topic is fluid mechanicsuse equation 1,2,3,4and all given quantities if required and obtain equation 8,9,10 Effects of Oscillatory Flow of a Couple stressFluid over a Semi-Infinite Vertical PermeablePlate in a Porous Medium with ViscousDissipation and Ohmic EffectSuresh Babu R¹, Rushi Kumar B², Dinesh P.A¹Department of Mathematics, Ramaiah Institute of Technology, Bengaluru, India¹School of Advanced Sciences, Fluid Dynamics Division, VIT University, Vellore, India ²ABSTRACT: This problem deals with unsteady, two-dimensional, Boussinesq, laminar, oscillatory flow of anincompressible couple stress fluid over an infinite vertical permeable plate through the porous medium under theinfluence of viscous dissipation and Ohmic effects is analytically. The highly non-linear coupled partial differentialequations are modelled for the boundary layer flow, heat and mass transfer using fully developed flow conditions andare transferred into a set of linear PDE's using non-dimensional parameters. The analytical solutions of the resultingboundary layer equations are assumed of oscillatory type and are solved using a two-term regular perturbationtechnique. Numerical computation are performed to analyze the effect of various non-dimensional parameters of thephysical model like Grashof number Gr Prandtl number Pr, Viscous ratio, Permeability parameter k, MaterialBparameter and Frequency number on velocity, temperature and angular velocity profiles and are presentedgraphically. The present results are compared with earlier works in the absence of some parameters of the model andfound to be in good arrangement.KEYWORDS: Oscillatory flow, Couple stress fluid, Viscous Dissipation, Ohmic effect, Porous medium.I. INTRODUCTIONIn the last five decades, convective heat and mass transfer flow in a porous medium has been a subject of greatinterest because of its numerous industrial, thermal and chemical engineering applications in different disciplines likemedicine, science and technology. For example, blood flow in a tube, packed-bed catalytic reactors, cooling of nuclearreactors and underground energy transport, enhanced oil recovery system, thermal insulation, geothermal reservoirs,drying of porous solids, tidal waves etc. It is very much interesting to study the effects of the free convection currentson the oscillatory flow type of boundary layer flow and it is very important for technological point of view. Such astudy was first initiated by Lighthill by considering an incompressible, 2-dimensional viscous fluid flow by assumingthat the flow is a regular fluctuating superimposed flow in a boundary layer.Soundalgekar [1] analysed a two-dimensional flow of an incompressible viscous fluid past an semi-infinitevertical porous plate in a porous medium with constant suction. Rapits et al. [2] studied free convection currents ofmass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction.analytically. An extensive study of free convection effects on the oscillatory flow of a couple stress fluid through aporous medium was discussed by Hiremath and Patil [3]. Raptis and Thakur [4] presented an analytical solutions forcouple stress fluid which is heated from below in porous medium. Magyari and Pop [5] also performed analyticalsolutions for unsteady free convection flow over a vertical plate through a porous media.Many transport processes occurring both in nature and in industries involves fluid flows with the combined heatand mass transfer and it plays an essential role in transport phenomena. Such flows are driven by multiple buoyancyeffects which are arising from the density variations due to temperature and species concentration. Mixed convectiveCopyright to UIRSETwww.ijirset.com111IJIRSETISSN(Online): 2319-8753ISSN (Print): 2347-6710International Journal of Innovative Research in Science,Engineering and Technology(An ISO 3297: 2007 Certified Organization)Vol. 6, Special Issue 13, July 2017flows in a porous medium have been examined during the last four decades due to its practical applications which canbe modelled or approximated of a boundary layer flow. Such a study was illustrated by Rudraiah et al.[6] on mixed.convective oscillatory flow in a vertical porous stream and Ogulu [9] On the oscillating plate temperature flow of apolar fluid past a vertical porous plate in the presence of couple stresses and radiation. Many authors like Youn Kim [7].Chamkha [8], Sahin Ahmed [10], Umamaheswari [11] and Raju et al. [12] have done their extensive research work byconsidering an additional MHD effect past a vertical plate in a porous medium. Comprehensive literature surveys,references regarding the subject of porous medium can be hand in recent books by Nield and Bejan [13].The main objective of this work is, therefore, to study effects of oscillatory flow of a couple stress fluid over asemi-infinite vertical permeable plate in a porous medium along the viscous dissipation and Ohmic effects analyticallywith oscillatory suction at the vertical porous plate and oscillatory free stream velocity away from the plate in a porous.medium. To achieve the objective