Flow through a Variable Permeability Brinkman Porous Core

Transcription

1 Journal of Applied Mathematics and Phsics, 06, 4, Published Online April 06 in SciRes Flow through a Variable Permeabilit Brinman Porous Core M S Abu Zatoon, T L Alderson, M H Hamdan * epartment of Mathematical Sciences, Universit of New Brunswic, Saint John, Canada Received 6 Februar 06; accepted 4 April 06; published 8 April 06 Copright 06 b authors and Scientific Research Publishing Inc This wor is licensed under the Creative Commons Attribution International License (CC BY Abstract In this wor, we consider the flow through composite porous laers of variable permeabilit, with the middle laer representing a porous core bounded b two arc laers Brinman s equation is valid in the middle laer and has been reduced to an Air s inhomogeneous differential equation Solution is obtained in terms of Air s functions and the Nield-Kuznetsov function Kewords Air s Functions, Variable Permeabilit Porous Laers Introduction Fluid flow through and over porous laers has been receiving increasing interest in the porous media literature for over half a centur, due to the importance of this tpe of flow in industrial and natural situations including lubrication problems, heating and cooling sstem design, groundwater flow, and the movement of oil and gas in earth laers []-[4] Much of the wor in this field has been devoted to the derivation of appropriate conditions at the interface between a fluid and a porous laer, or at the interface between two composite porous laers [5]-[9] Various flow models have also been tested to find the most appropriate model to use in a given flow situation, and the most appropriate model that provides compatibilit with the Navier-Stoes equations [] [0] [] Man excellent reviews are available in the literature which has been centred on the problem of flow through and over porous laers of constant permeabilit [] [0]-[3] More recentl, however, there has been increasing interest in the use of Brinman s equation with variable permeabilit due to the usefulness of Brinman s equation in modelling the flow in the transition laer between a arc porous medium and Navier-Stoes channel In fact, this has been extensivel analzed b Nield and Kuznetsov [4] in their introduction of the variable per- * Corresponding author How to cite this paper: Zatoon, MSA, Alderson, TL and Hamdan, MH (06 Flow through a Variable Permeabilit Brinman Porous Core Journal of Applied Mathematics and Phsics, 4,

2 meabilit transition laer Their analsis introduced the use of Brinman s equation to model the flow in which the chose a permeabilit function that reduced Brinman s equation into an Air s differential equation It is worth noting that there exist a large number of functions that can be used to model the variable permeabilit and result either in an Air s equation or in a different special differential equation In the current wor we will introduce a permeabilit function that is suitable for describing permeabilit variations in a Brinman laer bounded two arc laers of variable permeabilit This will be used in the analsis of the problem of flow through a variable permeabilit Brinman porous channel bounded on either side b a variable permeabilit arc laer The arc laers are terminated on their outer sides b solid, impermeable walls This problem is representative of flow in a porous channel with a porous core that is of different porosit and permeabilit than its bounding porous lining A main objective of this undertaing is to stud the ects of thin porous arc laers on the variable permeabilit flow in a Brinman laer In order to accomplish this wor, we choose a Brinman permeabilit function that reduces Brinman s equation to the well-nown inhomogeneous Air s differential equation [5] We provide an analtical solution to the resulting Air s inhomogeneous equation, and we provide computations using Maple s built-in functions to evaluate Air s functions Problem Formulation Consider the flow configuration in Figure The flow domain is a channel composed of three porous laers, where the flow in the middle laer is governed b Brinman s equation with variable permeabilit, and in the lower and upper laers with variable permeabilit arc law The channel is bounded b solid, impermeable walls at 0 and H In setting up the above flow problem, we mae the following assumptions that are essential for the current wor In the lower arc regiment, permeabilit is an increasing function of It starts at zero on the lower macroscopic wall and reaches a maximum,, at the lower interface ( of the Brinman laer In the upper arc regiment, permeabilit is a decreasing function of It starts at its maximum,, at L (the upper interface with the Brinman laer and drops to zero on the upper macroscopic wall ( H 3 > due to the choice of decreasing permeabilit distribution in the Brinman regime 4 All permeabilit functions are assumed continuous At each interface, the permeabilit of the lower channel is equal to permeabilit of the upper channel However, the rates of change of arc permeabilit are not necessaril equal at the interfaces 5 At each interface, we assume velocit continuit and shear stress continuit 6 Flow is driven b the same constant pressure gradient p x 7 Solutions below will depend on whose values will be determined from given permeabilit distributions in the arc regiments 8 We will choose L so that the Brinman permeabilit remains finite (for the function chosen in this wor Figure Representative setch 767

