Diffusion thermo effects on unsteady MHD free convection flow of a Kuvshinski fluid past a vertical porous plate in slip flow regime

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IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Diffusion thermo effects on unsteady MHD free convection flow of a Kuvshinski fluid past a vertical porous plate in slip

1 IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Diffusion thermo effects on unsteady MHD free convection flow of a Kuvshinski fluid past a vertical porous plate in slip flow regime To cite this article: Sivakumar Narsu and B Rushi Kumar 07 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancemes. Related coe - Unsteady MHD free convection flow of rotating Jeffrey fluid embedded in a porous medium with ramped wall temperature N A Mohd Zin, I Khan and S Shafie - Variable mass diffusion effects on free convection flow past an impulsively started infinite vertical plate B Rushi Kumar, R Jayakar and A G Vijay Kumar - Soret and Dufour effects on convective heat and mass transfer in stagnation-poi flow towards a shrinking surface Krishnendu Bhattacharyya, G C Layek and G S Seth This coe was downloaded from IP address on 7//08 at 8:37

2 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 Diffusion thermo effects on unsteady MHD free convection flow of a Kuvshinski fluid past a vertical porous plate in slip flow regime Sivakumar Narsu and B Rushi Kumar Departme of Mathematics, School of Advanced Sciences, VIT University, Vellore- 6304, India rushikumar@vit.ac.in Abstract. The main purpose of this work is to investigate the diffusion-thermo effects on unsteady combined convection magneto-hydromagnetic boundary layer flow of viscous electrically conducting and chemically reacting fluid over a vertical permeable radiated plate embedded in a highly porous medium. The slip flow regime is applied at the porous ierface a uniform magnetic field is applied normal to the fluid flow direction which absorbs the fluid with suction that varies with time. The dimensionless governing equations are solved analytically using two terms harmonic and non-harmonic functions. The expressions for the fields of velocity, temperature and conceration are obtained. For engineering ierest we also calculated the physical quaities the skin friction coefficie, Nusselt and Sherwood number are derived. The effects of various physical parameters on the flow quaities are studied through graphs and tables. For the validity, we have checked our results with previously published work and found good agreeme with already existing studies. Nomenclature. x, y dimensional distances Du Dufour number t dimensional time K dimensional chemical reaction parameter C p specific heat at consta pressure Sh Sherwood number u, v dimensional velocity compones Cs conceration susceptibility C f Skin friction Nu Nusselt number T dimensional temperature V0 suction velocity Dm molecular diffusivity Greek Symbols C dimensional conceration coefficie of viscosity D mass diffusivity thermal conductivity Coe from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maiain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd

3 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 Tw temperature at the wall Cw dimensional conceration at the wall T dimensional temperature at free stream A suction velocity parameter C free stream dimensional conceration Qo coefficie of dimensional Q radiation absorption K permeability of the porous medium V0 suction velocity h refraction parameter g acceleration due to gravity Gr thermal Grashof number Gm conceration Grashof number KT thermal diffusion ratio consta density electrical conductivity coefficie of Kuvshinski fluid thermal conductivity kinematic viscosity T thermal expansion coefficie c conceration expansion coefficie Superscripts dimensional properties 'differeiation with respect to y Subscripts w, wall and free stream condition. Iroduction Convective flows with concurre heat and mass transfer under the influence of a magnetic field, chemical reaction, and thermal radiation arise in many transport processes that has applications in many branches of science and engineering. This phenomenon plays a vital role in the chemical industry, chemical vapor deposition on surfaces, cooling of nuclear reactors, power and cooling industry for drying, and petroleum industries. Recely Anjalidevi and Kandasamy [], Elbashbeshy [], Shateyi and Motsa [3],Raju and Chamka et al. [4] are studied the Soret effect due to mixed convection on unsteady magnetohydrodynamic flow past a semi infinite vertical permeable moving plate in presence of thermal radiation, heat absorption and homogeneous chemical reaction. Ibrahim et al. [5] considered the effects of chemical reaction and radiation absorption on unsteady MHD free convection flow past a semi- infinite vertical permeable moving plate with heat source and suction. Vidya Sagar and Raju et al. [6] examined an unsteady magnetohydrodynamic (MHD) radiation absorption free convective boundary layer flow of Kuvshinski fluid in the presence of a porous medium. At the macroscopic level, it is well known that the boundary condition due to viscous fluid at the solid wall is one of the no slip is said the fluid velocity of the solid boundary. While the no-slip condition has been treated experimeally to be accurate for a number of macroscopic flows, it remains an assumption that is not based on physical principles. The particle at the surface has a finite tangeial velocity. In engineering applications, the study of magneto-micropolar fluid flows in the slip flow regimes with heat transfer, for example, the power generators, refrigeration coils, electric transformers, transmission lines, and heating elemes. Khandelwal et al. [7] proposed the effects of permeability variation on the Magnetohydrodynamic (MHD) an unsteady flow of micropolar fluid through a porous medium in a slip flow regime over an infinite porous flat plate. Sharma and Chaudhary [8] analyzed free convective heat and mass transfer effect of variable suction on viscous incompressible fluid flows past a vertical porous plate in a slip flow regime. Recely, Sharma [9] demonstrated the effects of periodic temperature and conceration on unsteady free stream consisting of a mean velocity over an

