Unsteady Flow of a Viscoelastic Fluid through a Porous Media between Two Impermeable Parallel Plates

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1 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (): 5-9 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (): 5-9 (ISSN: 4-706) Scholarlink Research Institute Journals, 00 (ISSN: 4-706) jeteas.scholarlinkresearch.org Unstead Flow of a Viscoelastic Fluid through a Porous Media between Two Impermeable Parallel Plates T.Gnana Prasuna, M.V.Ramana Murth,.Ch.Pattabhi Ramacharulu G.Venkateswara Rao Department of Mathematics, Osmania Universit, Hderabad, India Professor (Retired) in Mathematics, NIT, Warangal, India.. Department of Mathemtics, Govt. Degree College, Hderabad, India. Correspondence Author: T. Gnana Prasuna Abstract An unstead flow of a viscoelastic fluid through a porous media between two impermeable parallel plates is examined. Initiall, the flow is generated b a constant pressure gradient parallel to the bounding fluids. After attaining the stead state, the pressure gradient is suddenl withdrawn and the resulting fluid motion between the parallel plates is investigated. The problem is solved in two stages the first stage is a stead motion between the parallel plates under the influence of a constant pressure gradient. The plates however are impermeable i.e., no cross -flow is allowed across the boundar plates. The momentum equation does not involve the visco-elastic parameter however; the influence Darcian friction would appear in it. The solution of the momentum equation at this stage will be the initial condition for the subsequent flow. The second stage concern with an unstead motion for which the initial value for the velocit will be that obtained in stage one, together with the no slip condition on the boundar plates. The problem is solved emploing aplace transformation technique. Expressions for the flow rate and shear stress on the walls are computed. Kewords: Oscillator flow, Second order visco elastic fluid, Porous medium I TRODUCTIO Viscoelastic fluid flow through porous media has attracted the attention of scientists and engineers because of its importance notabl in the flow of oil through porous rock, the extraction of energ from geothermal regions, the filtration of solids from liquids and drug permeation through human skin. The knowledge of flow through porous media is useful in the recover of crude oil efficientl from the pores of reservoir rocks b displacement with immiscible water (vide, Rudraiah et al., 979). The flow through porous media occurs in the ground-water hdrolog, irrigation, drainage problems and also in absorption and filtration processes in chemical engineering. This subject has wide spread applications to specific problems encountered in the civil engineering and agriculture engineering, and man industries. Thus the diffusion and flow of fluids through ceramic materials as bricks and porous earthenware has long been a problem of the ceramic industr. The Scientific treatment of the problem of irrigation, Soil erosion and tile drainage are present developments of porous media. In hdrolog, the movement of trace pollutants in water sstems can be studied with the knowledge of flow through porous media. The principles of this subject are useful in recovering the water for drinking and irrigation purposes. Thurson (97) was the earliest to recognize the viscoelastic nature of blood and that the viscoelastic behavior is less prominent with increasing shear rate. amb (9) has discussed the viscous flow over an oscillator bottom earlier in his treatise on hdrodnamics. Berman (958) studied the problem of two dimensional stead state Newtonian laminar flow in a channel with porous walls. An exact, analtical expression for the dependence of velocit on the pressure gradient has been derived. The response of a Colemann-Noll (960) Second order visco elastic fluid occuping a semi infinite region due to harmonic oscillation of its bottom has been investigated later b Pattabhiramacharulu (964, 966). Vidanidhi, V.and Nigam S.D. (967) studied secondar flow in rotating channel. Oscillator motion of an electricall conducting viscoelastic fluid over a stretching sheet in a saturated porous medium was studied b K. Rajagopal (984). Visco-elastic effects of non- Newtonian flow through porous media was studied b Gupta R.K. and Sridhar T (985). Pascal and Pascal (989) studied the visco elastic effects in non - Newtonian stead flows through porous media. Flow of a visco elastic fluid over a stretching sheet was studied b Rajagopal (006). Ariel P.D. (994) studied the flow of a visco elastic fluid past a porous plate. Petrov A.G. (000) examined analticall the unstead 5