3 Equations governing the flow in the three regions in Figure are as follows For 0 < < : For < < L: For L< < H : ( v px ( µ d u µ px u ( d µ µ ( v px (3 µ where in v is the velocit in the lower arc laer, v is the velocit in the upper arc laer, u is the velocit in the Brinman laer, p x < 0 is the common driving pressure gradient, µ is the fluid viscosit and µ is the ective viscosit of the fluid in the porous medium associated with Brinman s flow Boundar conditions associated with the above flow are as follows Conditions on Upper and Lower Walls v 0 0, v H 0, 0 0, H 0 (4 Conditions at the Interfaces and L ; ; µ µ ; ; µ µ u v U u v u L v L U L L u L v L (5 3 Solution Methodolog arc s Equations ( and (3 are algebraic equations from which we can determine the velocities once the pressure gradient and viscosit are given, and the permeabilit distributions (ie (, ( are prescribed With this nowledge, we can determine the velocit distributions, v(, v(, and calculate the velocit and permeabilit at each interface The distribution ( is chosen as an increasing and differentiable function on 0 < <, with ( 0 0, and the distribution ( is chosen as an decreasing and differentiable function on L< < H, with ( H 0 ( L Once v ( and v ( are determined, we can calculate U v(, U v( L These are the velocities at the lower and upper interfaces, respectivel, that will be used as boundar conditions in the solution of Brinman s equation We will choose a Brinman permeabilit function and solve the Brinman equation for the velocit, u(, subject to nown interfacial velocities, u( U v(, and u( L U v( L Solution to Brinman s equation is given in terms of Air s functions These are computed in this wor using Maple s built-in functions 3 Solution to Brinman s Equation in the Middle Laer In this wor we consider the variable permeabilit distribution in the Brinman laer to be given b the follow- L : ing expression that satisfies and ( L ( ( + ( L Using (6 in ( reduces Equation ( to the form (6 d u β px α u d + α µ (7 where 768

4 µ α µ ( L (8 and then and Now, letting Equation (7 then becomes: β µ L µ ( L (9 Y β 3 α α (0 Y β u( u u 3 ( Y ( α α d u d d u dy 3 α β Y α α α ( 3 + u u Yu 3 α α 3 d u d p (4 x Yu 3 Y α µ Equation (4 is Air s inhomogeneous equation, which admits the following general solution for u [4] [5]: px u ( Y ca( Y + cb( Y π N ( Y (5 α µ i i 3 i where Ai ( Y and Bi ( Y are Air s functions of the first and second ind, and N etsov function, defined b [5] Y i Y is the Nield-Kuzn- d d (6 N Y A Y B t t B Y A t t i i i i i 0 0 Equation (5 taes the following form in terms of the original variable : px 3 u β β β β α ca i α cb i α π N i α + (7 α α α α µ α and the following form in terms of the original velocit u( : x i 3 i 3 3 i 3 α α α µ α Y α+ β α+ β p α+ β u( ca + cb π N (8 It is convenient at this stage to introduce the following dimensionless variables with respect to a characteristic length M, in which the quantities identified b an asteris (* are dimensionless: ( LH,, µ u µ v ; LH,, ; u ; v ; * * * * M M ( px M ( px M * i ( * * * * * max * 3 * i ( i ( M i ( ; max ; α M α; β M β M M (9 769

5 ropping the asteris (*, we obtain the following dimensionless equations: Permeabilit distribution in Brinman s laer: Velocit distribution in Brinman s laer: ( L ( + ( L α+ β α+ β πθ α+ β u( ca i cb 3 + i N i 3 α α α Shear stress distribution in Brinman s laer: ( α 3 3 u ( c A α + β i c 3 B α + β πθ i N α + α α β i 3 α α α ( α where prime notation denotes differentiation with respect to the respective arguments Velocit at the interfaces between laers: where in: α+ β α+ β πθ α+ β u( ca + cb + N U v ( α α α ( α i 3 i 3 3 i 3 αl+ β αl+ β πθ αl+ β u( L ca + cb + N U v ( L α α α ( α i 3 i 3 /3 i 3 α θ ( L L β θ ( L (0 ( ( (3 (4 (5 (6 µ θ (7 µ πθ πθ U N Γ B Γ U N Γ B Γ ( α c A B A B ( α ( Γ ( Γ ( Γ ( Γ 3 i i 3 i i i i i i πθ πθ U N Γ A Γ U N Γ A Γ ( α c A B A B ( α ( Γ ( Γ ( Γ ( Γ 3 i i 3 i i ( θ i i i i ( + ( L 3 ( ( ( L 3 3 Γ ( θ L ( + ( L 3 ( ( ( L 3 3 Γ (8 (9 (30 (3 3 arc Expressions in the Bounding Laers Solution to Brinman s equation, obtained above is predicated upon, which are dependent on 770

