INFLUENCE OF POROSITY AND RADIATION IN A VISCO ELASTIC FLUID OF SECOND ORDER FLUID WITHIN A CHANNEL WITH PERMEABLE WALLS

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1 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani INFLUENCE OF POROSITY AND RADIATION IN A VISCO ELASTIC FLUID OF SECOND ORDER FLUID WITHIN A CHANNEL WITH PERMEABLE WALLS *Ch. V. Ramana Murthy and Kancherla Rajani Lakireddy Balireddy College o Engineering, Mylavaram 5 (Andhra Pradesh) India Sanchar Communications, Vijayawada 5 (Andhra Pradesh) India *Author or Correspondence ABSTRACT As the Darcy s parameter increases the velocity also increases. The no slip condition is perectly satisied on the boundary. For a similar values o Darcy s parameter and when K =. the no slip condition is also satisied. However, a backward low is observed as we move away towards the other bounding surace. When the radiation parameter is held constant, it is observed that as the Darcy s number increases the no slip condition is satisied. But a backward low is observed. For such similar values o Darcy s number and when R=7, the velocity proiles are seen to be o same type. An interesting point in this case is that the velocity proiles are merged till 4% o the channel width. Thereater, the dispersion in the velocity proiles is observed. In this case also, the low appears to be more o backward and subsequently in the orward direction. Nomenclature: S : Cauchy stress tensor p : Scalar pressure : Coeicient o viscosity : Coeicient o elasticity : Coeicient o cross-viscosity : Density : Coeicient o thermal expansion g : Acceleration due to gravity k : Permeability o the porous medium : Transpiration cross low k : Viscoelastic parameter R : Cross low Reynolds number Re : Reynold s number Gr : Grasho number Pr : Prandtl number Da : Darcy number T : Wall temperature parameter : Angle o inclination A : Constant pressure gradient INTRODUCTION Non-Newtonian luid mechanics has had to be point o concern with the development o general constitutive equations or visco elastic luids. These constitutive equations should in principle 8

2 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani lead to the deinition o low properties that need to be measured to deine the visco elastic luid (rheometry) and to the development o the equivalent Navier-Stokes equations or the solution o all possible boundary value along with initial value problems that arises in several situations wherein heat and mass transer takes place. The solution presented need to be quite speciic about the experimental conditions pertaining to the relevant phenomena. Thereore, now the question that arises is to address the situation How do elastic liquids behave in complex lows? and it is immediately apparent that the answer must involve a consideration o how the same liquids behave in simple lows, so that obtaining rheometrical data on the test liquids is an essential part o the exercise. Such data, when available, serve more than one useul purpose they certainly provide a oundation set o data, which must be accommodated in the associated mathematical model or the test liquids. That is to deine a perect constitutive equation, which is an essential ingredient in any theoretical resolution o the experimental dilemmas, has to be consistent with the rheometrical data. Indeed, i the model cannot simulate behaviour in simple lows, what chance does it have in complex lows?! Clearly, the choice o constitutive equation is central to the whole operation and this choice is ar rom trivial or obvious. Indeed, a constitutive model which satisies the dual constraints o tractability and quantitative (or even semi quantitative) prediction may not exist! But that shouldn t and doesn t prevent a search or this missing link, but it is wise to be aware o the possibility o disappointment. The model that has been considered here is o second order luid whose constitutive relation has been proposed by Noll. The relation involves visco elasticity and also covers the concept o cross viscosity. In many chemical processing industries generally slurry adheres to the reactor vessels and gets consolidated. As a result o this, the chemical compounds within the reactor vessel percolates through the boundaries causing loss o production and then consuming more reaction time. The slurry thus ormed inside the reactor vessel oten acts as a porous boundary or the next cycle o chemical processing. A porous medium may be either an aggregate o a large number o particles such as sand or gravel or solid containing many capillaries such as a porous rock. In all such cases, one has to consider the gross eect o the phenomena represented by a macroscopic view applied to the masses o luid, large compared to the dimensions o the pore structure o the medium. The process can be described in terms o equilibrium o orces. The driving orce necessary to move a speciic volume o luid at a certain speed through a porous medium is in equilibrium with the resistance orce generated by internal riction between the luid and the pore structure. This resistance orce is characterized by Darcy s semi-empirical law established by Darcy (856). The simplest model or low through a porous medium is the one dimensional model derived by Darcy (856). Obtained rom empirical evidence, Darcy s law indicates that or an incompressible luid lowing through a channel illed with a ixed, uniorm and isotropic porous matrix, the low speed varies linearly with longitudinal pressure variation. Subsequently, Dupuit and Frochheimer presented empirical evidence that the Darcy law or the linearity between speed and pressure variation, breaks down or large enough low speed (a compilation o several experimental results) is presented by Mac Donald et al. (979). This was emphasized later by Joseph et al. (98) who stressed orce modeled by the Frochheimer acts in a direction opposite to the velocity vector. It ollows that, in multidimensional low, the momentum equations or each velocity component derived using the Frochheimer extended Darcy equation is at least speculative. Later, Knupp and Lage (995) analyzed the theoretical generalization to the tensor 9