2 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (): 5-9 (ISSN: 4-706) flow of Bingham fluid caused b a abruptl applied pressure gradient. With respect to the flows of non- Newtonian fluids between two parallel porous walls, Ariel (00) obtained exact analtical solutions of laminar flow of a second grade visco elastic fluid emploing two geometries. Oscillator plate temperature effects of free convection flow of dissipative fluid between long vertical parallel plates was studied b Narahari (009).Singh and Paul (006) presented an analsis of the transient free convective flow of a viscous incompressible fluid between two parallel vertical walls occurring as a result of asmmetric heating/cooling of the walls. MATHEMATICA FORMUATIO OF THE PROBEM This paper deals with an investigation of an unstead flow of a second order viscoelastic fluid between two impermeable horizontal parallel plates, separated b a distance h. The flow is governed b a generalized momentum equation which takes care of shear stress generating the flow medium and also the inertial convective acceleration apart from conventional Darc s resistive force. Initiall the flow is under a constant pressure gradient down the bounding plates. After attaining the stead state the pressure gradient is suddenl withdrawn and the subsequent fluid motion is investigated b emploing aplace Transform technique. The problem is solved in two stages; the first stage is the stead motion between the parallel plates under the influence of a constant pressure gradient. The momentum equation is free of the viscoelastic parameter while the Darcian friction would find its place in it. The solution of the momentum at this stage will be the initial condition for subsequent flow. The second stage is an unstead motion for which the initial velocit is the same as that obtained in earlier stage together with no slip condition on the boundar plates. The plates however are impermeable i.e. no cross flow is allowed across the boundar plates. The momentum equation of the fluid flowing through a generalized porous medium as suggested b Yamamoto and Yamamoto Iwamura (974) is given b ρ dq divs µ q () dt k and the continuit equation for incompressible homogeneous fluid div q 0 () Where, ρ - fluid densit (constant), q - fluid velocit, S - stress tensor,µ -the co-efficient of Newtonian viscosit, k - co-efficient of permeabilit. The stress strain rate relations of a second order viscoelastic fluid as given b Noll and Colemann is given b S PI + φ φ + φ () () ( ) () A + A A ( ) ( ) ( ) Where A E v + v ij, j j, and ( ) ( ) ( ) m ( ) ( ) ( ) m A Aij + v A j, m + A m v i, j + A t where S: the stress tensor, p: Isotropic mean pressure, v i : the velocit in the ith direction,, φ φ υ, υc, β are the material constants ( φ ) ( ), where ρ is the fluid densit. It can be noted that υ is the classical of the viscosit, υ is the cross viscosit and β is visco elasticit c coefficient. With reference to a sstem of cartesian coordinates (X, Y, Z) with the X-axis parallel to the plates placed midwa between them. The fluid velocit can be taken as (U(Y, T), 0, 0). Now the stress tensor is given b S XX U ρ + φ, U S YY ρ + ( φ + φ ) (4) U U S XY φ + φ, where φ νρ, T φ βρ. φ ρν c The momentum equation in the x direction reduces to U P U U υ (5) + υ + β U T ρ X T k The following non-dimensional quantities are introduced to make the basic equations and the boundar conditions dimensionless. Y h, U ϑ u, ρυ P p,[ β ] h, h h [ k ] h, T h t, ν Σ σ υ h Where h is characteristic length. (Which is taken to be half the distance between the plates). In view of the above dimensionless quantities the equation of motion (5) becomes p u u 0,in stead state (6) + x and u u u u + +,in unstead t t state and the boundar conditions are u 0 when and u 0 when - (7) ( ) mj v ( ) m j, i 6

