Indian Journal of Pure & Applied Physics Vol 4 June 5 pp 45-4 Convective flow of two immiscible viscous fluids using inkman model Atul umar ingh Department of Mathematics V D College anpur 8 Received July 4; revised February 5; accepted April 5 Generalised Couette flow of two viscous incompressible immiscible fluids with heat transfer through two straight parallel horizontal walls is studied A naturally permeable material of high porosity bound below the lower wall and the flow inside the porous medium is assumed to be of moderate permeability modeled by inkman equation Analytical solutions of the momentum and energy equations have been obtained for the flow domain by dividing it into three zones Appropriate matching conditions of velocity and temperature have been used to link various flow regimes The effects of inverse permeability parameter Reynolds number and viscous parameter on velocity fields and temperature fields in various zones along with mass flow rate skin-friction at the walls and rates of heat transfer have been presented eywords: Convective flow Heat transfer Porous medium IPC Code: GN Introduction Heat transfer in flows of immiscible incompressible viscous fluids in porous medium/ walls assume considerable importance due to their important applications in ground-water hydrology agricultural engineering petroleum industry Moreover the existence of a fluid layer adjacent to a layer of fluid saturated porous medium is a common occurrence in both geophysical and industrial environment including engineering applications such as thermal energy storage system a solar collector with a porous absorber etc In addition convection through a porous medium may be found in fiber and granular insulation including structures for high power density electric machines and course of nuclear reactors Darcy established the mathematical theory of the flow of viscous fluid through a porous medium Darcy's law describes the flow in the porous medium Generally this law is valid for the flows past porous bodies with low permeability Certain flows that pass through bodies with high porosity do not follow the Darcy's law and inkman's model is applicable for this type of flows Neale and Nader have introduced practical significance of inkman extension of Darcy law Beavers and Joseph 4 have discussed the rectilinear flow of a viscous fluid through a channel formed by an impermeable upper wall and a permeable lower wall defining a rigid surface below the porous material inha and Chadda 5 ingh and ingh 6 Venkataramana and Bathaiah 7 have studied problems using Darcy's equation and the boundary conditions at the interface (generally known as B-J conditions) of the free flow region and the fluid flow in the porous medium Using inkman's model Varma and Babu 8 have presented a stu on flow of viscous incompressible fluid through a porous channel They considered lower porous bed of high permeability (inkman's region) bounded below by a rigid wall while the upper wall of low permeability (Darcy region) Tong and ubramanian 9 Lauriat & Prasad Poulikakos and Renken have studied natural convection flow in vertical channels using the inkman extended Darcy model in different physical situations Comprehensive literature survey concerning with the subject is given in monographs of aviany and Nield and Bejan Bhargava and acheti 4 have discussed heat transfer in generalised Couette flow of two immiscible viscous incompressible fluids considering a rigid upper wall moving uniformly and a lower stationary wall of moderate permeability modeled by inkman equation and bounded below by a rigid wall Vafai and im 5 and Goyeau et al 6 have presented detailed stu on limitations of the inkman-forchheimer Darcy equation and natural convection in a porous cavity using Darcy-inkman model respectively Paul et al 7 have made an analytical solution of free convective flow in the interface region between a porous medium and a clear fluid using inkman
