LAW ANALYSIS OF THE FLOW OF TWO IMMISCIBLE COUPLE STRESS FLUIDS IN FOUR ZONES

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1 HEFAT04 0 th Iteratioal Coferece o Heat Trasfer, Fluid Mechaics ad Thermoamics 4 6 July 04 Orlado, Florida SECOND LAW ANALYSIS OF THE FLOW OF TWO IMMISCIBLE COUPLE STRESS FLUIDS IN FOUR ZONES Ramaa Murthy J V, *, Sriivas J ad Sai S Departmet of Mathematics, Natioal Istitute of Techology, Waragal, , Idia. Departmet of Mathematics, DMSSVH College of Egieerig, risha-500, Idia * jvrjosyula@yahoo.co.i ABSTRACT This work ivestigates the etropy geeratio i a stea flow of two immiscible couple stress fluids i a horizotal chael bouded by two porous beds at the bottom ad top. The flow is cosidered i FOUR zoes: zoe-iv cotais the flow of viscous fluid i the large porous bed with low permeability at the bottom, zoe-i ad II cotai free flow of two immiscible couple stress fluids ad zoe-iii cotais the flow of viscous fluid i the thi porous bed with high permeability at the top. The flow is assumed to be govered by Stokes s couple stress fluid flow equatios i the free chael. I zoe-iv, Darcy s law together with the Beavers-Joseph (B-J) slip coditio at the iterface is used whereas i zoe-iii Brikma s model is used for flow. The plates of the chael are maitaied at costat temperatures higher tha that of the fluid. The closed form expressios for etropy geeratio umber ad Beja umber are derived i dimesioless form by usig the expressios of velocity ad temperature. The effects of relevat parameters o velocity, temperature, etropy geeratio umber ad Beja umber are aalyzed ad preseted through graphs. INTRODUCTION Cotemporary egieerig thermoamics use a parameter called the rate of etropy geeratio (or productio) to gauge the irreversibility s related to heat trasfer, frictio, ad other o-idealities withi systems. The secod law of thermoamics should be cosidered to evaluate the sources of irreversibility i flow ad thermal systems. Coservig useful eergy depeds o desigig efficiet thermoamic heat-trasfer processes. Eergy coversio processes are accompaied by a irreversible icrease i etropy, which leads to a decrease i exergy (available eergy). Thus, eve though the eergy is coserved, the quality of the available eergy decreases because the eergy is coverted ito a differet form of eergy, from which less work ca be obtaied. Reduced etropy geeratio results i more efficiet desigs of eergy systems. Therefore, i recet years, the etropy miimizatio has become a topic of great iterest i the thermo-fluid area. Beja [,, ] focused o the differet reasos behid the etropy geeratio i applied thermal egieerig where the geeratio of etropy destroys the available work (exergy) of a system. Therefore, it makes good egieerig sese to focus o the irreversibility of heat trasfer ad fluid flow processes, ad try to uderstad the fuctio of associated etropy geeratio mechaisms. Beja [4] also carried out a extesive review o etropy geeratio miimizatio (EGM). The review traced the developmet ad adoptio of the method i several sectors of maistream thermal egieerig ad sciece. Furthermore, may researchers carried out studies o the etropy geeratio i various flow cases. Beja [] studied the heat trasfer problems i the pipe flow, boudary layer flow past a plate, ad flow i the etrace regio of a rectagular duct usig EGM. He demostrated that how the flow geometric parameters may be selected i order to miimize the irreversibility associated with a specific covective heat trasfer. There are may problems i the fields of hydrology ad reservoir mechaics i which systems ivolvig two or more immiscible fluids of differet desities/viscosities flowig i same pipe or chael or through porous media are ecoutered. Typical fluid flow examples of these systems are: air-water, water-salt water, oil-water, gas-oil, ad gas-oil -water systems. These are referred to as multi-phase flows i literature. Blood flow i arteries has bee studied by may researchers cosiderig blood as two phase flow. Several ivestigatios o multi-phase flows are reported by various researchers such as