Influence of thermophoresis and joule heating on the radiative flow of jeffrey fluid with mixed convection

PROCESS SYSTEMS ENGINEERING

Influence of thermophoresis and joule heating on the radiative flow of jeffrey fluid with mixed convection

S. A. ShehzadI, ^*; A. Alsaedi^II; T. Hayat^{I, II}

^IDepartment of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

^IIDepartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80257, Jeddah 21589, Saudi Arabia. E-mail: [e-mail]

ABSTRACT

The aim of the present study is to address the magnetohydrodynamic (MHD) radiative flow of an incompressible Jeffrey fluid over a linearly stretched surface. Heat and mass transfer characteristics are accounted for in the presence of Joule heating and thermophoretic effects. Series solutions by the homotopy analysis method are constructed for the velocity, temperature and concentration fields. A convergence criterion for the series solutions is discussed. In addition, the numerical values of the skin friction coefficient, local Nusselt and Sherwood numbers are first computed and then analyzed.

Keywords: Radiative flow; MHD Jeffrey fluid; Joule heating; Thermophoresis.

INTRODUCTION

The analysis of non-Newtonian fluids is significant because of several industrial and engineering applications. Such fluids are encountered in the process of manufacturing coated sheets, foods, drilling muds, cosmetic products, dilute polymer solutions, polymeric melts etc. Different models of such fluids have beenare suggested. This is due to the versatility of fluid characteristics in nature (Qi and Xu, 2007; Kothandapani and Srinivas, 2008; Nadeem and Akbar, 2010; Jamil and Fetecau, 2010a,b; Wang and Tan, 2011; Hayat et al., 2011a,b; Jamil et al., 2011; Nadeem et al., 2011; Hayat et al., 2012). Although much information is available on the boundary layer flow of viscous fluids, such attempts for non-Newtonian fluids are limited. In fact, the resulting differential equations in non-Newtonian fluids are more nonlinear than for a viscous fluid. To find the analytic/numerical solution of such equations is not an easy task.

Investigation of heat transfer in the boundary layer over a stretching sheet has key importance in extrusion processes, paper production, glass drawing, electronic chips, crystal growing, plastic manufacture, metal spinning and cooling of metallic plates in a cooling bath etc. For the quality of the final product in such processes, the cooling rate has great relevance (Bhattacharyya et al., 2001; Hayat et al., 2010; Rashidi and Pour, 2010; Vyas and Rai, 2010; Yao et al., 2011; Rashidi and Erfani, 2011; Rashidi et al., 2011; Makinde and Aziz, 2011; Sahoo and Poncet, 2011). The migration of small particles in the direction of decreasing thermal gradient is called thermophoresis (Hinds, 1982). This is used for particle collection. The velocity gained by the particle is called the thermophoretic velocity and the force experienced by the suspended particle due to the temperature difference is called the thermophoretic force (Tasai et al., 2004). Thermophoresis is a mechanism for particles on a cold surface. It is quite significant in aerosol particle sampling, deposition of silicon thin films, radioactive particle deposition in nuclear reactor safety simulations, scale formation on surfaces of heat exchangers, microelectronic manufacturing, particulate material deposition on turbine blades, modified chemical vapour deposition etc. Goren (1977) studied the laminar flow of a viscous fluid with thermophoresis in the presence of cold and hot plate conditions. The effect of thermophoresis in MHD flow of a viscous fluid over an inclined plate was anlyzed by Aslam et al. (2008). Hayat and Qasim (2010) investigated the radiative flow of Maxwell fluid in the presence of Joule heating and thermophoresis effects. Hayat and Alsaedi (2011) further examined the thermophoresis effects in MHD flow of Oldroyd-B fluid over a linearly stretching surface. Heat and mass transfer effects in the thermophoretic flow of viscous fluid over an inclined isothermal permeable surface have been explored by Noor et al. (2012).