46 INDIAN J PURE & APPL PHY VOL 4 JUNE 5 model Recently ingh 8 has discussed MHD free convection flow through a porous medium between two vertical parallel plates More Recently ingh 9 has studied heat transfer effects in generalized Couette flow using the inkman model In all the above studies on heat transfer quoted above heat source parameter is not taken into account which is often experienced as an important device in geothermal and industrial applications Hence the purpose of present analytical stu is to discuss the effects of various parameters on generalized convective Couette flow of two viscous incompressible immiscible fluids through two straight parallel horizontal walls partially filled with the fluids and partially with porous matrix having interface horizontally It is considered that the upper rigid wall is moving uniformly the lower wall is bounded below by a naturally permeable material of high porosity and the momentum transfer in porous medium is described by inkman model In the analysis the flow domain is divided into three zones taking into account a constant heat source in the upper fluid Appropriate matching conditions have been used to link various flow regions to obtain the solutions for discussion Mathematical Formulation The nomenclature used for the mathematical formulations is as follows: x' y' cartesian coordinates x y non-dimensional coordinates > viscosities of upper and lower liquids ( ) ρρ ( >ρ ) densities of upper and lower liquids ' ' ' u u u velocities in zones I II and III u u u non-dimensional velocities in zones I II and III ' ' ' T T T temperatures in zones I II and III T T T non-dimensional temperatures in zone I II and III T T thermal conductivity of the upper and lower liquids * ** T T constant temperatures of the upper rigid boundary and lower rigid boundary which bounds the porous φ φ medium diffusivity factors Re N M τ τ σ λ r h U H Q inkman number Reynolds number permeability of the porous medium (assumed constant) mass flow rate in free fluid region skin-friction factors at the upper boundary and at the interface rates of heat transfer at the upper boundary and at the interface permeability parameter conductivity parameter viscous parameter inverse permeability parameter non-dimensional thickness of the permeable material half width of the straight channel uniform velocity of the upper rigid moving wall thickness of the permeable material constant heat source non-dimensional constant heat source Consider the generalised Couette flow of two immiscible viscous incompressible fluids in a horizontal straight channel of height h bounded above by a rigid wall moving with a uniform velocity U and below by a naturally permeable material of moderate permeability (Fig ) In cartesian coordinate system let x' and y' be the axial and vertical coordinates with the origin at the centre of the channel and the positive direction of y' being directed towards the upper rigid wall Moreover the present analysis is based on the following assumptions: p' (i) A constant pressure gradient P acts at x' the mouth of the channel walls (ii) The upper fluid (viscosity density ρ ) occupies the upper half of channel walls (ie y' h) (iii) A constant point heat source (absorption type) Q( T T ) is taken into account in the upper fluid (iv) The lower fluid [viscosity ( > ) density ρ ( >ρ ) ] occupies the region [ h y' ]