3 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani permeability case (anisotropic medium) o the empirically obtained Frochheimer extended Darcy unidirectional low model. Heat transer in porous medium is gaining utmost importance due to its applicability in geothermal energy extraction, nuclear waste disposal, ossil uels detection, regenerator bed etc. Understanding the development o hydro dynamic and thermal boundary layers along with the heat transer characteristics is the basic requirement to urther investigate the problem. Cheng and Minkowycz (977) had analyzed the steady ree convection about a vertical plate embedded in porous dynamics in the orm o dissipative inequality (Clausius Duhem) and commonly accepted the idea that the speciic Helmholtz ree energy should be a minimum in equilibrium. From the point o medium applied to heat transer rom dike. Murthy and Singh (977) using method o similarity solution studied the inluence o lateral mass lux and thermal dispersion on non - Darcy natural convection over a vertical plate in porous medium. They have discussed the combined eect o thermal dispersion and luid injection on heat stratiication on non - Darcy mixed transer. Hassanien et al. (998) had studied the eects o thermal dispersion and dissipation eects on non Darcy mixed convection problems and established the trend o heat transer rate convection rom a vertical plate in porous medium and investigated the low and temperature ields. Subsequently, Murthy (998) had examined the dispersion while, Kuznetsov () investigated the eect o transverse thermal dispersion on orced convection in porous media and identiied the situations avorable to heat transer under dispersion eects. Mathematical Formulation We consider the laminar mixed convection low o a visco elastic luid through a porous medium in an inclined permeable channel, the space between the plates is h, as shown Figure : Geometry o the low ield when the channel is vertical It is assumed that the rate o injection at one wall is equal to the rate o suction at the other wall. A rectangular coordinate system x, y is chosen such that the x axis is parallel to the gravitational acceleration vector g, but with opposite direction and the y - axis is transverse to the channel walls. The let wall (i.e. at y ) is maintained at constant temperature T and the right wall (i.e. at y h ) is maintained at constant temperaturet, wheret T. The low is

4 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani assumed to be laminar, steady and is ully developed, i.e. the transverse velocity is zero. Then, u the continuity equation drops to. x The luid under consideration is assumed to be o Rivlin-Ericksen type whose constitutive equation is proposed as S pi A A A () The material constants, and can be determined rom viscometric lows or any real luid. A and A are Rivlin-Ericksen tensors and they denote respectively the rate o strain and acceleration. A and A are deined by A ) T V ( V () da T A AV ( V ) A () dt d where is the material time derivative and gradient operator and T ( ) transpose operator. dt The viscoelastic luids when modeled by Rivlin-Ericksen constitutive equation are termed as second grade luids. A detailed account o the characteristics o second grade luids is well documented by Dunn and Rajagopal (995). Later, Rajagopal and Gupta [] had studied the thermodynamics consideration and it is assumed that:, and (4) The basic equations o momentum and energy governing such a low, subject to the Boussinesq approximation, are du dp d u d u ( ) u g T T (5) dy dx dy dy k dt d T (6) dy dy dp here is a constant. dx The boundary conditions are given by u ( ) u( h), T( ) T and T( h) T Introducing the ollowing non-dimensional variables y u T T y =, u = h h and = T T into the Eqn. (5 ) and Eqn. ( 6 ), we obtain d u d u du Gr kr R u A GSin (8) dy dy dy Da Re (7)

5 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani d d dy dy where k is the visco elastic parameter, h h R (9) is the cross low Reynolds number, g ( T T ) h U Gr is the Grasho number, h Re is the Reynolds number, Pr is the T T dp U Prandtl number, r T is the wall temperature parameter and A ( ) is the T T dx h constant pressure gradient. The corresponding dimensionless boundary conditions are given by u ( ) u(), () r T and ( ) () Method o Solution We consider the irst order perturbation solution o the boundary value problem or small k. Since the constitute Eqn. () has been derived up to only the irst order o smallness o k, thereore, the perturbation solution obtained by retaining the terms up to the same order o smallness o k must be quite logical and reasonable. We write u u ku () and k () Substituting Eqn. () and Eqn. () into Eqn. (8) and Eqn. (9) and boundary conditions given by Eqn. () and then equating the like powers o k, we obtain Zeroth-order system ( k ) : d u dy du R u dy Da Gr AGSin Re d d dy dy (4) together with boundary conditions u ( ) u(), () r T and () (5) First-order system ( k ) : d u dy du dy Da d u dy Gr Re R u R (6) d d (7) dy dy together with boundary conditions u ) u () and ) () (8) ( ( ()