4 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 exponeially varying with time. Chaudhary and Jha [0] discussed the numerical study of chemically reactive MHD micropolar fluid flow past a vertical plate in slip flow regime. Jyothi et al. [] presely chemical and thermo-diffusion effects on MHD free convection flow through a porous medium in a slip flow regime. Effects on the magnetohydrodynamic free convective vertical porous plate due to diffusion-thermo and thermo-diffusion embedded in a non-darcian porous medium have been examined by Ramachandra Prasad et al. []. Heat and mass transfer free convective MHD flow of a porous diffusion-thermo over an inclined surface has been demonstrated by Durga Prasad et al. [3]. Krishna Reddy et al. [4] studied heat and mass transfer MHD flow of Kuvshinski fluid with temperature gradie depende heat source effects over a vertical porous surface in the presence of variable heat and mass flux. Varma et al. [5] developed an analysis of magnetohydrodynamic heat and mass transfer micropolar fluid with convective diffusion-thermo effects. Sekhar et al. [6] investigated the effects of heat and mass transfer MHD free convective diffusion-thermo and radiation absorption with a porous plate in a slip flow regime. Slip effects of viscous dissipation on unsteady MHD flow over a stretching sheet studied by Sekhar and Viswanatha Reddy [7]. As per the author s knowledge, the ieraction between the chemical reaction and diffusion-thermo in the presence of radiation absorption, porous medium effects has received little atteion. Hence, an attempt is made to study the diffusion-thermo effects on an unsteady heat and mass transfer MHD free convection flow of a viscous incompressible electrically conducting Kuvshinski fluid through a porous medium from a vertical porous plate with varying suction velocity in slip flow regime.. Mathematical formulation A two dimensional unsteady magnetohydrodynamic (MHD) flow past a vertical infinite plate embedded in a porous medium has been considered with the viscoelastic fluid model. A uniform transverse magnetic field B0 is applied normal to the direction of the fluid flow direction, let x -axis taken along the flow in the vertical direction and y -axis is taken perpendicular to it. The presence of chemical reaction, Dufour effect, and a heat source or sink parameters are also considered along with buoyancy effects past a vertical porous plate with a depende heat source in slip flow regime with uniform velocity u p. The physical model of the problem as shown in Figure. Figure. Physical Model 3