INGH et al: CONVECTIVE FLOW OF TWO IMMICIBLE VICOU FLUID 47 Fig Physical model of the problem comparing the lower half of the channel and the permeable material of thickness H (v) The governing equations in the region ( h y' h) are Navier-tokes equations together with energy equations (vi) In the permeable material ( ( ) ' h+ H y h ) the flow follows the inkman's equation along with energy equations following Bhargava & acheti 4 Under the above stated assumptions the momentum equations and energy equations in the three zones in non-dimensional form are: Zone - I ( y ): d u + Re du + T dt Zone - II ( y ): du () () Re + () Zone - III [ ( r) du u + y ]: λ Re (5) dt du + + 4σ Re (6) The non-dimensional boundary and matching conditions relevant to the problem are: y : u T (7) y : u u du (8) y : dt dt T T (9) y : du du u u φ () y : dt dt T T φ () ( r) y + : u T () Non-dimensionalisation of the governing Eqs [() - (6)] and the boundary and matching conditions in Eqs[ (7) - ()] have been achieved by defining the following non-dimensional variables: dt du + (4) y' u ' u ' u ' y u u u h U U U
48 INDIAN J PURE & APPL PHY VOL 4 JUNE 5 T T T T T T * ** T Qh T H r h λ σ T T T T * T ** U T * T ** T ( ) T T T T T * ** N σ H Ph Re U and olution of the Problem Momentum Eqs [() () and (5)] are ordinary second order differential equations These are solved under the corresponding boundary and matching conditions The solutions of these equations are : u Rey + y+ 4 () u Rey + y+ 4 (4) y y u Acosh + Asinh + Reσ σ σ (5) The energy Eqs () (4) and (6) are solved in usual du manner after substituting du and du in these equations respectively subject to relevant boundary and matching conditions referred above The solutions of these equations are: T cosh y+ sinh y 5 4 + ( ) 4 Re + 4 4 4 ( ) Re y Re Re y (6) 4 Re y Re y y T + y 4Re + 4 + + 5 (7) A + A T y λ y A A 8 σ 4 AA y + sinh + ( σ Rey) + 6 y+ 7 4 σ cosh + ( ) (8) + 6 7 9 7 8 5 where A A 5 8+ 6 9 5 9+ 6 8 coshλφλsinhλ φλ coshλsinhλ φλcoshλ 4φλ sinhλ 5 4 6 + Re + Re+ Re 8 coshλ ( + r) λ 7 Re 9 sinhλ ( + r) λ + 7 9 6 7 8 5 5 9+ 6 8 6 8+ 5 9 4 Re 4 + Re 4Re 4Re + 5 Re Re 6 + + sinh Re λ+ σ 4 7 λ + ( ) cosλ 4 8 4Re 8 7 6 9 λ λ ( ) cosλ φ 4 + + 4σ Re + λsinhλ 4 4Re Re + + 4Re ( ) 9 A + A coshλ + 8 λ + ( + r) ( A A ) 4 ( r)
INGH et al: CONVECTIVE FLOW OF TWO IMMICIBLE VICOU FLUID 49 5 6 sinhλ ( + r) + σ Re ( + r) 4 8 r 4 5 cosh sinh + ( + r) cosh + ( + ) and ( ) 5 r 4 + r 7 6 6 4 Let M and M be mass flow rate in the free fluid region due to the lower fluid and upper fluid respectively the total mass flow rate M is: Fig Velocity profiles in different zones M M + M u + u Re + 4 (9) Re + + + 4 The skin-friction factors τ and τ at the upper boundary and at the porous interface respectively are: du τ Re + y du τ Re+ y () () The rates of heat transfer and at the upper rigid boundary and at the porous interface respectively are: dt 8Re 4Re y + 5sinh 4cosh + dt 4Re Re + + y 4Re + () 4 () Fig Temperature distribution in different zones 4 Discussion Figure shows non-dimensional velocities (Zone - I ) u (Zone - II ) and u (Zone - III ) for a set of four combinations of permeability parameter ( σ ) in terms of λ Reynolds number (Re) and viscous parameter () and for fixed values of non-dimensional thickness of the permeable material (r) and viscosity factor ( φ ) Non-dimensional temperatures T (Zone - I ) T (Zone - II) and T ( Zone - III) for a set of six combinations of Reynolds number (Re) viscous parameter () permeability parameter ( σ ) heat source parameter ( ) and inkman number ( ) for fixed values of non-dimensional thickness of the permeable material (r) viscosity factor( φ ) conductivity parameter ( ) and diffusivity factor (φ ) are illustrated in Fig To be realistic the values of various parameters in various sets have been chosen following Nield and Bejan and Jha The effect of heat source parameter is observed because in many industrial and technological applications solar energy
