International Journal of Chemical Engineering

Volume 2016, Article ID 8190234, 10 pages

http://dx.doi.org/10.1155/2016/8190234

## Effects of Thermal Radiation on Mixed Convection Flow of a Micropolar Fluid from an Unsteady Stretching Surface with Viscous Dissipation and Heat Generation/Absorption

Department of Mathematics, Statistics and Computer Science, G.B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, India

Received 14 December 2015; Accepted 25 October 2016

Academic Editor: Jean-Pierre Corriou

Copyright © 2016 Khilap Singh and Manoj Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A numerical model is developed to examine the effects of thermal radiation on unsteady mixed convection flow of a viscous dissipating incompressible micropolar fluid adjacent to a heated vertical stretching surface in the presence of the buoyancy force and heat generation/absorption. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The model contains nonlinear coupled partial differential equations which have been converted into ordinary differential equation by using the similarity transformations. The dimensionless governing equations for this investigation are solved by Runge-Kutta-Fehlberg fourth fifth-order method with shooting technique. Numerical solutions are then obtained and investigated in detail for different interesting parameters such as the local skin-friction coefficient, wall couple stress, and Nusselt number as well as other parametric values such as the velocity, angular velocity, and temperature.

#### 1. Introduction

The micro polar fluids are those which contain microconstituents that can undergo rotation, the presence of which can affect the hydrodynamics of the flow. The classical Navier-Stokes theory does not describe the flow properties of micropolar fluids, for example, colloidal suspension, polymeric fluids, liquid crystals, fluids with additives, suspension solutions, animal’s blood, human blood, body fluids, biofluids, and fluids containing certain additives. Eringen [1] describes the theory of micropolar fluids, which show microrotation effects as well as microinertia. The theory of thermomicropolar fluids was developed by Eringen [2] by extending his theory of micropolar fluids. A good list of references for micropolar fluids is available in Łukaszewicz [3]. Many researchers [4–13] have studied the micropolar fluid flow for different fluid properties over different geometries.

Unsteady mixed convection flow plays an important role in chemical engineering, turbomachinery, aerospace technology, geophysics, and so forth; Zueco et al. [14] studied the unsteady free convection flow of an MHD micropolar fluid through two parallel infinite porous vertical plates. Unsteady mixed convection flow of a micropolar fluid adjacent to a heated vertical surface along with viscous dissipation and the buoyancy force is analyzed by Abd El-Aziz [15]. Hussainn et al. [16] reported the radiation effects on the unsteady boundary layer flow of a micropolar fluid over a stretching permeable sheet. Oahimire and Olajuwon [17] studied the heat and mass transfer effects on an unsteady flow of a chemically reacting micropolar fluid over an infinite vertical porous plate. Rashad [18] studied the unsteady MHD boundary-layer flow and heat transfer for an electrically conducting rotating fluid due to a stretching surface in porous medium in the presence of thermal radiation. Abd El-Aziz [19] investigated the effects of variable viscosity on mixed convection flow along a semi-infinite unsteady stretching sheet with viscous dissipation.

The heat transfer in the fluid flow due to a stretching sheet has attracted considerable attention during the last few decades due to its various applications in many industrial and engineering processes such as hot rolling, wire drawing, glass-fiber and paper production, drawing of plastic films, metal and polymer extrusion, and metal spinning. The pioneering work in this area was first made by Crane [20]; he studied boundary layer flow from a linearly stretching plate. Bhargava et al. [21] obtained a finite element solution for the mixed convection micropolar fluid flow in porous stretching sheet with suction. Eldabe and Ouaf [22] examined the heat and mass transfer flow of a micropolar fluid past a stretching surface with Ohmic heating and viscous dissipation. Pal et al. [23] analyzed the effects of thermal radiation and viscous dissipation on mixed convection flow of nanofluid over a nonlinear stretching/shrinking sheet. Singh and Kumar [24] studied boundary layer stagnation point flow of micropolar fluid towards a stretching/shrinking sheet in the presence of melting heat. Turkyilmazoglu [25] analyzed the micropolar fluid flow and heat transfer due to a porous stretching sheet.