Chaturai ad Samy [5], Bird et al. [6], Ramchadra Rao ad Sriivasa Usha [7], Bhattacharya [8], apur ad Shukla [9] etc. The flow ad heat trasfer i immiscible fluids are of special importace i the petroleum extractio ad trasport problem. Heat trasfer i immiscible flows were discussed by Bakhtiyarov ad Sigier [0], Chamkha [] etc. Fluid flow i porous media is a importat subject of widespread iterest i hydrology, geophysics, biology ad the petroleum idustry. The problem of water coig is ofte ecoutered i the oil idustry whe a layer of water forms uder a layer of oil. To uderstad this pheomeo, it is of iterest to examie the cotact layer at the flow of two immiscible fluids. So there has bee widespread iterest i the stu of flow through chaels ad tubes i recet years. Vajravelu et al. [] studiestea flow of two immiscible coductig fluids betwee two permeable beds. Vijayakumar ad Syam Babu [] discussed MHD viscous flow betwee two porous beds. Iyegar ad Puamchader [4] have studied the couple stress fluid flow betwee two porous beds. Etropy geeratio calculatios for differet systems which have differet geometries i porous or oporous chaels have bee restricted to the first law of thermoamics. Calculatios usig the secod law of thermoamics, which are related to etropy geeratio ad efficiecy calculatio, are more reliable tha first law-based calculatios. A great volume of iformatio is available dealig with secod law aalysis i 04

2 the flow field ad heat trasfer i a porous medium. Waqar ad Gorla [5] have aalyzed the secod law characteristics of heat trasfer ad fluid flow due to mixed covectio i o- Newtoia fluids over a horizotal plae. Hooma ad Ejlali [6] studied both the first ad the secod laws of thermoamics for thermally developig forced covectio i a circular tube filled with a saturated porous medium. Tamayol et al. [7] studied thermal aalysis of flow i a porous medium over a permeable stretchig wall. Morosuk [8] discussed etropy geeratio i coduits filled with porous medium totally ad partially. Mahmud ad Fraser [9, 0] have discussed cojugate heat trasfer iside a porous chael ad mageto-hydroamic free covectio ad etropy geeratio i a square porous cavity ad also mixed covectio radiatio iteractio i a vertical porous chael. Tasim et al. [] studied etropy geeratio i a porous chael with hydromagetic effect. amel Hooma [] discussed the secod-law aalysis of thermally developig forced covectio i a porous medium. Chauha ad Vikas umar [] described the effects of slip coditios o forced covectio ad etropy geeratio i a circular chael occupied by a highly porous medium govered by Darcy exteded Brikma-Forchheimer model. Paresh Vyas ad Archaa Rai [4] ivestigated the etropy geeratio i radiative MHD Couette flow of a Newtoia fluid i a parallel plate chael with a aturally permeable base. I recet years, the fluid flow ad etropy geeratio i two immiscible fluids i a chael have received cosiderable attetio by researchers. amisli ad Oztop [5] cosidered the fluid flow ad etropy geeratio i two immiscible fluids i a chael. These authors explaied very icely the thermoamic iterface coditios ivolved i a flow of immiscible fluids ad made a sigificat observatio that miimum temperature gradiet i the trasverse directio of the flow offers miimum etropy geeratio ear the plates. Recetly, Ramaa Murthy ad Sriivas [6] have studied the secod law aalysis for the flow of two immiscible micropolar fluids betwee two parallel plates. They observed that the etropy geeratio is more ear the plates tha at the iterface of the chael. Couple stress fluids: To the extet the preset authors have surveyed the flow of immiscible icompressible couple stress fluids betwee two porous beds has ot bee studied so far. The cosideratio of couple-stress, i additio to the classical Cauchy stress, has led to the recet developmet of theories of fluid micro cotiua. This ew brach of fluid mechaics has attracted a growig iterest durig recet years maily because it possesses the mechaism to describe such rheologically complex fluids as liquid crystals, polymeric suspesios, ad aimal blood for which