5 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 In the analysis, the following assumptions are made. The flow is unsteady and laminar.. The plate is taken that sufficiely long enough is very small, so the all the physical quaities are taken as the functions of y and t only. 3. The induced magnetic field is neglected because magnetic Reynolds number is very small. 4. The effects of Viscous and Joule's dissipation are neglected. 5. Now assumed that all the fluid flow properties to be consta except that of the impact of density variation temperature. v y 0 u u u B t t y y K t T T k T Q0 DmKT c v T T Q C C t t y C y C C C y 0 v g T T T g CC C u p p s p C C C v D K C C (4) t t y y The appropriate boundary conditions are for the velocity, temperature and species diffusion fields are du u up L, T Tw Tw T e dy C Cw Cw C e at y 0 u 0, T T, C C at y Where n is consta velocity of the wall. From Eq. () it is clear that the suction velocity at the plate is a function of time t only. Let us assume that v V Ae (6) 0 Where V0 is the suction velocity and A. The negative sign indicated that the suction velocity directed towards the plate. By iroducing the non-dimensional quaities are () () (3) (5) 4

6 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 u v Vy Vt u n T T C C u, v, y, t, u, n,, C, V V V V T T C C 0 0 p p w w g T T g C C B K V K Gr, Gm, M, K,, T w c w V0 V0 V0 V0 Q CwC V DmK 0 T Cw C LV C 0 p TwTV0 CsKTw T Q,, Du, h,,pr, Q0 Sc, D C V p 0 By using the above non-dimensional quaities, the Eqs. ()-(4) can be expressed in a dimensionless form as u u u u u Ae Gr GmC M u M t t y y K K t C Ae Q C Du t t y Pr y y C C C C Ae C t t y Sc y The corresponding initial and boundary conditions in Eq. (5) in a non-dimensional form are given below: du uup h, e, C e at y0 dy u0, 0, C0 as y 3. Solution of the problem The Eqs. (8)-(0) are coupled and non linear partial differeial equations (PDEs) whose solutions in closed form are difficult to obtain, to solve these coupled non-linear partial differeial equations, we assume that the unsteady flow, so that in the neighborhood of the plate, we have uyt (, ) u( y) eu( y) O( ) 0 ( yt, ) 0( y) e ( y) O( ) C y t C y e C y O (, ) 0( ) ( ) ( ) By substituting the Eq.() io the Eq.(8)-(0)the equations are reduced to harmonic and nonharmonic terms and neglecting the higher order terms in, we obtain u o u 0 M u0 Gr 0 GmC0 (3) K uu n n M n u Gr GmC Au K 0 Pr Pr PrQC Pr DuC (5) (7) (8) (9) (0) () () (4) 5

7 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 Pr Pr n n Pr QC Pr DuC APr (6) 0 0 C ScC Sc C (7) C ScC Sc n n C AScC (8) Now, the corresponding boundary conditions are: du0 du u0 up h, u h, 0,, C0, C at 0 dy dy y u 0, u 0, 0, 0, C 0, C 0 as y (9) uyt (, ) Be Be Be m9 y m y m5 y 9 0 m y m y m3 y m5 y m7 y m 9 y e B8e B3e B4e B5e B6e B7e m5 y m 3 3 y m y m 8 5 y m y m y 6 7 ( yt, ) ( B) e Be e Be Be Be Be () m y m3y my (, ) ( ) Cyt e e B e Be () For engineering purpose, the friction factor, local Nusselt number and local Sherwood number are given by u w at y 0 y Where Vu 0 (0) f w C m B m B m B e m B m B m B m B m B m B (3) qw x Nux Tw T k Nu 0 ( B3) m5 mb3e m7b8 mb5 m3b6 m5b7 (4) Rex V0 x Where Re x c y Sh x C w C at y 0 Sh c at y 0 Re x y m e ( m )( B ) m B (5) = 3 4. Results and Discussion (0) 6

8 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 The analytical solutions are transformed for velocity, temperature and conceration profiles for differe values of physical parameters as Magnetic field parameter M, Dufour number Du, thermal Grashof number Gr, mass Grashoff number Gm, Schmidt number Sc, radiation absorption parameter Q, porosity parameter K, chemical reaction parameter, Visco-elastic parameter, Prandtl number Pr, and heat source parameter are depicted from figures -8. We assign values to the parameters as n, A ; Sc 0., t, U p.5, 0.0,Pr 0.7, Gr 5, Gm 5. The effect of magnetic field on velocity profile is plotted in Figure. 5. It is clear that the increasing values of magnetic field parameter decrease the velocity profiles. This validates the general behavior of magnetic field. Physically, the rising values of magnetic field generate opposing force to the flow direction, this force is called Lorez force. Due to this cause, we have seen a decreme in the velocity field. Figure. Effects of on Velocity profiles Figure 3. Effects of Gr on Velocity profiles Figure 4. Effects of Gm on Velocity profiles Figure 5. Effects of M on Velocity profiles 7

