**2. Mathematical formulation**

ously from a die, the cooling of a metallic plate in a bath, the aerodynamic extrusion of plastic sheets, the continuous casting, rolling, annealing and thinning of copper wires, the wires and fibre coating. During its manufacturing process, a stretched sheet interacts with ambient fluid thermally and mechanically. Both the kinematics of stretching and the simultaneous heating or cooling during such processes have a decisive influence on the quality of the final product. In [1],theeffectsofchemicalreactionandmagneticfieldonviscousflowoveranon-linearstretching sheet were reported. Mabood et al. [2] studied numerically MHD flow and heat transfer of nanofluid over a non-linear stretching sheet. Abel et al. [3] investigated the steady buoyancydriven dissipative magneto-convective flow from a vertical non-linear stretching sheet. In [4], an analysis of heat transfer over an unsteady stretching sheet with variable heat flux in the presence of heat source or sink was made. Several other studies have addressed various aspects

Micropolar fluids are fluids with microstructure and asymmetrical stress tensor. Physically, they represent fluids consisting of randomly oriented particles suspended in a viscous medium. These types of fluids are used in analysing liquid crystals, animal blood, fluid flowing in brain, exotic lubricants, the flow of colloidal suspensions, etc. The theory of micropolar fluids was first proposed by Eringen [11]. In this theory, the local effects arising from the micro‐ structure and the intrinsic motion of the fluid elements are taken into account. The compre‐ hensive literature on micropolar fluids, thermomicropolar fluids and their uses in engineering and technology was presented by Kelson and Desseaux [12]. Gorla and Nakamura [13] discussed the combined convection from a rotating cone to micropolar fluids with an arbitrary variation of surface temperature. Prathap Kumar et al. [14] studied the effect of surface conditions on the micropolar flow driven by a porous stretching sheet. In [15], the case of mixed convection flow of a micropolar fluid past a semi-infinite, steadily moving porous plate with varying suction velocity normal to the plate in the presence of thermal radiation and viscous dissipation was discussed. Mansour et al. [16] studied heat and mass transfer effects on the magnetohydrodynamic (MHD) flow of a micropolar fluid on a circular cylinder. El-Hakiem [17] proposed the dissipation effects on the MHD-free convective flow over a non-isothermal surface in a micropolar fluid. Joule heating and mass transfer effects on the MHD-free convective flow in micropolar fluid are investigated by El-Hakiem et al. [18] and El-Amin [19], respectively. In [20], the derivation of the unsteady MHD-free convection flow of micropolar fluid past a vertical moving porous plate in a porous medium was presented. Many researchers

The study of heat source/sink effects on heat transfer is another important issue in the study of several physical problems. The effect of non-uniform heat source, only confined to the case of viscous fluids, was also included in [24–27], while Mabood et al. [28] investigated nonuniform heat source/sink effects and Soret effects on MHD non-Darcian convective flow past

Combined heat and mass transfer problems with chemical reactions are important in many processes of interest in engineering and have received significant attention in recent years. These processes include drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler [29]. Chemical reactions are classified as

of regular/nanofluids [5–10].

290 Numerical Simulation - From Brain Imaging to Turbulent Flows

investigated different aspects of micropolar fluid [21–23].

a stretching sheet in a micropolar fluid with radiation.

We consider an unsteady two-dimensional, mixed convection flow of a viscous incompressible micropolar fluid, heat and mass transfer over an elastic vertical permeable stretching sheet in the presence of a heat source/sink and chemical reaction. Following [39], the sheet is assumed to emerge vertically in the upward direction from a narrow slot with a velocity,

$$U\_w(\mathbf{x}, t) = \frac{a\mathbf{x}}{1 - at},\tag{1}$$

where both *a* and α are positive constants with dimension per unit time. We measure the positive *x* direction along the stretching sheet with the top of the slot as the origin. We then measure the positive *y* coordinate perpendicular to the sheet and across the fluid flow. The surface temperature (*Tw*) and the concentration (*Cw*) of the stretching sheet vary along the *x* direction and in time *t* as

