Journal of Applied Science and Engineering, Vol. 20, No. 3, pp. 277 282 (2017) DOI: 10.6180/jase.2017.20.3.01 Effect of Heat Absorption on MHD Flow Over a Plate with Variable Wall Temperature U S Rajput* and Gaurav Kumar Department of Mathematics and Astronomy, University of Lucknow, Lucknow U.P, India Abstract The present study is carried out to examine the effects of heat absorption on flow model. The model consists of unsteady flow of a viscous, incompressible and electrically conducting fluid. The flow is along an impulsively started oscillating vertical plate with variable wall temperature and mass diffusion. The magnetic field is applied perpendicular to the plate. The plate temperature and the concentration level near the plate increase linearly with time. The fluid model under consideration has been solved by Laplace transform technique. The model contains equation of motion, diffusion equation and energy equation. To analyze the solution of the model, desirable sets of the values of the parameters have been considered. The numerical data obtained is discussed with the help of graphs. We found that the values obtained for velocity and temperature are in concurrence with the actual flow of the fluid. Key Words: MHD Flow, Heat Absorbing, Variable Temperature, Mass Diffusion, Hall Current 1. Introduction The unsteady flow under the action of strong magnetic field plays a decisive role in different areas of science and technology. Unsteady heat and mass transfer from a rotating vertical cone with a magnetic field and heat generation or absorption effects was investigated by Chamkha and Mudhaf [1]. MHD effects on heat transfer over stretching sheet embedded in porous medium with variable viscosity, viscous dissipation and heat source/ sink was analyzed by Dessie and Naikoti [2]. Effect of viscous dissipation and heat source on unsteady MHD flow over a stretching sheet was developed by Reddy et al. [3]. Shehzad et al. [4] have investigated three-dimensional MHD flow of Casson fluid in porous medium with heat generation. Seth et al. [5] have studied Hall effects on unsteady MHD natural convection flow of a heat absorbing fluid past an accelerated moving vertical plate with ramped temperature. Effects of thermal dissipation, *Corresponding author. E-mail: rajputgauravlko@gmail.com heat generation/absorption on MHD mixed convection boundary layer flow over a permeable vertical flat plate embedded in an anisotropic porous medium was considered by Adeniyan and Aroloye [6]. Similar studies were done by Siva and Suram [7], Narayana et al. [8], Ibrahim and Suneetha [9]. Rajput and Kanaujia [10] etc. Chemical reaction effect on unsteady MHD flow past an impulsively started oscillating inclined plate with variable temperature and mass diffusion in the presence of Hall current was studied by us [11]. The main purpose of the present investigation is to study the effects of heat absorption on unsteady MHD flow past an oscillating vertical plate with variable wall temperature and mass diffusion in the presence of Hall current. The model has been solved using the Laplace transform technique. The results are shown with the help of graphs and table. 2. Mathematical Analysis The geometrical model of the problem is shown in Figure 1.
278 U S Rajput and Gaurav Kumar dimensionless form: (6) Figure 1. Physical model. The unsteady flow of an electrically conducting, incompressible, viscous fluid through porous medium in a vertical plate has been considered. The x axis is taken in the direction of motion and z normal to it. A transverse magnetic field B 0 of uniform strength is applied on the flow. Initially it has been considered that the plate as well as the fluid is at the same temperature T. The species concentration in the fluid is taken as C. At time t > 0, the plate starts moving with a velocity u 0 in its own plane, and temperature of the plate is raised to T w. The concentration C near the plate is raised linearly with respect to time. The governing equations are as under: (1) The dimension less flow model becomes The corresponding boundary conditions become (7) (8) (9) (10) (2) (11) (3) (4) Dropping bars in the above equations and combining equation (7) and (8) by using (q = u + iv), the model becomes The initial and boundary conditions are (12) (5) (13) (14) The following non-dimensional quantities are introduced to transform equations (1), (2), (3) and (4) into Finally, the boundary conditions become:
Effect of Heat Absorption on MHD Flow Over a Plate with Variable Wall Temperature 279 (15) The dimensionless governing equations (12) to (14), subject to the boundary conditions (15), are solved by the usual Laplace transform technique. The solution obtained is as under: are also decreased. Effect of heat absorption on fluid flow behavior is shown by Figures 2 and 3. It is seen here that when heat absorption parameter H increases, velocities get decreased throughout the boundary layer region. Further, from Figures 4 and 5, it is observed that if phase angle t isincreased then velocities of fluid are decreased. This means that the frequency parameter or the phase angle have a retarding effect on velocities of flow along the plate. From Figure 7 it is observed that the temperature increases with time. This is due to the reason that heat is transported to the system continuously. The values of skin friction are given in Table 1. The values of x and y decrease with the increase in heat absorption parameter and phase angle. 4. Conclusions In this paper a theoretical analysis has been done to study effects of heat absorption on unsteady MHD flow past an oscillating vertical plate with variable wall temperature and mass diffusion in the presence of Hall current. It is observed that the primary and secondary velocities decrease with increasing the values of heat absorption and phase angle parameter. The symbols involved in the above equations are mentioned in the appendix. 3. Results and Discussion In the present paper we have studied the effects of heat absorption on the flow. The behavior of other parameters like magnetic field, Hall current and thermal buoyancy is almost similar to the earlier model studied by us [11]. The analytical results are shown graphically in Figures 2 to 7. The numerical values of skin-friction x and y are presented in Table 1. It is observed that the temperature decreases when heat absorption parameter is increased (Figure 6). The tendency of heat absorption is to diminish the temperature distribution of the fluid along the plate. This causes the buoyancy force is decrease, therefore the primary and secondary flow along the plate Figure 2. u vs. z for different values of H. Figure 3. v vs. z for different values of H.
