Incompressible Viscoelastic Flow of a Generalised Oldroyed-B Fluid through Porous Medium between Two Infinite Parallel Plates in a Rotating System

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1 International Jornal of Compter Applications ( ) Volme 79 No., October Incompressible Viscoelastic Flow of a Generalised Oldroed-B Flid throgh Poros Medim between Two Infinite Parallel Plates in a Rotating Sstem Dhiman Bose Department of Applied Mathematics Universit of Calctta Kolkata 79, India Uma Bas Department of Applied Mathematics Universit of Calctta, Kolkata 79 ABSTRACT Incompressible viscos flid flow throgh a poros medim between two infinite parallel plates with moving pper plate in a rotating sstem has been stdied here. The exact soltion of the governing eqation for the velocit field has been obtained b sing Laplace and finite Forier sine transformations in series form in terms of Mittage-Leffler fnction. It can be fond that the flid velocit decreases with the increasing vales of fractional calcls parameter α and the permeabilit of the poros medim K. It can be also observed that the flid velocit increases with the higher vales of the viscosit of the poros medim. The dependence of the velocit field on fractional calcls parameters as well as material parameters has been illstrated graphicall. Kewords Capto operator; Generalised Oldroed-B flid; Laplace transformation: Finite Forier sine transformation; poros medim.. INTRODUCTION In flid dnamics the std of non-newtonian flid flow throgh poros medim has applications in different fields sch as prification of crde oil, petrolem indstr, polmer technolog, electrostatic precipitation, irrigation, sanitar engineering, food indstr etc. The flow behavior of non- Newtonian flids cannot be described b Newtonian flid model. For this reason varios tpes of constittive eqations have been proposed and Oldroed-B flid model is one of them that has some sccess in describing non-newtonian flids. In recent ears fractional calcls approach is fond to be qite flexible in describing the viscoelastic flids. In the approach the time derivative of integer order in the constittive eqation is replaced b Capto fractional calcls operator. Charl and Ram [] have investigated laminar flow of an incompressible micro polar flid between two parallel plates with poros lining. Feteca et al [] have stdied nstead flow of a second grade flid between two side walls perpendiclar to a plate. Ganapath [] have stdied oscillator Coette flow in a rotating sstem. Jana et al [] have stdied nstead flow of viscos flid throgh a poros medim bonded b a poros plate in a rotating sstem. Khan et al [] discssed exact soltions for some oscillating flows of a second grade flid with a fractional derivative model. Rajagopal [] investigated nstead nidirectional flows of a non-newtonian flid. Tan et al. [7] discssed exact soltion for nstead coette flow of the generalized second grade flid. Wenchang et al [8] have stdied nstead flows of viscoelastic flid with the fractional Maxwell model between two parallel plates. In present work we have stdied the viscoelastic flow of a generalized Oldroed-B flid throgh poros medim between two infinite parallel plates in a rotating sstem. Here we have sed fractional calcls approach in finding exact soltion for the velocit field b replacing the time derivative of integer order with Capto Fractional calcls operator. The exact soltions for the velocit fields are obtained b tilizing the integral transformations in series form in terms of Mittage-Leffler fnction. We have focsed on the behavior of the velocit fields with change in vales of porosit parameter and fractional calcls parameters.. CONSTITUTIVE AND GOVERNING EQUATION The constittive relation involving the Cach Stress tensor T in a homogeneos and incompressible Oldroed-B flid with fractional calcls model can be proposed as where p is the hdrostatic pressre, I is the identit tensor, λ is the time of relaxation, is the time of retardation, μ is the coefficient of viscosit of the flid, S is the extra stress tensor, α and β are the fractional calcls parameters, V is the flid velocit, is the Rivlin-Erickson tensor. and are Capto fractional calcls operators of order α and β respectivel defined b Where is the Gamma Fnction. We choose a Cartesian co-ordinate sstem with x-axis along the lower plate in the direction of the flow, -axis normal to the plates and z-axis perpendiclar to the x-plane. We assme the velocit field of the form where and are the velocit components in the x- and z-coordinate directions that are taken along the ()