6 the choice of permeabilit functions in the arc laers We illustrate the dependence of the solution of Brinman s equation on b choosing linear, quadratic and exponential permeabilit functions in the arc laers, whose dimensionless forms together with the velocit distribution and shear stress terms (the first derivative of velocit functions, are summarized in Table, below, after dropping the asteriss (* 4 Results and Analsis The dimensionless forms of linear, quadratic and exponential permeabilit distributions and associated velocit distributions for the arc laers, together with the shear stress terms, summarized in Table, above, are used to generate Table, which lists the values of permeabilit, velocit and shear stress term at the lower and upper interfaces between laers in terms of the permeabilit at the lower interface, and that at the upper interface for all chosen permeabilit distributions Similarl, the velocit at the lower interface is U and at the upper interface U for all chosen permeabilit distributions These values are independent of the dimensionless thicness of each laer The shear stress terms at the interfaces, on the other hand, are dependent on the dimensionless permeabilit values at the interfaces and the dimensionless thicnesses of the porous laers ependence of permeabilit profiles on the thicness of the porous laers is illustrated in Table 3 b taing 0, L 09 and H for a thic middle laer, and in Table 4 with 04, L 06 and H for a thin middle laer Permeabilit distributions in the lower and upper bounding arc laers are given in Table 3 and Table 4, as functions of, and permeabilit distribution in the middle, Brinman laer is calculated b the dimensionless expression of equation (0 and given in Table 3 and Table 4 for chosen values of Tables 5-0 document the values of velocities and shear stresses at the interfaces between the porous laers, and list values of parameters involved in velocit computations It should be emphasized here that some of the computed values of velocit and shear stresses become inaccurate or extremel large for small values a, hence not listed in this wor This ma be attributed to inaccurac in computations and approximations of Air s functions when a is small (that is, when a < 000 Graphs illustrating linear, quadratic, and exponential permeabilit profiles are illustrated in Figures (a-(c These figures show the relatives shapes of the permeabilit distribution in each of the laers, and the decreasing permeabilit in the middle laer How the permeabilit distributions affects the velocit profiles across the laers is illustrated in Figures 3(a-(e These figures show regions of expected increase and decrease in the velocit across the laers in a manner that is reflective of the increase and decrease in the permeabilit profiles 5 Conclusion In this wor we considered flow through composite porous laers of variable permeabilit The problem considered is that of a porous core the flow through is governed b Brinman s equation for variable permeabilit media, while the core is bounded b two arc laers of variable permeabilit Various tpes of variable arc Table imensionless permeabilit, velocit, and shear stress terms for darc laers Permeabilit istribution Velocit istribution Shear Stress Term max ( ( exp( ( ( e ( H ( H L max v ( exp( v ( ( e ( H v ( ( H L v ( H H L max max max v ( v ( max max ( H ( H L ( H ( LH exp e ( exp e max v ( exp( v ( e v ( H L max ( v ( H H L exp( H e exp( H exp( LH e H exp( LH e v ( v 77

7 (a (b (c Figure (a Permeabilit distribution for linear permeabilit functions,, /3, L /3, and different values of max (b Permeabilit distribution for quadratic permeabilit functions,, /3, L /3, and different values of max (c Permeabilit distribution for exponential permeabilit functions,, /3, L /3, and different values of max 77

8 Table imensionless permeabilit, velocit, and shear stress terms at the lower and upper interfaces Permeabilit at the Interface Velocit at the Interface Shear Stress Term at the Interface max U v ( v ( max max max U v ( v ( max max U v ( max max U v ( L v ( L L U v ( L v ( L L U v ( L v ( L L e v e ( max ( H L ( H L ( LH exp H exp LH e Table 3 ependence of permeabilit profiles on the thicness of the porous laers: thic laer Lower Laer max U ( Linear: 0 Quadratic: Exponential: Middle Laer L max U ( max 0 U U 0 Upper Laer H ( exp( 0 ( e 8 90 U ( Linear: 0( Quadratic: 00( Exponential: Table 4 ependence of permeabilit profiles on the thicness of the porous laers: thin laer Lower Laer max U ( ( exp e 0 exp 09 e Linear: 5 Quadratic: Exponential: Middle Laer L max U ( max 0 U U 0 Upper Laer H ( exp( 04 ( e U ( Linear: 05( Quadratic: Exponential: ( 0 6 ( exp e 0 exp 06 e 773