6 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani Zeroth-order solution (or Solution or a Newtonian luid): Solving Eqn. () and Eqn. (4) using the boundary conditions (5), we get y ( rt e ) ( rt ) e ( e ) (9) ay by Gr y u ce c e ( e ) ADa GDaSin () Re where R R 4 / Da R R 4 / Da ( rt e ) Da a, b,, ( e ) ( rt ) Gr, ( ) ADa DaGSin, ( e )( / Da) Re b a Gr 4 e e 4 ( e ) ADa DaGSin, c, c. b a b a Re e e e e 4 First-order solution (or Solution or a second-grade luid): Solving Eqn. (7) with corresponding boundary conditions, we obtain () Substituting the Eqn. () and Eqn. () into the Eqn. (6) and then solving the resulting equation with the corresponding conditions, we get where u 5 ay by ay by ce c4e 6 ye 7 ye 5 e y 4 Gr R Pr Rca, 6, Re ( / Da) a R Rcb b R 7, b a a b 8 5e 5e 8 8 5e 6e 7e, c, c. b a 4 b a e e e e It can be veriied that when k, R and Da our results reduces to those given by Aung and Worku (986) Finally, temperature is given by () y ( rt e ) ( rt ) e () ( e ) and velocity is given by u ay c kc e c kc by y 4 e Gr( e ) Re ADa GDaSin ky 6 e ay ky 7 e by k 5 e y (4)

7 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani Figure : Velocity Proiles with respect to Darcy s Parameter Figure : Inluence o Darcy s Parameter on Velocity Proiles 4

8 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani Figure : Eect o Darcy s Parameter on Velocity Proiles Figure 4: Variation in Velocity Proiles with respect to Darcy s Parameter 5

9 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani Figure 5: Velocity Proiles with respect to Radiation Parameter Figure 6: Variation o Velocity with respect to Darcy s parameter and Radiation Parameter 6

10 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani Figure 7: Inluence o Radiation parameter and Darcy s Radiation parameter Figure 8: Eect o Darcy s parameter on Velocity proiles with respect to Radiation parameter 7

11 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani RESULTS AND CONCLUSIONS. The nature o velocity proiles with respect to the Darcy s and porosity o the luid bed are illustrative in Fig-, Fig-, Fig-, Fig-4. In Fig-it is noticed that, or a constant value o porosity, as the Darcy s parameter increases the velocity also increases. The no slip condition is perectly satisied on the boundary. For a similar values o Darcy s parameter and when K =. the no slip condition is also satisied. However, a backward low is observed as we move away towards the other bounding surace. Similar such a trend as stated above is noticed when K =.6. The velocity proiles are observed to be more dispersed as we approach the other boundary. Even or, K =. not much o signiicant trend in the velocity proiles is observed. Moreover, there seems to be no change in the nature o velocity proiles.. The inluence o radiation and Darcy s parameter on the nature o velocity proiles is shown in Fig-5, Fig-6, Fig-7 and Fig-8. In Fig-5, when the radiation parameter is held constant, it is observed that as the Darcy s number increases the no slip condition is satisied. But a backward low is observed. For such similar values o Darcy s number and when R=7, the velocity proiles are seen to be o same type. However, the magnitude o the backward low is noticed to be slight altered. In Fig-7, when the radiation parameter is (R = ), and or same values o Darcy s parameter, a slight change in the velocity proiles is noticed. As usual and as discussed above, more o backward low is observed in this case. An interesting point in this case is that the velocity proiles are merged till 4% o the channel width. Thereater, the dispersion in the velocity proiles is observed. Fig-8 depicts the variation in the velocity proiles when R= and the Darcy s number ranging rom. to.4. In this case also, the low appears to be more o backward and subsequently in the orward direction. The conditions o the no slip are satisied in this situation also. REFERENCES Cheng P and Minkowycz W (977). Free convection about a vertical lat plate embedded in porous medium with application to heat transer rom a dike. Journal o Geophysical Research Darcy H. (856). Les Fountaines publiques de la ville de Dijon, victor Dulmont, Paris. Dunn JE and Rajagopal KR (995). Fluids o dierential type critical review and thermodynamic analysis. International Journal o Engineering Sciences Hassanien IA, Bakier AY and Gorla RSR (998). Eects o thermal dispersion and stratiication on non-darcy mixed convection rom a vertical plate in a porous medium. Heat and Mass Transer 4 9. Joseph DD, Nield DA and Papanicolaou G.(98). Non linear equation governing low in a saturated porous medium. Water Resources Research Knupp PM and Lage JL (995). Generalization o the Forchheimer - extended Darcy low model to the tensor permeability case via a variational principle. Journal o Fluid Mechanics Kuznetsov AV (). Investigation o the eect o transverse thermal dispersion on orced convection in porous media. Acta Mechanica

12 International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani Mac. Donald, FEl-Sayed I, Mow MS and Dullien (979). Flow through porous medium the Ergun equation revisited. Industrial Chemistry Fundamentals Murthy PVSN and Singh P (977). Thermal dispersion eects on non-darcy natural convection with lateral mass lux. Heat and Mass Transer -5. Murthy PVSN (998). Thermal dispersion and viscous dissipation eects on non-darcy mixed convection in a luid saturated porous medium. Heat and Mass Transer 95-. Rajagopal KR and Gupta AS (984). An exact solution or the low o a non-newtonian luid past an ininite porous plate. Mecanica