2 International Jornal of Compter Applications ( ) Volme 79 No., October direction of the parallel plates and normal to the x-plane respectivel. Frther we assme the stress of the form () Sbstitting the Eqations () and () in Eqation () and taking accont the initial condition we get the following Eqations with The eqations of motion are given b and where are viscosit of the poros medim, flid densit, viscosit of the flid and permeabilit of the poros medim respectivel. Eliminating between the Eqations () and (8) we get the following governing eqation with the bondar conditions and () and initial conditions () Combining the Eqations () and (), we have Again eliminating between the Eqations (7) and (9) we get the following governing eqation where (7) sbject to the bondar and initial conditions for t > is the kinematic viscosit.. FORMULATION OF THE PROBLEM Let s consider the nstead flow of a generalized Oldroed-B flid throgh a poros medim bonded b two infinite parallel plates in a rotating sstem. The plates and the flid are initiall at rest and at the entire sstem begins to rotate with anglar velocit abot the -axis and at the same time the pper plate moves with constant velocit in the x-direction. We take the velocit field of the form where, w are the velocit components along the x-direction and perpendiclar to x-plane respectivel. The governing eqations are We introdce the non-dimensional variables Then the Eqation () can be written in terms of nondimensional variables as are non- where dimensional variables.

3 International Jornal of Compter Applications ( ) Volme 79 No., October Dropping the asterisk sign we get the dimensionless governing eqation as Inserting we get from the Eqation () and sing the binomial theorem constant where is complex The dimensionless bondar and initial conditions are for t > Mltipling both sides of the Eqation () b and then integrating with respect to from to and sing the bondar conditions (a) we get the following eqation where is the finite Forier sine transformation of defined b Now we have an important Laplace transformation of the Mittage-Leffler fnction Taking Laplace transformation of both sides of Eqation () and sing we get Where is the Mittage-Leffler fnction () Where is the Laplace transformation of defined b, s is the Laplace transform parameter. In order to avoid the length calclation of contor integration and resides we rewrite the Eqation () in series form as the following Taking inverse Laplace transformation we get from the Eqation()

4 International Jornal of Compter Applications ( ) Volme 79 No., October The non-dimensional shear stresses at the stationar plate ( ) de to the primar and secondar flows is given b Taking inverse Forier sine transformation of the Eqation (9) and comparing the real and imaginar parts of both sides of the reslting eqation we get Separating the real and imaginar parts of both sides of the Eqation () we get the shear stress components de to the primar and secondar flows at the stationar plate as

5 International Jornal of Compter Applications ( ) Volme 79 No., October Figre. It is seen from the figre that w increases with the increase in β bt the parabolic natre of the flow pattern remains fixed. Figre depicts the shear stress at the stationar plate de to primar flow against the kinematic viscosit for different vales of the permeabilit parameter of the poros medim. It is evident from the figre that shear stress decreases with increase in. It is observed from the Figre that as α takes higher vales the shear stress decreases. Figre shows the dependence of on the fractional calcls parameter β. As β takes higher vales, the shear stress increases. Figre depicts the shear stress against kinematic viscosit at the stationar plate de to the secondar flow for different vales of α. It is seen from the figre that decreases with the increase in α. Figre reveals that the shear stress increases with the increase in the fractional calcls parameter β.. =. =. =.... RESULTS AND DISCUSSION Figre depicts the behavior of the primar velocit component against the distance from the lower plate measred along -axis for different vales of the fractional calcls parameter α. From the figre it is observed that as α takes higher vales the flow velocit decreases and the profile of the velocit crve changes from parabolic slope. Figre depicts the primar velocit for three vales of the rotational parameter. The flow velocit decreases with increase in and this is similar to the case in Figre. Figre explains that the flow velocit decreases with the increasing vales of permeabilit parameter that is porosit prodces a resistance force in the flow field. The natre of flow patterns are slightl deviated from the parabolic tpe. Figre depicts the velocit component against the distance from the lower plate measred along -axis for different vales of the fractional calcls parameter β. As β increases the flow velocit also increases and the natre of the velocit crves are more parabolic with increase in β. It is observed from the Figre that the flow velocit increases with the increase in the parameter of viscosit The velocit field is plotted in Figre 7 against distance from the lower plate at different time t. It is noted from the figre that the velocit field decreases and the natre of flow pattern are less parabolic with increase in time t. Figre 8 depicts the velocit field against for different vales of the kinematic viscosit The flow velocit, increases with the increase in. It is observed from the Figre 9 that the secondar velocit w decreases with increase in the parameter α and the parabolic natre of the flow pattern is not effected b α. The inflence of the parameter β on the secondar flow velocit w is illstrated in Figre :The velocit field is depicted against the distance from the lower plate for different vales of the fractional calcls parameter α. 7 =. =.7 = Figre : The velocit field is depicted against the distance from the lower plate for different vales of 7