the Navier-Stoke s theory is iadequate. Oe such couple stress theory of fluids was developed by Stokes [7], ad represets the simplest geeralizatio of the classical theory which allows for polar effects such as the presece of couple stresses ad bo couples. The couple stress fluid is a special case of a o- Newtoia fluid which is iteded to take ito accout the particle size effects. A review of couple stress (polar) fluid amics was reported by Stokes [8]. Arima [9] discussed the applicatios o couple stress fluids ad compared with that of micropolar fluids. A umber of studies for such a fluid have bee reported [0, ]. The mai objective of this paper is to stu the etropy geeratio aalysis for the flow of two immiscible couple stress fluids betwee two porous beds. PROBLEM FORMULATION AND GOVERNING EQUATIONS Cosider the flow of two immiscible couple stress fluids betwee two parallel plates distat h apart, bouded by two porous beds of differet permeability s. The lower porous bed i zoe-iv has low permeability with ifiite thickess whereas the upper porous bed i zoe-iii is highly permeable with fiite thickess H (Figure ). The permeability s of lower apper beds are ad respectively. Let X ad Y are the axial ad vertical coordiates respectively with the origi at the cetre of the chael. Fluid flow is geerated due to a costat pressure gradiet which acts at the mouth of the chael. The lower fluid (viscosity µ, desity ρ ad thermal coductivity k ) occupies the regio ( h Y 0) comprisig the lower half of the chael ad this regio will be referred to as zoe I. The upper fluid (viscosity µ, desity ρ (<ρ ) ad thermal coductivity k ) is assumed to occupy the upper half of the chael (i.e., 0 Y h), ad this regio is called zoe II. The two walls of the chael are held at differet temperatures T I ad T II (with T I < T II ). The equatios for the flow i zoe I ad II (i.e., h Y h) are assumed to be govered by couple stress fluid flow equatios (eglectig bo forces except gravity force ad bo couples) of Stokes [7, 8] ad eergy equatio dρ +div(ρq)=0 () dt dq ρ = curl(ρ l)- P+µcurl(curlq)) dt () -ηcurl(curl(curl(curlq)))+(λ+µ)grad(divq) de ρ =Φ+k T () dt where Φ=µ (gradq):(gradq) T +(gradq):(gradq) + 4η (gradω):(gradω) T +4 η '[(gradω):(gradω) ] The equatios () () represet coservatio of mass, balace of liear mometum ad eergy equatio respectively. The scalar quatity ρ is the desity ad P is the fluid pressure at ay poit. The vectors q, ω, f ad l are the velocity, rotatio, bo force per uit mass ad bo couple per uit mass, respectively. The material costats λ ad µ are the viscosity coefficiets ad η ad η' are the couple stress viscosity coefficiets satisfyig the costraits µ 0 ; λ+µ 0; η 0, η' η. There is a legth parameter l= η/µ which is a characteristic measure of the polarity of the couple stress fluid ad this parameter is idetically zero i 05

3 the case of o-polar fluids. I the eergy equatio Φ is the dissipatio fuctio of mechaical eergy per uit mass, E is the specific iteral eergy, h = - k T is the heat flux, where k is the thermal coductivity ad T is the temperature. The force stress tesor t ij ad the couple stress tesor M ij that arises i the theory of couple stress fluids are give by t = (-P+λdiv(q)) δ +µd + δ ij ij ij ijk m,k +4ηw k,rr +ρc k (4) M ij = mδ ij +4ηω j,i +4ηω ' j,i I the above ω= curl(q) (5) is the spi vector, ω i,j is the spi tesor ad ρc k is the bo couple vector. d ij is the compoets of rate of shear strai, δ ij is the roecker symbol, δ ijk is the Levi-Civita symbol ad comma deotes covariat differetiatio. Darcy s law is valid for the flows through porous bodies with low permeability. I applicatios where fluid velocities are low, such as movemets of groudwater ad petroleum, etc., Darcy s law well describes the fluid trasport i porous media. Darcy law is the simplest ad, by far, the most popular oe, due to its simplicity. It states that the filtratio velocity of the fluid is proportioal to the differece betwee the bo force ad the pressure gradiet. The costat appearig i the relatio is called the permeability of the medium. Sice Darcy law