4 INDIAN J PURE & APPL PHY VOL 4 JUNE 5 problems or in problems of space science the temperature dependent heat source plays an important role The numerical values of the mass flow rate M in free fluid region skin-friction factor τ at the rigid wall and skin-friction factor τ at the porous bed due to free fluid region have been represented in Table and respectively Rates of heat transfer in terms of sselt number namely and due to free fluid region and at the interface indicating the effects of σ Re φ and φ for fixed values of r 5 5 and 5 are shown in Table while the effects r and on rates of heat transfer and for fixed values of σ 4 5 Re 5 φ 4 and φ 7 have been shown in Table 4 5 Conclusions (i) An increase in λ or results in a decrease in velocities in all the three zones while an increase No σ Table Mass flow rate (M) in the free fluid region φ Re r M 4 6 4 5 4 5 4 84 8 6 4 5 4 5 4 6 4 5 4 5 4 4856 4 4 6 4 5 6 5 4 598 5 4 6 4 5 4 4 85 6 4 6 4 5 4 5 8 87 Table kin-friction factors ( τ ) and ( τ ) due to free fluid region No σ φ Re r τ τ 4 5 4 5 8 444 7 8 5 4 5 8 49 58 4 4 5 8 547 4964 4 4 5 6 5 8 8577 48 5 4 5 4 8 5554 497 6 4 5 4 5 4 897 464 Table Rates of heat transfer ( ) and ( ) due to free fluid region (r 5 5 and 5 ) No σ Re φ φ 4 5 5 4 7 6668 86 8 5 5 4 7 787 7676 4 5 4 7 97 6697 4 4 5 4 7 99 868 5 4 5 5 6 7 7658 7546 6 4 5 5 4 9 7894 457 Table 4 Rates of heat transfer ( ) and ( ) due to free fluid region ( σ 4 5 Re 5 φ 4 and φ 7) No r 4 5 5 6668 86 8 5 5 566 869 4 8 5 7769 759 4 4 5 5 5 57664 8495 5 4 5 5 6856 65
INGH et al: CONVECTIVE FLOW OF TWO IMMICIBLE VICOU FLUID 4 in Re results in an increase in velocities in these zones (ii) The velocity increases and approaches to maximum in Zone-I ( y ) thereafter decreases in Zone-II ( y ) and ultimately tends to zero in Zone-III [ ( + r) y ] (iii) An increase in Re or σ increases temperature in all the three Zones while an increase in or decreases the temperature fields in these Zones (iv) The temperature decreases more steeply in Zone-I ( y ) in comparison to the temperature in Zone-II ( y ) and ultimately attains a constant value satisfying boundary condition in Zone-III [ ( + r) y ] (v) An increase in σ or φ decreases mass flow rate in the region ( y ) while an increase in Re or r increases it in the said region (vi) An increase in or Re decreases skin-friction but increases skin-friction τ while an increase in σ φ or r increases τ as well as τ (vii) An increase in σ φ φ or increases heat transfer rate and while an increase in r decreases and (viii) An increase in Re or increases but decreases (ix) An increase in heat source parameter decreases heat transfer rate but reverse effect is observed for Acknowledgement The author is extremely grateful to the University Grants Commission for financial assistance [No F5 ()/4(MRP/NRCB)] in the form of minor research project References Darcy H Les Fountains Publique De La Ville De Dijon Dalmont Paris 97 inkman H C Appl ci Res A (97) 7 Neale G & Nader W Canadian J Chem Engng 5 (974) 475 4 Beavers G & Joseph D D J Fluid Mech (967) 9 5 inha A & Chadda G C Ind J Pure Appl Math5 (984) 4 6 ingh B & ingh N P J M A C T 9 (986) 4 7 Venkataramana & Bathaiah D Ind J Pure Appl Math 7 (986) 5 8 Varma V & Babu M Ind J Pure Appl Math 6 (986) 796 9 Tong T W & ubramaniam E Int J Heat Mass Transfer 8 (985) 56 Lauriat G & Prasad V J Heat Transfer 9 (985) 688 Poulikakos D & Renken AME J Heat Transfer 9 (987) 88 aviany M Principles of Heat Transfer in Porous Media pringer-verlag New York 99 Nield D A & Bejan A Convection in Porous Media pringer-verlag New York 99 4 Bhargava & acheti N C Indian J Tech 7 (989) 5 Vafai & im J Int J Heat Fluid Flow 6 (995) 6 Goyeau B ongbe J P & Gobin D Int J Heat Mass Transfer 9(996) 6 7 Paul T Jha B & ingh A J Heat Mass Transfer (998) 55 8 ingh Atul umar Indian J Pure Appl Phys 4 () 79 9 ingh Atul umar J MANIT 6 () 58 Jha B Astrophys pace ci 75 (99)