6 w International Jornal of Compter Applications ( ) Volme 79 No., October. K=. K=. K=. t=. t=.7 t= Figre : The velocit field is depicted against the distance from the lower plate for different vales of parameter of permeabilit Figre 7: The velocit field is depicted against the distance from the lower plate at different time t. =.8 =.8 =.9. =. =. = Figre : The velocit field is depicted against the distance from the lower plate for different vales of fractional calcls parameter Figre 8: The velocit field is depicted against the distance from the lower plate for different vales of material parameter.. =. =. =... =. =. = Figre : The velocit field is depicted against the distance from the lower plate for different vales of viscosit parameter Figre 9: The velocit field w is depicted against the distance from the lower plate for different vales of fractional calcls parameter. 8

7 w International Jornal of Compter Applications ( ) Volme 79 No., October.. =. =.7 =.8 8 =. =. =. =.7 =.8 =.9 =.... x Figre : The velocit field w is depicted against the distance from the lower plate for different vales of fractional calcls parameter Figre : The shear stress at the stationar plate de to the primar flow is depicted against the kinematic viscosit ν for different vales of parameter β. 7 K=. K=.7 K=.8 K=.9 K= K= K= K= 7 =. =. =. =. =.8 =. x Figre : The shear stress at the stationar plate de to the primar flow is depicted against the kinematic viscosit ν for different vales of permeabilit of poros medim K Figre : The shear stress at the stationar plate de to the secondar flow is depicted against the kinematic viscosit ν for different vales of parameter x 7 =. =. =. =. =. =.7 =. 7 =. =. =.7 =.8 = Figre : The shear stress at the stationar plate de to the primar flow is depicted against the kinematic viscosit ν for different vales of parameter α Figre : The shear stress at the stationar plate de to the secondar flow is depicted against the kinematic viscosit ν for different vales of parameter. 9

8 International Jornal of Compter Applications ( ) Volme 79 No., October. CONCLUSION The flow of an incompressible generalised Oldroed-B flid throgh a poros medim between two infinite parallel plates in a rotating sstem is considered. The exact soltion for the velocit field is obtained b tilizing Laplace and finite Forier sine transformation in series forms in terms of Mittage-Leffler fnction. The inflence of the fractional calcls parameters as well as material parameters on the velocit field has been illstrated graphicall. Moreover the effects of permeabilit parameter of the poros medim, fractional calcls parameters α and β on the shear stresses and de to the primar and secondar velocit components respectivel have been discssed graphicall.. ACKNOWLEDGEMENTS The athors wish to thank the reviewers for sefl comments which have lead to the improvement of or work in the present form. 7. REFERENCES [] Charl, V.N. and Ram, M.S.. Laminar flow of an incompressible micro polar flid between two parallel plates with poros lining. Int. J. Applied Math and Mech., 8-9. [] Feteca, C., Haat, T., Feteca, C. and Ali, N. 8. Unstead flow of a second grade flid between two side walls perpendiclar to a plate. Nonlinear Anal: Real World Appl. 9, -. [] Ganapath, R. 99. A note on oscillator Coette Flow in a rotating sstem. Jornal of Applied Mechanics., 8-9. [] Jana, M., Maji, S.L.,Das, S.and Jana, R.N.. Unstead flow of viscos flid throgh a poros medim bonded b a poros plate in a rotating sstem. Jornal of Poros Media., -. [] Khan, M., Ali, S.H and Qi, H. 9. Exact soltions for some oscillating flows of a second grade flid with a fractional derivative model. Mathematical and Compter Modelling. 9, 9-. [] Rajagopal, K.R.98. A note on nstead nidirectional flows of a non-newtonian flid. International Jornal of Non-Linear Mechanics. 7, 9-7. [7] Tan, W.C., Xian, F. and Wei, L.. Exact soltion for the nstead Coette flow of the generalized second grade flid. China Sci. Bll. 7,-8. [8] Wenchang,T.,Wenxiao, P. and Ming, X.. A note on nstead flows of a viscoelastic flid with the fractional Maxwell model between two parallel plates. International Jornal of Non-Linear Mechanics. 8, -. IJCA TM :

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