Influence of thermal radiation on flow and heat transfer study is much more important in different industries. The heat transfer and temperature profile of a fluid flow over different geometries can be affected significantly at high temperature. Mohamed and Abo-Dahab [26] studied thermal radiation effect on hydromagnetic free convection heat and mass transfer flow of a micropolar fluid over vertical porous plate. The micropolar fluid flow and heat transfer from a porous shrinking sheet were investigated by Bhattacharyya et al. [27]. Prakash and Muthtamilselvan [28] analyzed the effect of thermal radiation on fully developed flow of MHD micropolar fluid through two infinite parallel porous vertical plates.

The effect of heat generation on heat transfer is an important issue in view of various physical problems. Ziabakhsh et al. [29] proposed micropolar fluid flow with heat generation. Bakr [30] investigated the effects of heat source on heat and mass transfer flow of a micropolar fluid in a rotating frame of reference. The heat generation/absorption effects on MHD flow of micropolar fluid through a stretching surface have been studied by Mahmoud and Waheed [31]. Abbasi et al. [32] investigated the Maxwell nanofluid fluid flow and heat transfer in the presence of heat generation/absorption. Mliki et al. [33] examined the influence of Brownian motion and heat generation/absorption over linear/sinusoidally heated cavity. Elgazery [34] analyzed the chemical reaction effect on MHD flow in the presence of temperature dependent viscosity and thermal diffusivity.

The purpose of the present work is to study the effects of thermal radiation on mixed convection flow of a micropolar fluid through an unsteady stretching surface with viscous dissipation and heat generation/absorption. This problem is important in the processing of chemical engineering fluids including polymeric suspensions, lubricant manufacture, and so forth. The nonlinearity of basic equations associated with their inherent mathematical difficulties has led us to use numerical method. Thus the transformed dimensionless pertinent equations are solved numerically by using the Runge-Kutta-Fehlberg fourth fifth-order method along with shooting technique. The velocity, angular velocity, and temperature profiles are shown and the influences of the micropolar parameter, the thermal radiation parameter, the unsteadiness parameter, and the buoyancy parameter on the flow and heat transfer characteristics are discussed in detail. To the best of author’s knowledge such a study does not appear in the scientific literature.

#### 2. Mathematical Formulation

We consider a two-dimensional unsteady mixed convection boundary layer flow of a viscous incompressible micropolar fluid over an elastic, vertical, and impermeable stretching sheet which emerges vertically in the upward direction from a narrow slot with velocity: where both and are positive constants with dimension per time. The problem is in the presence of thermal radiation and heat generation/absorption. The flow configuration of this problem is illustrated in Figure 1. The positive coordinate is measured along the stretching sheet with the slot as the origin and the positive coordinate is measured normal to the sheet in the outward direction towards the fluid. The surface temperature of the stretching sheet varies with the distance from the slot and time as where is a constant with dimension temperature over length and is the temperature of the ambient fluid. The expressions for and in (1) and (2) are valid only for time unless . Further, expression (1) for velocity of the sheet reveals that the elastic surface which is fixed at the origin is stretched by applying force in the positive *-*axis and the effective stretching rate increases with time. With the same analogy expression (2) for the surface temperature depicts a situation in which the surface temperature increases (or decreases) if is positive (or negative) from at the slot in proportion to and such that the amount of temperature increase (or decrease) along the sheet increases with time. It is further assumed that the fluid properties are taken to be constant except for the density variation with the temperature in the buoyancy term. Under these assumptions, the governing equations of boundary layer are given in the following form:where and are the components of velocity along and directions, respectively. Further is dynamic viscosity, is vortex viscosity, is fluid density, is the gravitational acceleration, is the volumetric coefficient of the thermal expansion, is kinematic viscosity, is microinertia density, is the spin-gradient viscosity, is the component of microrotation whose direction of rotation lies in the plane, is the temperature, is the thermal conductivity of the fluid, is the heat capacity at constant pressure , is the radiative heat flux, is the heat generation/absorption coefficient, and is the temperature of the ambient fluid.