9 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 Figure 6 Effects of K on Velocity profiles Figure 7. Effects of on Velocity profiles Figure 8. Effects of h on Velocity profiles Figure 9. Effects of Du on Velocity profiles Figure 0. Effects of on Velocity profiles Figure. Effects of Q on Velocity profiles 8

10 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 Figure. Effects of on Temperature profiles Figure 3. Effects of on Temperature profiles 9

11 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 Figure 4. Effects of Q on Temperature profiles Figure 5. Effects of Du on Temperature profiles Figure 6. Effects of Pr on Temperature profiles Figure 7. Effects of Sc on Conceration profiles Figure 8. Effects of on Conceration profiles Figure 9. Effects of on Conceration profiles Figures 3 and 4 show the effect of thermal and conceration buoyancy forces on velocity field It is noted from that the increasing values of Gr and Gm improves the velocity field. Generally, raising the values of Gr and Gm keeps more pressure on the flow due to this we saw an increme in velocity field. 0

12 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 The porosity parameter on velocity distributions is depicted in Figure. 6. It is observed that the enhancing value of porosity parameter increases the velocity distributions. The improving values of K increases the permeability; this can lead to improving the velocity profiles. Figures, and 8 show the effect of on velocity, temperature and conceration profiles. It is observed that the increasing values of decreases the velocity, temperature, and conceration fields. This may happen due to the domination of viscous nature in the flow. The similar behavior was observed in the presence of Prandtl number it is shown in Figures. 6. The effect of on velocity and temperature fields is shown in Figures 7 and 3. It is observed that the heat source parameter depreciates the velocity and temperature fields. The quite similar behavior was observed in Figures. 7. It shows the effect of Schmidt number on conceration field. From Figures and 4 show the increase in velocity and temperature profiles with increasing values of Q. The effect of refraction parameter h on velocity profiles is shown in Figure. 8. This validates the general behavior of slip. Basically, increasing values of slip parameter improve the thickness of the boundary layer this can help to enhance the velocity profiles. Figures 0 and 9 show the effect of chemical reaction parameter on velocity and conceration profiles. It is clear that the increasing the values of chemical reaction parameter depreciate the velocity and conceration profiles. The effect of Dufour number Du on velocity and temperature profiles is plotted in Figures 9 and 5. It is observed that the increasing values of Du increases the velocity as well as temperature profiles, due to the domination of ierfacial mass transfer in the flow. Table. The variations in Skin friction, Nusselt and Sherwood number for various values of A, Gr 5, Gm0, Pr 0.7 Sc Q M K Harinath Reddy et al. [8] Prese Work ( Du 0, h 0) Cf Nu Sh Cf Nu Sh

13 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/ Comparison of the results In order to explain the accuracy of the prese results of the study, it is considered that the analytical solutions obtained are computed the numerical results for friction coefficie, local Nusselt and Sherwood numbers respectively. These calculated and compared results are preseed in Table. From this Table, it is ieresting to watch that the prese results in the absence of diffusion-thermo and refraction parameters ( h ) are in good agreeme with the corresponding results is obtained from Harinath Reddy et al. [8]. 6. Conclusion The velocity of the plate for the Kuvshinski fluid which is exposed to a transverse magnetic field can be maiained constaly, the obtained partial differeial equations (PDEs) are expressed in the dimensional form using non-dimensional quaities. These equations are solved analytically by using perturbation technique through graphs are obtained by to study the effects of differe physical