280 U S Rajput and Gaurav Kumar Figure 4. u vs. z for different values of t. Figure 6. vs. z for different values of H. Figure 5. v vs. z for different values of t. Figure 7. vs. z for different values of t. Table 1. Skin friction for different parameter M m Pr Sc Gm Gr H t t x y 2 1.0 0.71 2.01 100 10 1 60 0.4 10.699520-3.87084 2 1.0 7.01 2.01 100 10 100 60 0.4 8.69831-0.29772 2 1.0 0.71 2.01 100 10 5 30 0.4 8.46596-1.16007 2 1.0 0.71 2.01 100 10 5 45 0.4 8.30705-1.16017 2 1.0 0.71 2.01 100 10 5 60 0.4 8.09995-1.16028 Nomenclature a * absorption constant b acceleration parameter b dimensionless acceleration parameter C species concentration in the fluid C the dimensionless concentration C P specific heat at constant pressure C w species concentration at the plate C the concentration in the fluid D mass diffusion u, v velocity of the fluid in x and z-direction uv, dimensionless velocity in x and z-direction T temperature of the fluid the chemical reaction parameter K 0 M the magnetic parameter m the Hall parameter (m = e e ) Pr Prandtl number R radiation parameter Sc Schmidt number e cyclotron frequency of electrons e electron collision time volumetric coeff. of thermal expansion * volumetric coeff. of concentration expansion the kinematic viscosity the fluid density electrical conductivity the magnetic permeability the dimensionless temperature the coefficient of viscosity
Effect of Heat Absorption on MHD Flow Over a Plate with Variable Wall Temperature 281 G m G r k T g T w K t mass Grashof number thermal Grashof number the thermal conductivity the temperature of the fluid gravity acceleration temperature of the plate permeability of the medium time Appendix References [1] Chamkha, A. J. and Mudhaf, A. A., Unsteady Heat and Mass Transfer from a Rotating Vertical Cone with a Magnetic Field and Heat Generation or Absorption
282 U S Rajput and Gaurav Kumar Effects, International Journal of Thermal Sciences (Elsevier), Vol. 44, pp. 267 276 (2005). doi: 10.1016/ j.ijthermalsci.2004.06.005 [2] Dessie, H. and Naikoti, K., MHD Effects on Heat Transfer over Stretching Sheet Embedded in Porous Medium with Variable Viscosity, Viscous Dissipation and Heat Source/Sink, Ain Shams Engineering Journal (Elsevier), Vol. 5, pp. 967 977 (2014). doi: 10. 1016/j.asej.2014.03.008 [3] Reddy, M. G., Padma, P. and Shankar, B., Effects of Viscous Dissipation and Heat Source on Unsteady MHD Flow over a Stretching Sheet, Ain Shams Engineering Journal (Elsevier), Vol. 6, pp. 1195 1201 (2015). doi: 10.1016/j.asej.2015.04.006 [4] Shehzad, S. A., Hayat, T. and Alsaedi, A., Three-dimensional MHD Flow of Casson Fluid in Porous Medium with Heat Generation, Journal of Fluid Mechanics, Vol. 9, No. 1, pp. 215 223 (2016). [5] Seth, G. S., Sharma, R. and Hussain, S. M., Hall Effects on Unsteady MHD Natural Convection Flow of a Heat Absorbing Fluid Past an Accelerated Moving Vertical Plate with Ramped Temperature, Emirates Journal for Engineering Research, Vol. 19, No. 2, pp. 19 32 (2014). [6] Adeniyan, A. and Aroloye, S. J., Effects of Thermal Dissipation, Heat Generation or Absorption on MHD Mixed Convection Boundary Layer Flow over a Permeable Vertical Flat Plate Embedded in an Anisotropic Porous Medium, Gen. Math. Notes, Vol. 30, No. 2, pp. 31 53 (2015). [7] Siva, R. S. and Suram, A. K., Finite Element Analysis of Heat and Mass Transfer Past an Impulsively Moving Vertical Plate with Ramped Temperature, Journal of Applied Science and Engineering, Vol. 19, No. 4, pp. 385 392 (2016). [8] Narayana, P. V. S., Venkateswarlu, B. and Devika, B., Chemical Reaction and Heat Source Effects on MHD Oscillatory Flow in an Irregular Channel, Ain Shams Engineering Journal (Elsevier), pp. 1079 1088 (2016). doi: 10.1016/j.asej.2015.07.012 [9] Ibrahim, S. M. and Suneetha, K., Heat Source and Chemical Effects on MHD Convection Flow Embedded in a Porous Medium with Soret, Viscous and Joules Dissipation, Ain Shams Engineering Journal (Elsevier), pp. 811 818 (2016). doi: 10.1016/j.asej. 2015.12.008 [10] Rajput, U. S. and Kanaujia, N., MHD Flow Past a Vertical Plate with Variable Temperature and Mass Diffusion in the Presence of Hall Current, International Journal of Applied Science and Engineering, Vol. 14, No. 2, pp. 115 123 (2016). [11] Rajput, U. S. and Kumar, G., Chemical Reaction Effect on Unsteady MHD Flow Past an Impulsively Started Oscillating Inclined Plate with Variable Temperature and Mass Diffusion in the Presence of Hall Current, Applied Research Journal, Vol. 2, No. 5, pp. 44 253 (2016). Manuscript Received: Nov. 1, 2016 Accepted: Feb. 22, 2017