Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 8521580, 12 pages

http://dx.doi.org/10.1155/2016/8521580

## Heat and Mass Transfer in a Thin Liquid Film over an Unsteady Stretching Surface in the Presence of Thermosolutal Capillarity and Variable Magnetic Field

School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China

Received 20 January 2016; Accepted 30 June 2016

Academic Editor: Mohamed Abd El Aziz

Copyright © 2016 Yan Zhang et al. 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

The heat and mass transfer characteristics of a liquid film which contain thermosolutal capillarity and a variable magnetic field over an unsteady stretching sheet have been investigated. The governing equations for momentum, energy, and concentration are established and transformed to a set of coupled ordinary equations with the aid of similarity transformation. The analytical solutions are obtained using the double-parameter transformation perturbation expansion method. The effects of various relevant parameters such as unsteady parameter, Prandtl number, Schmidt number, thermocapillary number, and solutal capillary number on the velocity, temperature, and concentration fields are discussed and presented graphically. Results show that increasing values of thermocapillary number and solutal capillary number both lead to a decrease in the temperature and concentration fields. Furthermore, the influences of thermocapillary number on various fields are more remarkable in comparison to the solutal capillary number.

#### 1. Introduction

In the recent years, researches on the flow and heat transfer of a liquid film on an unsteady stretching sheet have got more and more attentions for its wide applications. For example, during mechanical forming processes, such as polymer extrusion, melt spinning process, the process of shaping by forcing through a die, wire and fiber coating, and food stuff process, the flow of a liquid film on an unsteady stretching sheet will be met.

In 1970, Crane [1] studied the analytic solutions of two-dimensional boundary layer flow due to a stretching flat elastic sheet. Munawar et al. [2] considered time-dependent flow and heat transfer over a stretching cylinder. Shehzad et al. [3] investigated thermally radiative flow with internal heat generation and magnetic field. Furthermore, many scholars discussed other effects on the flow over a stretching sheet, such as three-dimensional flow [4, 5], heat and mass transfer [6–9], MHD [10], first-order chemical reaction [11], non-Newtonian fluids [12, 13], or different possible combinations of these above effects [14–16]. All of the above studies mainly focus on infinite fluid. In fact, the flow and heat transfer of a finite film are more suitable to describe industrial engineering. The hydrodynamics of the thin liquid film over a stretching sheet were first considered by Wang [17] who reduced the unsteady Navier-Stokes equations to the coupled nonlinear ordinary differential equations by similarity transformation and solved the problem using a kind of multiple shooting method (see Roberts and Shipman [18]). Subsequently, Wang [19] obtained the analytical solutions of a liquid thin film and confirmed the validity of the homotopy analysis method. On the basis of Wang’s work [17], several authors [20–28] explored finite fluid domain of both Newtonian and non-Newtonian fluids using various velocity and thermal boundary conditions. The combined effect of viscous dissipation and magnetic field on the flow and heat transfer in a liquid film over an unsteady stretching surface was presented by Abel et al. [29]. The thermocapillary effect in finite fluid domain was first discussed by Dandapat et al. [30]. Noor and Hashim [31] extended the flow problem to hydromagnetic case.

Marangoni convection is caused by surface-tension gradient at a free liquid-gas or liquid-liquid interface that occurs due to gradient of temperature or concentration in the course of heat or mass transfer. The surface-tension variation on the free liquid surface resulting from the temperature gradient or concentration gradient can induce motion within the fluid called thermocapillary flow or solutal capillary flow (thermal Marangoni convection or solutal Marangoni convection). Pop et al. [32] investigated thermosolutal Marangoni forced convection boundary layers. On the flow field of power-law fluid, Lin et al. [33] analyzed the effect of radiation on Marangoni convection flow and heat transfer in the fluids with variable thermal conductivity. Then, Lin et al. [34] dealt with thermosolutal Marangoni convection flow in the presence of internal heat generation. The surface tension plays an important role on the free liquid surface. Other studies about thermocapillary effect on a thin film can be found in [35–38].

The main objective of our study is to extend previous research to the solutal Marangoni effect and mass diffusion. By means of an exact similarity transformation, governing PDEs are reduced into coupled nonlinear ODEs. And the analytical solutions are obtained using the double-parameter transformation perturbation expansion method [11]. The influences of various relevant parameters such as unsteadiness parameter , Hartmann number Ma, the Prandtl number Pr, the Schmidt number Sc, the thermocapillary number , and the solutal capillary number on the flow field are elucidated through graphs and tables.

#### 2. Mathematical Formulation

##### 2.1. Governing Equations and Boundary Conditions

Consider the thin elastic sheet that emerges from a narrow slit at origin of the Cartesian coordinate system shown in Figure 1. A variable magnetic field normal to the stretching sheet is applied, where is a positive constant. And in the above model, we take concentration into consideration. A thin liquid film with uniform thickness rests on the horizontal sheet. By applying the boundary layer assumptions [39], the governing time-dependent equations for mass, momentum, energy, and concentration are given byThe boundary conditions arewhere and are the velocity components of the fluid in the - and -directions, is the time, is the kinematic viscosity, is the electrical conductivity, is the magnetic field, is the density, is the temperature, is the thermal diffusivity, is the concentration, is the mass diffusivity, is dynamic viscosity, and is the uniform thickness of the liquid film. The dependence of surface tension on the temperature and concentration can be expressed as [40]where and . The fluid motion within liquid film resulted from not only the viscous shear arising from the stretching of the elastic sheet but also the surface-tension gradient. The stretching velocity is assumed to be of the same form as that considered by Wang [17]:where is a positive constant and denotes the initial stretching rate. With unsteady stretching (i.e., ), however, becomes the representative time scale of the resulting unsteady boundary layer problem. The adopted formulation of the velocity sheet in (5) is valid only for times unless . Also the temperature and the concentration of the surface of the elastic sheet are assumed to vary both along the sheet and with time, respectively, as [22, 30]where and are the temperature and the concentration at the slit, respectively, and and are the reference temperature and the reference concentration in the case of , respectively. Equation (6) represents a situation in which the sheet temperature and concentration decrease from and at the slot in proportion to . The nonuniform distributions of temperature and concentration cause surface-tension gradient which leads to the fluid flow from lower surface tension to higher surface tension. According to the temperature and concentration boundary layer conditions (6), one can conclude that thermosolutal Marangoni convection flow is in accord with the flow direction of the thin film.