3 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (): 5-9 (ISSN: 4-706) SOUTIO OF THE POBEM Case (i) Stead State; On solving the equation (6) with the boundar conditions (8), we get the initial velocit (as the viscoelastic parameter disappears) as u(,0) u * () (9) c Flow rate is given b Q c tanh And the skin friction on the plates u c tanh at () Case (ii) unstead State u n ( ) (0) On appling aplace transformation, equations (7) and (8) becomes d u m d c u + c s s + + Where ( + s ) m ( + s ) u ( ±, s) 0 On solving equation () together with the boundar conditions (), we get (4) c m c c u + s(+ s) m (+ s) s On appling the inverse aplace Transformation for equation (4),we get the velocit as t S t n n (, t ) e cos + c e c + ( ) π π n n Where S n + π + π Flow rate is given b Q u d π 8 n n Snt ( ) n n ( n ) e { S And the skin friction on the plates u u τ x ] ± + ] ± t ( ) n e S n t c { e } t DISCUSSIO OF THE RESUTS Intiall the flow is under a constant pressure gradient and hence, the velocit profiles are perabolic tpe and smetric about the channel central line. The pressure gradient is suddenl with drawn and subsequent flow is investigated. It is noticed from the figures (-5) that the initial velocit profile is contracted pushedbackward in a core aound the centrl line. This results in a raising of the velocit near the channel boundaries. The phenomina of the velocit profiles is predominantl noticed as time increases and this can be atributed to the resistance due to the prosit of the medium. In flow rate (figure 6) it is observed that it } slowl decreasing almost linearl as the co-efficient of visco elasticit and porosit increases. As time increeases it comes to rest. (5) ACK OWEDGEME T Authors express their sincere thanks to Universit Grants Commission; New Delhi for sanctioning DRS/SAP to the Department of Mathematics whose resources are used for this research work. REFERE CES Ariel P. D. 00. On exact solutions of flow problems of a second grade fluid through two parallel walls, International Journal of Engg. Science. 40:9-94. Ariel P.D The flow of a visco elastic fluid past a porous plate, Acta Mech.07: Berman A.S., 958. aminar flow in an annulus with porous walls. Journal. of Appl. Ph. 9:7-75. Coleman, B. D., Mankovitz, H. and Noll, W Viscometric flows of non Newtonian fluids theor and experiment, Springer Verlog:09-7. Colemann B. D.and Noll W 960. Archieves Rational Mechanics and analsis. 6: (6) (7) 7

4 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (): 5-9 (ISSN: 4-706) Gupta R. K. and Sridhar T 985. Visco-elastic effects in Non-Newtonian flow through porous media, Rheol, Acta. 4:48-5. APPE DIX amb. H. 9. Hdrodnamics 6 th Edition, Dover: medium, Journal of Engg Mathematics.0:4-54. Narahari M Oscillator plate temperature effects of free convection flow of dissipative fluid between long vertical parallel plates, International. Journal. of Appl. Math.and Mech, 5:0-46. Pascal. H and Pascal 989. On Visco elastic effects in non Newtonian stead flows through porous media.transport in Porous Media A:7 5. Pattabhi Ramacharulu N.Ch 964. Applied Science Research, 4(A):68. Figure : Unstead state velocit for different values of t with, 0. Pattabhi Ramacharulu N.Ch 966. The problem in the stud of Non- Newtonian fluid dnamics, Phd Thesis,submitted O.U., Hderabad, India, Petrov A.G The development of the flow of viscous and viscoelastic media between two parallel plate. Journal of Appl. Mathematics Mech. 64:-. Rajagopal K, Veena P.H., and Pravin V. K Oscillator motion of an electricall conducting viscoelastic fluid over a stretching sheet in saturated porous medium with suction/blowing, Mathematical problems in Engg.-4. Rajagopal K. R., Na.T. Y and Gupta. A. S Flow of a visco elastic fluid over a stretching sheet. Journal of Rheolog..:-5. Figure : Unstead state velocit for different values of t with, 0. Singh A.K,Paul T 006. Transient natural convection between two vertical walls heated/cooled asmmetricall. International Journal.Appl. Mech.and Engg ():4-54. Vidanidhi, V, and Nigam, S.D Secondar flow in rotating channel. Journal of Maths and Phsical Sciences : Yamamoto K and Iwamura N 976. convective acceleration through a porous Flow with Yamamoto K and Yosida Z 974. Flow through a porous wall with convective acceleration, Journal.of Phsical Societ of Japan.7: Figure : Unstead state velocit for different values of t with 4, 0. 8

5 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (): 5-9 (ISSN: 4-706) Figure 4: Unstead state velocit for different values of t with, 0. Figure 5: Unstead state velocit for different values of t with, 0.5 Figure 6: Flow rate for different values of Visco elasticit with., 9