is a first order PDE for the velocity, it caot sustai the o-slip coditio o a impermeable wall or a trasmissio coditio of the cotact with free flow. That motivated Brikma to modify the Darcy law i order to be able to impose the o-slip boudary coditio o a obstacle submerged i porous medium. Certai flows that pass through bodies with high porosity do ot follow the Darcy law. For this type of flows Beavers ad Joseph [] coditio is ot applicable. The Darcy law fails to describe the presece of a impermeable (solid) boudary. Brikma model is applicable for this type of flows. Brikma model is used i may applicatios as it allows oe to resolve problems with boudary coditios o impermeable boudary as well as o the iterface betwee porous medium ad a ope fluid domai. I this paper, Darcy law [] for flow i zoe IV ad the Brikma law [4] for flow i zoe-iii are take. The flow i the ifiite porous bed (i.e. i zoe IV) is govered by Darcy law q= (f - P) µ The flow i the fiite porous bed (i.e. i zoe III) is govered by Brikma equatio µ P= - q+µ q The boudary coditios for the flow through porous beds eed special attetio. Geerally the o-slip coditio is valid o the boudary whe a fluid flows past impermeable surfaces. But whe it flows past permeable surfaces, the o-slip coditio is o loger valid sice there will be a migratio of fluid, tagetial to the boudary withi the permeable surfaces. The velocity withi the permeable beds will be differet from the velocity of the fluid i the chael ad we have to match the two velocities at the iterface. Beavers ad Joseph [] based o their experimetal ivestigatios proposed that the slip velocity is related to the tagetial stress (kow as the Beavers ad Joseph (BJ) coditio). du α* = (Us -Q) (6) dy Here Us is the slip velocity, i.e., the local averaged tagetial velocity just outside the porous medium, Q is the velocity iside the porous bed, is the permeability of the porous medium ad the slip coefficiet α * is a dimesioless costat depedig o the material properties of the iterstices ad the derivative du is takig positive whe the ormal to the dy boudary ito the fluid medium is i a positive directio. α * varies from 0 to 5 for differet porous material surfaces (Nield [5]).If α * =0, it idicates a perfect slip coditio ad α *, we get o slip coditio i.e., U s = Q. Whe, the surface will be impermeable ad the velocity of the fluid i the ormal directio to the surface is zero. Figure Schematic of the ivestigated problem Herei the velocity vector q is take i the form q=(u(y),0,0). No-dimesioal variables are itroduced X Y U P through: x=, y=, u= ad p= where U h h Uo o is ρ U o the maximum velocity of the fluid i the chael. Neglectig bo forces ad bo couples from the equatio (), we get the followig set of o-dimesioal form of goverig equatios ad boudary coditios correspodig to the flow i two zoes. The goverig equatios i the correspodig zoes are: Zoe I:(- y 0) 06

4 4 dp -s = - Re s 4 dx Zoe II:(0 y ) 4 ρ dp -s = - Re s 4 µ dx Zoe III:( y (+δ)) (7) (8) ρ dp u = Re (9) Da k µ dx Zoe IV:(y -) u 4 where dp = Da Re (0) dx ρ Uoh Re= is the Reyolds umber, Da= µ h Darcy umber, = µ µ is the viscosity ratio, µ = ρ desity ratio, = is the permeability ratio, the couple stress parameter, (i=,) ad H δ= h. s = i ρ ρ κ i h γ is the is the Boudary ad iterface coditios: A characteristic feature of the two-layer flow problem is the couplig across liquid-liquid iterfaces. The liquid layers are mechaically coupled via trasfer of mometum across the iterfaces. Trasfer of mometum results from the cotiuity of tagetial velocity ad a stress balace across the iterface. V.. Stokes has proposed two types of boudary coditios (A) ad (B) respectively ad the vaishig of couple stresses o the boudary is referred to as coditio (A) [8]. This coditio is adopted here as this is appropriate i the preset cotext. At the lower porous boudary, couple stress vaishes: Beavers-Joseph (BJ) slip coditio is take at the lower porous bed i.e., * du α h dp = (us -u p ) where u p =-Da Re () dx u(-)=u ad s =0 (coditio( A)) at y= - () At the fluid iterface velocity, vorticity, shear