Motivated by the above mentioned studies, the aim here is to investigate the effects of thermophoresis in MHD flow of a non-Newtonian fluid over a stretching surface. Mathematical modelling is presented using constitutive equations of a Jeffrey fluid. Thermal radiation and Joule heating effects are taken into account. The present study is structured in the following fashion. The mathematical formulation is completed in the next section. Series solutions by the homotopy analysis method (HAM) (Liao, 2003; Yao, 2009; Rashidi et al., 2011b; Hayat et al., 2011c; Vosughi et al., 2011; Aziz et al., 2012; Hayat et al., 2011d) are developed in the subsequent sections. Then convergence analysis and discussion are presented. Important results are summarized in the last section.

FLOW FORMULATION

We consider Cartesian coordinate system in such a way that the axis is along the stretching surface and the axis is perpendicular to the axis. Magnetohydrodynamic boundary layer flow of a Jeffrey fluid is considered. A uniform magnetic field is applied parallel to the axis (see Figure 1). The induced magnetic field is neglected for small magnetic Reynolds number. Heat and mass transfer characteristics are taken into account in the presence of thermal radiation and thermophoresis effects. The uniform temperature of the surface is larger than the ambient fluid temperature

The species concentration at the surface and ambient concentration are constants. Invoking the Rosseland approximation [24], the resulting equations take the following forms:

Here ( are the velocity components along the and axes, and are the ratio of relaxation/retardation times and retardation time, respectively, the dynamic viscosity, the density of the fluid, the electrical conductivity, the gravitational acceleration, and the thermal expansion coefficients, the temperature, the specific heat, the Stefan-Boltzmann constant, the mean absorption coefficient, the diffusion coefficient and the thermophoretic velocity.

The associated boundary conditions are:

where is the stretching velocity, is the temperature at the wall, is the concentration at the wall and and are the ambient fluid temperature and concentration, respectively.

The term in Eq. (4) can be defined as follows:

in which is the thermophoretic coefficient and is the reference temperature. A thermophoretic parameter is defined by the following relation:

The wall temperature and concentration fields are:

where and are the positive constants.

Through the following transformations:

Equation (1) is automatically satisfied and Eqs. (2)-(4) are reduced as follows:

In the above expressions, is the Deborah number the Hartmann number, the local buoyancy parameter, the local Grashof number, the constant dimensionless concentration buoyancy parameter, the Prandtl number, the radiation parameter, the Eckert number and the Schmidt number.

The skin friction coefficient, local Nusselt number and local Sherwood number are transformed to the following:

SERIES SOLUTIONS

The homotopic solutions for and in a set of base functions:

can be written to the following expressions:

in which , and are the coefficients. Initial guesses and the auxiliary linear operators are

The operators in Eq. (19) satisfy the following properties:

where denote the arbitrary constants. The zeroth order deformation problems are expressible in the form

Here is the embedding parameter, , and the non-zero auxiliary parameters and the nonlinear operators , and are given by:

When and , one has

Note that, when increases from to , then , and approach to , to and to According to Taylor's series, one has:

The convergence of series (22)-(24) is closely associated with and The values of , and are chosen such that the series (22)-(24) converge at . Thus

The general solutions for , and in terms of special solutions , and are

CONVERGENCE ANALYSIS AND DISCUSSION

The auxiliary parameters , and play an important role in controlling and adjusting the convergence of the series solutions. To find a suitable value for each of these auxiliary parameters, the curves at the order of approximation are displayed. Figure 2 indicates that the admissible values of , and are , and Our series solutions converge in the whole region of for .

