the form (, ) f ( ) g ( ). (37) where f ( ) and g( ) are four lines differentiable arbitrar functions of. 3. Reduction of the governing partial differential equations to ordinar differential equations Substituting equation (37) into equation (33) gives ( ) ( ) ' ' '' '' '' f v g v f g f f f g f '' g '' 0 (38) Equating the coefficients of similar powers of, leads to the following coupled set of fourth order ordinar differential equations : ( v ) f f '' [ f ' f '' f f '' ] 0 (39) 3.3 Particular Solution ( v ) g g '' [ g ' f '' f g '' ] 0 (40) In the absence of general solutions for the coupled equations (39) and (40), it is customar to determine f ( ) and g ( ) in accordance with the following algorithm: 8

9 ISSN : Volume Issue A particular solution is found for f ( ) satisfing equation (39). The function f ( ) is then substituted into equation (40) and a general solution is then found for g ( ). With the nowledge of f ( ) and g ( ) the stream function (, ) can then be obtained from equation (37). The velocit components and the vorticit function can be calculated from () and (9), the pressure function can be determined from equation (35) and (36) and the pressure follows from equation (34). 3.4 The different tpes of possible flows and their solutions A solution to the sstem of equations (39) and (40) is considered for the case. In this case, equations (39) and (40) give the coupled set of fourth order ordinar differential equations v [ ' '' ''] 0 (4) ( ) f f f f f v [ ' '' ''] 0 (4) ( ) g g f f g Equation (4) admits the three particular solutions 6 f( ) (43) f ( ) ( e ) (44) and f 3( ) c c. (45) where,, c and c are arbitrar constants. The solution f 3( ) is independent of the viscocit. Upon substituting for f( ) from equation (43) into equation (4) gives g ( ), (46) where 0,, and 3 are arbitrar constants. The corresponding stream function (, ) and the velocit components tae the following forms respectivel: 3 (, ) (47) u (, ) (48) 3 v (, ) (49) 9

10 ISSN : Volume Issue Similarl, substituting from f ( ) equation (3.4) into equation (4) ields g ( ) c c e c e d ep [ e ] d 4 3 c e d ep[ e ] d ep[ ] d, (50) where c, c, c 3 and c 4 are arbitrar constants. The corresponding stream function with the velocit components are 4 3 (5) (, ) ( e ) c c e c e d ep [ e ] d c e d ep ( e ) d ep( ) d u (, ) ( e ) v (, ). (5) (53) Liewise, substituting f 3( ) from equation (45) into equation (4) gives g 3( ) c 3 ep[ { c c }] ddd (54) where c, c and c 3 are arbitrar constants. The corresponding stream function and the velocit components tae the following forms respectivel : (, ) ( c c) c3 ep[ { c c }] ddd (55) u (, ) c c, (56) v (, ) ( c c 3 ep[ { c c }] dd ) (57) Conclusion In the current wor we have discussed the analtical solution of the magnetohdrodnamic flow through porous media in the presence of a magnetic field. We have implemented the Riabouchins method. The solutions obtained using the above procedure involve a number of arbitrar constants, and restrictive 0

11 ISSN : Volume Issue assumptions are needed to determine them. It is also clear that determining g( ) is not an eas tas, and some integrals are rather involved. In the absence of a sstematic procedure for determining the arbitrar constants, Riabouchins [4] assumed that g ( ) 0, and obtained the special form of the stream function (, ) f ( ). This amounts to assigning the value of zero to each of the arbitrar constants g( ) and g( ) and g3( ) above. References [] Auirault, J. : One the cross-effects of coupled macroscopic transport equations in porous media, Transport in Porous Media, 6, 3--5, (994). [] Ahmed, N., Sarmah, H. : MHD transient flow past on impulsivel started infinite horizantal porous plate in a rotating sstem with hall current, Int. J. of Appl. Math. and Mech. 7 (), --5, (0). [3] Breward, C. : The role of cell-cell interactions in a two-phase model for a vascular tumour growth, J.Math. Biol. 45 (), 5--5, (00). [4] Fitt, A. : Multiphase flow in a roll press nip, Euro. Jnl. Appl. Maths. 3, , (00). [5] Fowler, A. : Fast and slow compaction in sedimentar basins, SIAM J. Appl. Math. 59, , (998). [6] Fowler, A. : A generalized secondar frost heave model, SIAM J. Appl. Math. 54, , (994). [7] Hamdan, A. : Single-phase flow through channels, A review : Flow models and entr conditions, J. Appl. Math. Comp. 6, 03--, (994). [8] Hamdan, M., Ford, A. : Single-phase flow through channels, J. Appl. Math. Comput. 69, 4--54, (995). [9] Kaloni, P., Huschilt, K. : Semi-inverse solution of a non-newtonian fluid, inter. J. Non-linear Mechanics, 9(4), , (984). [0] Kingston, J., Power, G. : An analsis of two-dimensional aligned-field magnetohdrodnamic flows, J. Appl. Math. Phsics (ZAMP), 9, , (968). [] Lapropulu, L. : Riabouchins flows in magnetohdrodnamics, M.Sc Thesis, Universit of Windsor, (987). [] Lemaitre, R. : Fractal porous media : Three dimensional Stoes flow through random media and regular fractals. Transport in Porous Media, 5, , (990). [3] Nabil, T. : Darc Lapwood Bricman field flow and heat transfer through porous medium over a stretching porous sheet, Bull. Cal. Math. Societ, 9--, 33, (000). [4] Riabouchins, D. : Some considerations regarding plane irrotational motion of liquid, C.r. heed. Séance. Acad. SCI, Paris, 79, , (94). [5] Talor, G. : On the deca of vortices in viscous fluid, Phlosophical Magazine 46, , (93).

12 ISSN : Volume Issue [6] Waterloos, J. : Plane magnetohdrodnamic flow with constantl inclined magnetic and velocit fields, J. Fluid Mech. 4, , (073).