stress ad couple stress are cotiuous: u = =, + u (0 ) (0 ) is du du du du =, s - = s - η (0- ) (0 + ) - (0 ) (0 + ) ad Where = η (0 ) (0 + ) η η= is the couple stress coefficiet ratio. η () At the upper plate boudary, velocity ad shear stress are cotiuous ad couple stresses vaish due to o slip ad hyperstick coditios: u du du = = u, s s ( ) ( + ) = ad ( ) (+ ) = 0(coditio(A)) at y = (4) No slip coditio: u =0 at y=+δ (5) where u s a p are respectively the dimesioless slip velocity ad Darcy s velocity. SOLUTION OF THE PROBLEM Velocity distributios: Solvig equatios (7) ad (0), we see that the velocity compoets i the zoes as: Zoe I: (- y 0) u (y) = c + c y + c coshs y + c 4 sihs y + Re By (6) Zoe II: (0 y ) u (y) = c + c y + c coshs y + c 4 sihs y + ρ µ Re By (7) Zoe III: ( y +δ) ρ u (y)=c cosh y+c sih y- Da ReB Da Da µ (8) Zoe IV: (y -) u 4 (y)=-da Re B (9) dp where B = (costat). Here we take the solutios as: zoe I: dx u = u (y), zoe II: u = u (y), zoe III: u = u (y) ad zoe IV: u = u 4 (y). These ivolve costats c, c, c, c 4, c, c,c, c 4, c, c a s. These costats are foud from the boudary coditios give i () - (5) ad these are obtaied usig Mathematica. As the expressios are cumbersome ad they are ot preseted here. 07

5 Heat trasfer aalysis: Oce the velocity distributios are kow, the temperature distributios for the two zoes are determied by solvig the eergy equatio () i the respective zoes, subject to the appropriate boudary ad iterface coditios. Thermal couplig is achieved through cotiuity of temperature at the iterface ad the balace of heat flux across the iterface. I the preset problem, it is assumed that the two walls are maitaied at costat temperatures T I ad T II (T I < T II ). The goverig equatio for the temperature T I of the coductig fluid i zoe I is the give by d T du d U k µ η = + = 0 (0) dy dy dy The goverig equatio for the temperature TII of the coductig fluid i zoe II is the give by d T du d U k = µ + η = 0 dy dy dy () I order to o-dimesioalize the above equatios (0)-(), the followig trasformatio is used for o-dimesioal T TI temperature θ: θ =. TII T I The equatios (0) ad () are the reduced to the followig form: d θ du + Br + = 0 () s d θ Br µ du + + = 0 () s k µ cp where Br = Ek Pr is the Brikma umber, Pr= is the k U Pradtl umber, Ek = o c p (TII is the Eckert umber ad TI ) k k = is the thermal coductivity ratio. k I the o-dimesioal form, the boudary coditios for temperature ad heat flux at the walls ad iterface become: (i) at the lower apper plate boudaries the temperatures are respectively, θ (y) = 0 ad y = - ad θ (y) = at y = (4) (ii) at the fluid iterface temperature (θ) ad heat flux ( h ) are cotiuous: dθ θ (y) = θ (y) ad dθ = k at y=0 (5) The solutios of equatios () ad () with boudary ad iterface coditios are solved aalytically ad they are legthy ot show here. The solutio ivolves 4 costats c 5, c 6, c 5 ad c 6 ad these are foud from the 4 boudary coditios (equatios (4) ad (5)) ad are obtaiesig Mathematica. ENTROPY GENERATION ANALYSIS Oce the velocity ad temperature fields have bee obtaied, oe ca determie the etropy geeratio distributio i a flow chael. This fuctio, which characterizes the irreversible behavior of system, will be used to optimize (miimize) the etropy geeratio rate by evaluatig parameters as well as fluid properties. The covectio process i a chael is iheretly irreversible. No-equilibrium coditios arise due to the exchage of eergy ad mometum withi the fluid ad at the solid boudaries. This causes the cotiuous etropy geeratio. Oe portio of this etropy productio is due to heat trasfer i the directio of fiite temperature gradiets. Aother portio of the etropy productio arises due to fluid frictio irreversibility. The volumetric rate of etropy geeratio for icompressible couple stress fluid is give as follows k (S i ) G = ( T) + Φ T o T o where Φ is the viscous dissipatio fuctio. The volumetric rate of etropy geeratio reduces to k T µ U η U (S i ) G = i i i i i i + + (6) To Y To Y To Y where the value of i ca be either or that represets fluid I or fluid II, respectively. O