$$T\_w(\mathbf{x}, t) = T\_\alpha + \frac{b\mathbf{x}}{\left(1 - \alpha t\right)^2}, \quad C\_w(\mathbf{x}, t) = C\_\alpha + \frac{c\mathbf{x}}{\left(1 - \alpha t\right)^2},\tag{2}$$

where *b* and *c* are constants with dimension temperature and concentration respectively, over length. It is noted that the expressions for *Uw*(*x*, *t*), *Tw*(*x*, *t*) and *Cw*(*x*, *t*) are valid only for *t* <*α* <sup>−</sup><sup>1</sup> . We also remark that the elastic sheet which is fixed at the origin is stretched by applying a force in the *x-*direction and the effective stretching sheet rate *a* /(1−*αt*) increases with time. Analogously, the sheet temperature and concentration increase (reduce) if *b* and *c* are positive (negative), respectively, from *T∞* and *C∞* at the sheet in the proportion to *x*. We assume that the radiation effect is significant in this study. The fluid properties are taken to be constant except for density variation with temperature and concentration in the buoyancy terms. Under those assumptions and the Boussinesq approximations, the governing two-dimensional boundary layer equations are given as:

$$
\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial \mathbf{y}} = \mathbf{0},
\tag{3}
$$

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \nu \frac{\partial u}{\partial y} = \left(\frac{\mu + \kappa}{\rho}\right) \frac{\partial^2 u}{\partial y^2} + \frac{\kappa}{\rho} \frac{\partial N}{\partial y} + \mathbf{g} \,\beta\_\varepsilon (T - T\_\alpha) + \mathbf{g} \,\beta\_\varepsilon (C - C\_\alpha) - \frac{\sigma B\_0^2 u}{\rho} - \frac{\nu u}{\rho K\_\rho^\*},\tag{4}$$

$$\frac{\partial N}{\partial t} + \mu \frac{\partial N}{\partial \mathbf{x}} + \nu \frac{\partial N}{\partial \mathbf{y}} = \frac{\gamma}{\rho j} \frac{\partial^2 N}{\partial \mathbf{y}^2} - \frac{\kappa}{\rho j} \left(2N + \frac{\partial u}{\partial \mathbf{y}}\right), \tag{5}$$

$$
\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial \mathbf{x}} + \nu \frac{\partial T}{\partial \mathbf{y}} = \alpha\_0 \frac{\partial^2 T}{\partial \mathbf{y}^2} + \left(\frac{\mu + \kappa}{\rho c\_p}\right) \left(\frac{\partial u}{\partial \mathbf{y}}\right)^2 \pm \frac{\mathcal{Q}}{\rho c\_p} (T - T\_\kappa), \tag{6}
$$

$$
\rho \frac{\partial C}{\partial t} + \mu \frac{\partial C}{\partial x} + \nu \frac{\partial C}{\partial y} = D \frac{\partial^2 C}{\partial y^2} - \kappa\_c \left( C - C\_o \right). \tag{7}
$$

where *u* and *v* are the velocity components along the *x* and*y* axes, respectively, *T* is the fluid temperature, *μ* is the component of the microrotation vector normal to the *x y* plane, *γ* is the spin gradient viscosity, *α*0 is the thermal conductivity, *Cp* is the heat capacity at constant pressure, *g* is the acceleration due to gravity, *β<sup>t</sup>* and *βc* are the coefficients of thermal expression and concentration expansion, respectively, *β<sup>0</sup>* is the transverse magnetic field, *C* is the concen‐ tration of the solutes, *T∞* and *C∞* denote the temperature and concentration far away from the plate, respectively, and *j* is the microinertia density or microinertia per unit mass. The appropriate boundary conditions for the current model are:

$$\mu = U\_w(\mathbf{x}, t), \quad \nu = V\_w, \ N = 0, \ T = T\_w(\mathbf{x}, t), \ C = C\_w(\mathbf{x}, t) \quad \text{at } \mathbf{y} = \mathbf{0}, \tag{8}$$

$$
\mu = 0, \ T \to T\_{\alpha}, \ C \to \mathcal{C}\_{\alpha} \quad \text{as} \; y \to \infty. \tag{9}
$$