14 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 parameters on temperature, velocity, and species diffusion profiles. In the curre study, the following conclusions can be drawn:. Velocity decreases for an increase in Visco-elastic parameter, the radiation parameterq, the Schmidt number Sc, the permeability parameter K, the magnetic field parameter M, chemical reaction parameter and increases for an increase in mass Grashof number Gm.. Temperature profiles increased for an increase in, decreased due to in heat absorption coefficie, radiation parameter Q and chemical reaction parameter as well as Schmidt number Sc. 3. Velocity and temperature distributions are increased for an increase in diffusion-thermo effect Du. Appendix Sc Sc 4Sc Sc Sc 4ScB Pr Pr 4Pr m m3 m5 Pr Pr 4 Pr B4 4M 4B 7 9 m m m AScm Pr Q Pr Dum B nn B B 3 m Scm ScB m Pr m Pr Pr QPr Dum3 B m Pr mpr B4 B n n B B 6 m3 Pr m3pr B4 APr mb Pr QB Pr DuB m APr m B GrB Gm B B B B B B B m5 Pr m5pr B4 m mm Gr B up B9B0B9hm B0hm 5 B 3 0 m5 m5m hm 9 AB m GrB GmB B n n M n B B m mb GrB Gm( B ) AB m GrB B m3 m3b m5 m5 B. Acknowledgeme: We are very much thankful to anonymous reviewers for their valuable commes and suggestions to improve this manuscript. References [] Anjalidevi SP, Kandasamy R 000 J. Appl. Math. Mech [] Elbashbeshy EMA 003 Appl. Math. Comput

15 4th ICSET-07 IOP Conf. Series: Materials Science and Engineering (07) 0606 doi:0.088/ x/63/6/0606 [3] Shateyi S, Motsa SS 00 Adv. Top. Mass Trans [4] Raju MC, Chamka A J, Philip J, Varma SVK, I. J. Appl. Comput. Math. [5] Ibrahim FS, Elaiw AM, Bakar AA 008 Communications in Non-Linear Science and Numerical Simulation [6] Vidya Sagar B, Raju MC, Varma SVK, Venkataramana S 03, Review of Adv. in Phy. Theories and Applications 48-6 [7] Khandelwal K, Anil Gupta, Poonam, Jain N C Ganita () 03 [8] Sharma P K, Chaudhary R C Emirates 008 J. of Engg. Research, 8() [9] Sharma P K J. Matematicas 005 XIII() 5-6 [0] Chaudhary R C, Abhay Kumar Jha 008 Appl. Math. Mech. Engl. Ed., 9(9) [] Jyothi P, Viswanatha Reddy G, VijayakumarVarma S 03 I. J. of Adv. Eng. Tech [] Ramachandra Prasad V, Vasua B, Anwar O 03 Chemical Engineering Journal [3] Durga Prasad P, Kiran Kumar RVMSS, Mamatha B, Varma SVK 06 Global Journal of Pure and Applied Mathematics.() 4-50 [4] Krishna Reddy V, Viswaha Reddy G, Kiran Kumar RVMSS, Durga Prasad P, Varma SVK 06 East Journal of Scieific Research 4 (3) [5] Kiran Kumar RVMSS, Raju VCC, Durga Prasad P and Varma SVK Applications And AppliedMathematics, [6] Sekhar KR, Reddy GV, Varma SVK 05 I. J. of Sci. and Inno. Math.Research 3(3) [7] Sekhar KR, Viswanatha Reddy G 07 i-manager s journal on mathematics, 6() 7-34 [8] Harinath Reddy S, Raju MC, Keshava Reddy E 07 I. J. Eng. Res. in Africa

UNSTEADY MHD FREE CONVECTIVE FLOW PAST A MOVING VERTICAL PLATE IN PRESENCE OF HEAT SINK

Journal of Rajasthan Academy of Physical Sciences ISSN : 097-6306; URL : http:raops.org.in Vol.16, No.1&, March-June, 017, 1-39 UNSTEADY MHD FREE CONVECTIVE FLOW PAST A MOVING VERTICAL PLATE IN PRESENCE