the right had side of the above equatio, the first term is the etropy geeratio due to heat coductio ad the remaiig two terms represet the etropy geeratio due to the viscous dissipatio fuctio Φ for a icompressible couple stress fluid. The characteristic etropy geeratio rate S G,C is defied as, ( h) k ( T) SG,C = = (7) kto h To I the above equatio, h is the heat flux i zoe I, T o is the average, characteristic, absolute referece temperature of the medium, T = T II T I ad h is the half of trasverse distace of the chael. The dimesioless form of etropy geeratio is the etropy geeratio umber (Ns) is the ratio of the volumetric etropy geeratio rate (S i ) G to a characteristics trasfer rate S G,C. (S ) Ns i G i = (i=,) S G,C The etropy geeratio umber for each fluid with dimesioless variables are give by 08

6 dθ Br du Ns = + + (8) Ω s dθ Br du Ns = k + µ + (9) Ω s µ U where o Br= is the Brikma umber, which determies k T the importace of viscous dissipatio because of the fluid frictios relative to the coductio heat flow resultig from the impressed temperature differece ad Ω = ( T/T o ) is the dimesioless temperature differece. It is desirable to cosider the Ek ad Pr i a group that is called the Brikma umber (Br = Ek.Pr) for evaluatig the relative importace of the eergy due to viscous dissipatio to the eergy because of heat coductio. It was reported that Br is much less tha uity for may egieerig processes []. Etropy geerates i a process or system due to the presece of irreversibility [, 6]. I covective problem both fluid frictio ad heat trasfer have cotributios to the rate of etropy geeratio. Expressio of the etropy geeratio umber (Ns) is good for geeratig etropy geeratio profiles but fails to give ay idea about the relative importace of frictio ad heat trasfer effects. The domiatio of the irreversibility mechaisms is physically importat sice the etropy geeratio umber is uable to overcome this problem. Two alterate parameters, irreversibility distributio ratio (ϕ) ad Beja umber (Be), are itroduced for this purpose ad they are gaiig a icreasig popularity amog researchers stuig the secod law. The idea of irreversibility distributio ratio ϕ ca ehace the uderstadig of the irreversibility s associated with the heat trasfer ad the fluid frictio. It is defied as the ratio of etropy geeratio due to fluid frictios (Nf) to heat trasfer i the trasverse directio (Ny) i.e., SG,fluid frictio Nf ϕ = = (0) SG,heat trasfer Ny ϕ ca be iterpreted as follows: If 0 ϕ <, the ϕ idicates that heat trasfer irreversibility domiates ad if ϕ > the fluid frictio domiates. For the case of ϕ =, both the heat trasfer ad fluid frictio have the same cotributio for etropy geeratio. A alterative irreversibility distributio parameter, called Beja umber Be, was defied by Paoletti et al., [7] as ratio of etropy geeratio due to heat trasfer to the total etropy geeratio ad it is give by Ny Ny Be = = = () Ns Ny + Nf + φ This is employed to uderstad the etropy geeratio mechaisms i thermal systems. Clearly, the value of the Beja umber rages from 0 to. For Be>0.5, the etropy geeratio due to the heat trasfer domiates, ad this correspods to the case of ϕ 0. O the other had, Be<0.5 refers to the etropy geeratio effects due to the fluid frictio. This correspods to ϕ. Whe Be=0.5, the cotributios of the heat trasfer ad fluid frictio i the etropy geeratio are equal, ad this correspods to the case of ϕ=. RESULTS AND DISCUSSION The closed form solutios for the flow of two immiscible couple stress fluids betwee two porous beds are obtaied ad reported i the previous sectio. Numerical work is udertake ad the variatios of velocity, temperature, etropy geeratio rate ad Beja umber for differet values of parameters are show through figures. Flow Field: Figure Effect couple stress parameter s o velocity u for δ=0., α*=0.6, B=-0.8, Da=0.08, ρ =0.9, =0.9, =., µ =0.9, Re=, s =.5. Figure Effect of slip parameter α* o velocity u for δ=0., B=-0., Da=0.08, ρ =0.9, =0.9, =., µ =0.9, Re=, s =., s =.. The ifluece of the couple stress parameter s o the velocity field is show i Figure. It is see that as s icreases, the velocity icreases. As s we get the case of Newtoia fluid. It ca be cocluded that the velocity of viscous fluid is more tha that of couple stress fluid. Thus, the presece of couple stresses i the fluid icreases the velocity. Figure depicts the effects of the slip parameter α* o the velocity field. As α* icreases, the velocity decreases. This chage i velocity is see to be more ear the lower porous bed where Darcy law is applicable. The effect of the Darcy umber 09