9 Table 5 Values of dimensionless α, β, U, and U for /3, L /3 and different values of max 0 00 α 7 β U U 0 α 97 α 70 β β U U U 000 U 0 U 00 U 00 α 997 α 970 β β U 0 U 000 ` α 700 β U 00 U 000 Table 6 Values of dimensionless Γ, Γ for /3, L /3 and different values of max Γ 03 Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Table 7 Values of dimensionless c and c for /3, L /3 and different values of max c c c c c c c c c c c c Table 8 Value of v, v at the interfaces for linear permeabilit functions, /3, L /3 and different values of 0 00 max u v u L v 03 u v 3 u L v 003 u v 03 u L v u v 3 u L v 0003 u v 03 u L v 0003 u v 003 u L v

10 (a (b (c 775

11 (d (e Figure 3 (a Velocit profiles for linear permeabilit function, θ, /3, L /3, max, and different values of max (b Velocit profiles for linear permeabilit function, θ, max 0, /3, L /3, and different values of max (c Velocit profiles for quadratic permeabilit functions, θ, max, 0, L 09, and different values of max (d Velocit profiles for quadratic permeabilit functions, θ, max 0, 05, L 075, and different values of max (e Velocit profiles for exponential permeabilit functions, θ, max, 05, L 075, and different values of max Table 9 Value of v, v at interfaces for Quadratic permeabilit functions, /3, L /3 and different values of 0 max u v u L v u v 6 u L v 006 u v 06 u L v u v 6 u L v 0006 u v 06 u L v 0006 u v 006 u L v

12 Table 0 Values of v, v at interfaces for exponantial permeabilit functions, /3, L /3 and different values of max u v u L v u v u L v u v u L v u v u L v u v u L v u v u L v permeabilit have been considered and solution to flow through the Brinman laer is cast in terms of Air s and the Nield-Koznetsov functions References [] Alazmi, B and Vafai, K (00 Analsis of Fluid Flow and Heat Transfer Interfacial Conditions between a Porous Medium and a Fluid Laer International Journal of Heat and Mass Transfer, 44, [] Alazmi, B and Vafai, K (000 Analsis of Variants within the Porous Media Transport Models Journal of Heat Transfer,, [3] AbuZatoon, MS (05 Flow through and over Porous Laers of Variable Thicnesses and Variable Permeabilit Ph Thesis, Universit of New Brunswic, Saint John, Canada [4] Vafai, K and Thiagaraja, R (987 Analsis of Flow and Heat Transfer at the Interface Region of a Porous Medium International Journal of Heat and Mass Transfer, 30, [5] Vafai, K and Tien, CL (98 Boundar and Inertia Effects on Flow and Heat Transfer in Porous Media International Journal of Heat and Mass Transfer, 4, [6] Beavers, GS and Joseph, (967 Boundar Conditions at a Naturall Permeable Wall Journal of Fluid Mechanics, 30, [7] Ochoa-Tapia, JA and Whitaer, S (995 Momentum Transfer at the Boundar between a Porous Medium and a Homogeneous Fluid: I Theoretical evelopment International Journal of Heat and Mass Transfer, 3, [8] Ochoa-Tapia, JA and Whitaer, S (995 Momentum Transfer at the Boundar between a Porous Medium and a Homogeneous Fluid: II Comparison with Experiment International Journal of Heat and Mass Transfer, 3, [9] Vafai, K and Kim, SJ (990 Fluid Mechanics of the Interface Region between a Porous Medium and a Fluid Laer: An Exact Solution International Journal of Heat and Fluid Flow,, [0] Parvazinia, M, Nassehi, V, Waeman, RJ and Ghoreish, MHR (006 Finite Element Modelling of Flow through a Porous Medium between Two Parallel Plates Using the Brinman Equation Transport in Porous Media, 63, [] Joseph,, Nield, A and Papanicolaou, G (98 Nonlinear Equation Governing Flow in a Saturated Porous Medium Water Resources Research, 8, [] Chandesris, M and Jamet, (006 Boundar Conditions at a Planar Fluid-Porous Interface for a Poisueille Flow International Journal of Heat and Mass Transfer, 49, [3] Sahraoui, M and Kavian, M (99 Slip and No-slip Velocit Boundar Conditions at Interface of Porous, Plain Media International Journal of Heat and Mass Transfer, 35, [4] Nield, A and Kuznetsov, AV (009 The Effect of a Transition Laer between a Fluid and a Porous Medium: 777

13 Shear Flow in a Channel Transport in Porous Media, 87, [5] Hamdan, MH and Kamel, MT (0 On the Ni(x Integral Function and Its Application to the Air s Non-Homogeneous Equation Applied Mathematics and Computation, 7,