Figures 3-18 plot the effects of the various interesting parameters on the velocity , temperature and concentration . The fluid velocity and momentum boundary layer thickness increase with an increase in Deborah number (see Figure 3). In Figure 4, the influence of the ratio of relaxation to the retardation times is sketched for the fluid velocity. It shows that the fluid velocity decreases upon increasing Figure 5 shows that an increase in the local buoyancy parameter yields an increase in the velocity. In fact, the local buoyancy parameter depends on the buoyancy force and an increase in buoyancy force gives rise to the fluid velocity increase. An increase in Hartmann number decreases the fluid velocity (see Figure 6). The Hartmann number is a consequence of the Lorentz force. Obviously an increase in the Lorentz force opposes the flow and thus the fluid velocity decreases. Figure 7 shows that the velocity and momentum boundary layer thickness are decreasing functions of the Prandtl number Figure 8 displays the effects of the ratio of buoyancy parameter The effects of the ratio of the buoyancy parameter are similar to that of in a qualitative way. The difference we noticed is that the fluid velocity increases more rapidly in the case of increasing local buoyancy parameter when compared with the ratio of the buoyancy parameter.

Effects of different parameters on the temperature are seen in Figures 9-14. Figure 9 depicts the effects of the local buoyancy parameter on the temperature field. The temperature field and thermal boundary layer thickness are reduced with an increase in Figure 10 shows that the temperature increases when the Hartmann number is increased. In fact, large corresponds to lower permeability and hence the temperature and the thermal boundary layer thickness increase. Figures 11 and 12 illustrate the influences of Eckert number and Prandtl number on the temperature. It is observed from these figures that and have quite opposite effects on the velocity and the associated thermal boundary layer thickness. The temperature profile decreases more quickly for in comparison to The ratio of the buoyancy parameter and the radiation parameter correspond to a decrease and an increase in the temperature, respectively (see Figures 13 and 14).

The effects of , and on concentration profile are seen through Figures 15-18. Figure 15 shows that the concentration and boundary layer thickness are decreasing functions of It is found from Figure 16 that the Hartmann number increases the concentration profile and associated boundary layer thickness. The effects of and on are similar. Increases in and decrease the concentration and related boundary layer thickness (see Figures 17 and 18). It is worth mentioning here that the effects of M on the temperature are more pronounced when compared with the concentration. This same observation also holds for the influences of , N and R on the temperature and concentration.

Table 1 shows the convergence of the series solutions through computations of numerical values. This table indicates that our series solutions converge from the 20th-order of deformations for velocity and temperature and 25th-order of approximation for the concentration. Table 2 includes the numerical values of the skin-friction coefficient for the different values of and when and The skin-friction coefficient increases with increasing and It decreases when and are increased. Interestingly the variation in the values of the skin-friction coefficient is very small with increasing thermophoretic parameter . Table 3 consists of the values of the local Nusselt and Sherwood numbers. The values of the local Nusselt and Sherwood numbers increase slowly in comparison to the increase in the values of the skin-friction coefficient when increases. The increase in the values of the local Nusselt number is very small upon increasing , but the increase in the values of the local Sherwood number is large. A comparative study of Tables 2 and 3 shows that the values of the skin-friction coefficient and local Sherwood number are larger than the values of the local Nusselt number.

CLOSING REMARKS

In this study we discussed the effects of thermal radiation, Joule heating and thermophoresis in stretched flow of a Jeffrey fluid. The main observations are as follows:

• Effects of and on the fluid velocity are quite opposite.

• An increase in the ratio of the buoyancy parameter corresponds to an increase in

• The fluid temperature rises with an increase in Eckert number

• Effects of the ratio of the buoyancy parameter and radiation parameter are qualitatively similar.

• Variations in temperature are more pronounced than those in concentration when increases.

• The increase in the values of the skin-friction coefficient and the local Nusselt number is very small in comparison to the increase in the local Sherwood number when increases.

• Numerical values of the local Nusselt number increase and the values of the local Sherwood number decrease with increasing Prandtl number

ACKNOWLEDGMENTS

The research of Alsaedi was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. We are also thankful to the reviewer for his/her useful suggestions.

(Submitted: June 9, 2012 ; Revised: November 8, 2012 ; Accepted: December 12, 2012)

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To whom correspondence should be addressed

Publication Dates

• Publication in this collection
  10 Jan 2014
• Date of issue
  Dec 2013

History

• Received
  09 June 2012
• Accepted
  12 Dec 2012
• Reviewed
  08 Nov 2012