7 Da o the velocity field is show i Figure 4. It is see that as Da icreases, the velocity icreases i all regios of the chael. distributio. As the slip parameter α* icreases, temperature decreases. The effect of a icrease i Darcy parameter Da o temperature field is foud to icrease it i both zoes of the chael as show i Figure 7. Figure 8 idicates that the temperature icreases with icreasig the Brikma umber Br. This may be due to viscous dissipatio. Figure 4 Effect of Darcy umber Da o velocity u for δ=0., α*=0.5, B=-0., ρ =0.8, =0.8, =., µ =0.9, Re=, s =, s =. Figure 7 Effect of Darcy umber Da o temperature θ for δ=0., α*=0., B=-0., Br=0., ρ =0.9, =0.8, k=0.9, =., µ =0.8, Re=., s =, s =. Figure 5 Effect of couple stress parameter s o temperature θ for δ=0., α*=0.5, B=-0.9, Br=0., Da=0.0, ρ =0.9, =0.9, k=0.9, =., µ =0.9, Re=, s =. Figure 8 Effect of Brikma umber Br o temperature θ for δ=0., α*=0.8, B=-0., Da=0.08, ρ =0.8, =0.8, k=0.8, =0.8, µ =0.8, Re=., s =, s =. Figure 6 Effect of slip parameter α* o temperature θ for δ=0., B=-0., Br=0., Da=0.08, ρ =0.9, =0.7, k=0.9, =., µ =0.9, Re=, s =, s =. Thermal field ad Heat trasfer: Figure 5 displays the effect of the couple stress parameter s o the temperature field. As the couple stress parameter s icreases, the temperature icreases. Figure 6 presets the effects of slip parameter α*o the temperature Figure 9 Effect of couple stress parameter s o Etropy geeratio umber Ns for δ=0., α*=, B=-0., Br=.5, Da=0.0, ρ =0.9, =0.9, k=, =0.9, µ =0.9, Re=5, s =5, Ω=. 040

8 Figure 0 Effect of couple stress parameter s o Beja umber Be for δ=0., α*=, B=-0., Br=0., Da=0.0, ρ =0.9, =0.9, k=, =., µ =0.9, Re=., s =5, Ω=. Figure Effect of slip parameter α* o Etropy geeratio umber Ns for δ=0., B=-0.5, Br=0., Da=0.0, ρ =0.6, =0.6, k=, =., µ =0.9, Re=, s =, s =,Ω=. Figure Effect of Reyolds umber Re o Beja umber Be for δ=0., α*=0., B=-0.8, Br=0., Da=0.0, ρ =0.9, =0.9, k=, =0.9, µ =0.9, s =5, s =5, Ω=. Figure 4 Effect of slip parameter α* o Beja umber Be for δ=0., B=-0.5, Br=0., Da=0.0, ρ =0.6, =0.6, k=, =., µ =0.9, Re=,s =, s =,Ω=. Figure Effect of Reyolds umber Re o Beja umber Be for δ=0., α*=0., B=-0., Br=0.9, Da=0.0, ρ =0.6, =0.6, k=0.8, =0.8, µ =0.9, s =.5, s =.5, Ω=. Etropy geeratio ad Heat trasfer irreversibility: Figure 9 demostrates the effect of couple stress parameter s o the etropy geeratio umber Ns. As the couple stress parameter icreases, the etropy geeratio ear the plates icreases more rapidly i the fluid for the correspodig parameter. Ns is more o values ear the plates i zoe-i, tha i the zoe-ii. This may be due to the more viscous ature of the fluid i zoe-i. Figure 5 Effect of Darcy umber Da o Etropy geeratio umber Ns for δ=0., α*=0.8, B=-0.5, Br=0., ρ =0.6, =0.6, k=, =0.8, µ =0.9, Re=, s =, s =,Ω=. Figure 0 illustrates the effect of couple stress parameter s o Beja umber Be. As s icreases Beja umber decreases. A slight icrease i couple stress parameter s, icreases Beja umber Be huge at the iterface ad is early zero ear the plates. Hece we coclude that ear the plates the etropy geeratio rate due to coductio i the trasverse directio is almost zero ad etire etropy geeratio rate is due to fluid frictios oly. From the limitig case of s, we 04