of the present work, the plan of this work is as follows. We consider in the nextsection a vertical porous plate embedded in a fluid saturated porous medium with oscillatory suction velocity at thevertical plate. In this section we also give the conservation equations for momentum and energy with suitable boundaryand inertia effect by considering Darcy-Lapwood-Brinkman equation. The relevant physical parameters are alsodiscussed in this section. These basic equations involve variable coefficients which are solved analytically using regularperturbation technique. These solutions are numerically computed and the results are discussed in the last section.II. MATHEMATICAL FORMULATIONWe consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi-infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite tothe direction of gravity and the v-axis is taken exactly perpendicular to the porous plate. We assume the following We consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi-infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite tothe direction of gravity and the y-axis is taken exactly perpendicular to the porous plate. We assume the followingassumptions in order to find the analytical solution for the above problem. x varies between to + allphysical quantities are independent of x except pressure; density is constant throughout the momentum equationexcept for body force; density pis a function of temperature only, hence P=Pa [1-B(T-T₂)].. oscillatorysuction velocity and the free stream velocity far away from the porous plate oscillates about a mean constant value ina direction parallel to the x-axis.T--50-A²)|₁Fig.1: Physical model.Under these assumptions, the basic governing equations of motion aredu ду+= 0,dx ду(1)du du1 opd²udo+V =ct dy+ Bg(T-T).Po ox-u+2vdy² kдуCopyright to UJIRSET112IJIRSETISSN(Online): 2319-8753ISSN (Print): 2347-6710International Journal of Innovative Research in Science,Engineering and Technology(An ISO 3297: 2007 Certified Organization)Vol. 6, Special Issue 13, July 2017ar OT 8²TU duU+v- =K++Ətdy dy² CaykCP(3)dodo a d¹o+V=ôt дуI by²(4)The associated boundary conditions for the physical model aredoy=0u = 0,==T = T₁,dydy²yoo: u=U (1),@→0, T→T.(5)Making these equations dimensionless using the known physical quantities asuv²tv²kvnWT-TVy² = 10)u U =t' =n' =k=.y =2VVVoVoT-TBug (T-T)Pr=0,UoGr=@U.V.E=2©* =UovoC, (T.-T.)D(6)The fluctuating free-stream and suction velocities respectively defined as,U(t)=U₁ (1+ A&ent) v(t)=-v₁ (1+ Beeint)(7)andUsing Eq.(6) and (7),du du du λrewriting equations (2), (3) and (4) in the dimensionless form are,du- V₁ (1+ Bε e¹¹)Bɛ=до+ +- (U−u)+2dt oy² kду+Gr 0,Otду(8)2де80-VVo(1+ BE 0818²0+Eby Proy2 dv=+atdodo1d²w-Vo(1+Beeint)=diby Boy²The corresponding dimensionless boundary conditions takes the form,doy=0:u = 0,==0 = 1,дуÔy ²yo: u(1+& Ae), @→ 0,0→0.III. METHOD OF SOLUTION+UCT↑T-T.2VoOktyE-E₂0-B²)ýwww.ijirset.com.0=(2)(9)(10)(11)

From the given journals: ρ=ρ01-βT-T∞ Assume: γ=1γ=λβ=1 Governing equations of motion: ∂u∂x+∂v∂y=0…

We consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi- infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite to the direction of gravity and the y-axis is taken exactly perpendicular to the porous plate. We assume the following assumptions in order to find the analytical solution for the above problem. x varies between 0 to +00; all physical quantities are independent of x except pressure; density is constant throughout the momentum equation except for body force; density pis a function of temperature only, hence P=Pa [1-B(T-T.)]; oscillatory suction velocity and the free stream velocity far away from the porous plate oscillates about a mean constant value in a direction parallel to the x-axis. du du 1 др +V at ду Po êx Fig.1: Physical model. Under these assumptions, the basic governing equations of motion are ди av +- = 0, dx ду Copyright to IJIRSET ar at y = aT +V=K Pr= == dt 20 at dy dy Voy V K * 8²T V du + Cay +U- u= d²u V dy² k do do a do +1== at dy 1 y2 The associated boundary conditions for the physical model are do 0²u y=0:u=0, T = T₁, dy dy² U U₁ IJIRSET International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization) Vol. 6, Special Issue 13, July 2017 -(1+Bee = . 150-A 1 CT 1-1. t' = yo: u=U (1), @→0. T→T. Making these equations dimensionless using the known physical quantities as vn v²k Vo @U.V. Gr=Bug (T-T.) Gr= + v²t V 00 10²0 by Pr oy u+ 2v² U . +E KC p do -u², == dy . 11 = E= 20 ду U² C, (T.-T.)' The fluctuating free-stream and suction velocities respectively defined as, U (1)=U₁ (1+ A&et) v(t) = -v₁ (1+ Bee) Bɛent) Vo do 8²u ду dy² yo: u(1+& Aet), @→0. + Bg(T-T₂). The +(U-u)+2- E-E₂0-B +: y = 0=1, www.ijirset.com 0→0. V Vo do -V₁ (1+Bɛ y Boy² dt -Vo(1+Bee)_10² The corresponding dimensionless boundary conditions takes the form, y = 0: u = 0, . ; k= and (7) Using Eq.(6) and (7), rewriting equations (2), (3) and (4) in the dimensionless form are, du до - ₁ (1+ Beem) ou du ¹²u 2 +Gr 0, ду dt oy² k ду D 0= (1) ISSN(Online): 2319-8753 ISSN (Print): 2347-6710 T-T T-T (2) 112 (3) (4) (5) (6) (8) (9) (10) (11)