9 see that for viscous fluids, values of Be are less tha the values of Be i couple stress fluids. With this we coclude that eergy dissipatio is more for viscous fluids tha for couple stress fluids. Figure shows the effect of Reyolds umber Re o the etropy geeratio umber Ns. The etropy geeratio ear the plates icreases more rapidly i the fluid I tha i the fluid II. This is due to the fact that i zoe-i fluid is more viscous. Figure shows the effect of Reyolds umber Re o Beja umber Be. As Re icreases, Be decreases. The variatio of Be ear the plates is more tha the variatio at the iterface. Figure 8 Effect of viscous dissipatio parameter (Br/Ω) o Beja umber Be for δ=0., α*=, B=-0., Da=0.0, ρ =0.6, =0.6, k=0.8, =0.8, µ =0.9, Re=, s =.5, s =.5. Figure 6 Effect of Darcy umber Da o Beja umber Be for δ=0., α*=0.5, B=-0., Br=0., ρ =0.6, =0.6, k=, =., µ =0.9, Re=, s =.5, s =.5, Ω=. Figure 7 Effect of dissipatio parameter (Br/Ω) o Etropy geeratio umber Ns for δ=0., α*=0., B=-0., Da=0.0, ρ =0.6, =0.6, k=, =0.8, µ =0.9, Re=, s =, s =. Figure predicts the etropy geeratio for differet values of slip parameter α*. As the slip parameter α*is icreasig, the etropy geeratio is icreasig i zoe-i oly where the slip coditio is applied. α* idicates o-slip coditio, velocity decreases as α* icreases. The same is observed i Figure. Hece whe frictio icreases, etropy geeratio rate icreases. Figure 4 describes the Beja umber Be profiles for differet values of slip parameter. As the slip parameter α* icreases, the Beja umber decreases. This is i good agreemet with the previous observatio i Figure for etropy geeratio rate. Figure 5 demostrates the etropy geeratio for differet values of Darcy umber Da. As the Darcy umber Da is icreasig, the etropy geeratio is decreasig. Figure 6 describes the Beja umber Be profiles for differet values of Darcy umber Da. As the Darcy umber Da is icreasig, the Beja umber is icreasig. From Figure 7, we observe that as the viscous dissipatio parameter (Br/Ω) icreases, etropy geeratio umber Ns icreases. The more the viscosity of the fluid is, the more is the etropy geeratio. It is observed that Ns 0 at the iterface of the chael. This implies that at the iterface, etropy geeratio is miimum (almost zero) i.e., exergy (available eergy) is maximum ad hece at the iterface almost dissipatio of eergy is zero. From Figure 8, we observe that the Beja umber is maximum at the iterface of the chael ad decreases as we move towards the chael walls i either directio. The Beja umber decreases as the viscous dissipatio parameter (Br/Ω) icreases. CONCLUSIONS The first ad secod laws (of thermoamics) aspects of fluid flow ad heat trasfer i a chael of two immiscible couple stress fluids betwee two porous beds is ivestigated aalytically. The velocity ad temperature profiles are foud aalytically. The exergy loss distributio is studied i terms of the secod law of thermoamics. The effect of viscous dissipatio parameter (Br/Ω) o the etropy geeratio umber (Ns) ad Beja umber (Be) are studied aalytically. The computatioal results are preseted through figures. The followig observatios are made from the above aalysis:. The presece of couple stresses i the fluid icreases the velocity ad temperature.. Viscous dissipatio parameter has a sigificat effect o the etropy geeratio rate.. The values of Ns ear the plates are more tha they are at the iterface, idicatig that frictio due to surface o the fluids icreases etropy geeratio rate. 4. The values of Ns i zoe-i are more tha they are i the zoe-ii ear the plates. This idicates the more is the viscosity of the fluid, the more is the etropy geeratio rate. 04

10 5. The Beja umber is maximum ad irreversibility ratio ϕ is miimum at the iterface of the chael. This idicates that the amout of exergy (available eergy) is maximum ad irreversibility is miimum at the iterface. 6. The maximum etropy geeratio rate shifts to each plate as viscous effects become more importat sice the plates act as strog irreversibility producers due to more fluid frictios i plate regios. 7. As the slip parameter icreases, etropy geeratio rate icreases. REFERENCES [] Beja A., A stu of etropy geeratio i fudametal covective heat trasfer, Joural of Heat Trasfer., Vol. 0(4), 979, pp [] Beja A., Secod law aalysis i heat trasfer, Eergy, Vol. 5, 980, pp [] Beja A., Etropy Geeratio Miimizatio, CRC Press, Boca Rato, New York, 996. 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