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.
2.2. Similarity Transformation
The special forms of the stretching velocity, surface temperature, and fluid concentration in (5) and (6), respectively, are chosen to allow (2) to be converted into a set of ordinary differential equations by means of similarity transformations:where the similarity variable is given by Wang [19]and is yet an unknown constant denoting the dimensionless film thickness, defined by
Substituting (7)–(9) into (1)–(3), (1)–(3) can be transformed into the following nonlinear ordinary differential equations:subject to the boundary conditionswhere is Hartmann number, is the dimensionless unsteadiness parameter, is the Prandtl number, is the Schmidt number, is the dimensionless film thickness, and the thermocapillary number and the solutal capillary number are defined as and , respectively.
The skin friction coefficient and the Nusselt number are defined aswhere the wall skin friction and the sheet heat flux areSubstituting (14) into (13), then we getwhere is the local Reynolds number.
2.3. Solution Approach
In order to solve (10), We used the double-parameter transformation perturbation expansion method that confirmed the validity by Zhang and Zheng [11]. We introduced an artificial small parameter ; the equations can be expanded to the form of power series. We can obtain analytical solutions in the form of series by comparing the coefficients of same power and solving the equation sets.
We transform the dependent variable and independent variable as follows:where and is an artificial small parameter. We assume , , and ; here , , and are constants; the boundary conditions become Equation (10) reduces toAnd we can assumewith the boundary conditions Let and , and we can obtainSubstituting (21) into (12), we can get four equations involving the four variables such as , , , and . Analytical solutions of (10)–(12) in the form of series can be obtained with this method.
3. Results and Discussion
The effects of physical parameters on velocity, temperature, and concentration fields are presented in Tables 1–7, respectively. Based on Table 1, the values of dimensionless film thickness and skin friction coefficient obtained are compared with the results by Wang [19] for the case , and good agreement can be seen. From Table 2, it can be observed that increasing the value of unsteadiness parameter will decrease the film thickness , the skin friction coefficient , the heat flux , and the diffusion flux but increase the free velocity , free temperature , and free concentration . In Table 3, as the magnetic parameter Ma increases, the hydrodynamics behavior is the same as that in Table 2 except for the skin friction coefficient .
From Tables 4 and 5, it can be seen that and have the same effects on velocity, temperature, and concentration fields. Increasing the value of the thermocapillary number or solutal capillary number can cause a descent in the free temperature and free concentration but a rise in the film thickness , the skin friction coefficient , the heat flux , and the free velocity . Table 6 shows that the values of the free concentration and the heat flux increase while the film thickness , the skin friction coefficient , the diffusion flux , the free velocity , and free temperature decrease with an increment in the Prandtl number Pr. Table 7 shows the effect of Schmidt number on velocity, temperature, and concentration fields.
The velocity, temperature, and concentration profiles for the hydromagnetic flow in a thin film over an unsteady stretching sheet with thermocapillary are presented graphically in Figures 2–7. Figures 2(a)–2(c) show the impacts of unsteadiness parameter on the velocity, temperature, and concentration, respectively. We can find the enhancement in the velocity is more remarkable compared with the temperature and concentration when increment of is the same. And unsteadiness parameter has the similar effects on the temperature and concentration fields. Figures 3(a)–3(c) show the effects of Hartmann number on the velocity, temperature, and concentration fields, which are the same as Table 3. From Figure 3(a), it can be seen that when the Hartmann number increases, at the beginning, the velocity decreases slightly due to the fact that magnetic field produces a drag, and then the velocity near the surface of the thin film rises up gradually under the Marangoni effect, which explains the emergence of the intersection. Meanwhile, the temperature and concentration rise significantly and monotonously as increases (see Figures 3(b) and 3(c)). It can be proved that the magnetic field promotes heat and mass transfer in a thin liquid film.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
Figures 4(a) and 5(a) show the velocity profiles for different values of thermocapillary number and solutal capillary number . Similar to Figure 3(a), with increasing value of thermocapillary number or solutal capillary number, one can see that the velocity decreases initially mainly due to the magnetic field effect; then the velocity near the surface of the thin film obviously rises up mainly because of the Marangoni effect; furthermore, the two factors are balanced at the intersection. The effects of thermocapillary number on the temperature and concentration field are shown in Figures 4(b) and 4(c). Results demonstrate that the temperature and concentration both reduce when the thermocapillary number increases. Figures 5(b) and 5(c) illustrate that the fluid temperature and concentration are the decreasing functions of solutal capillary number. The lower value of thermocapillary number has higher temperature and concentration. Furthermore, decrement in the temperature and concentration profiles due to thermocapillary number is more remarkable in comparison to the solutal capillary number .
The effect of Pr on temperature is presented in Figure 6. It can be observed that heat transfer behaviors are strongly dependent on the value of the Prandtl number. Figure 6 shows that the temperature of the fluid decreases monotonously with the increasing of Pr. That is to say, Pr plays an opposite effect on the temperature field compared with and Ma. In addition, the effect of Schmidt number on the concentration distribution is displayed in Figure 7. Increasing the Schmidt number causes a decrease in the concentration profile, and it is clear that weaker mass transfer ratio is obtained with bigger Sc values.
4. Conclusions
In this study, a similarity analysis for thermosolutal capillarity and a magnetic field in a thin liquid film on an unsteady elastic stretching sheet has been studied. The effects of various relevant parameters on the velocity, temperature, and concentration fields have been discussed. The main points of presented analysis are listed below:(1)The thermocapillary number and solutal capillary number both restrict the heat and mass transfer. Furthermore, the effects of the thermocapillary number on the temperature and concentration are more remarkable than solutal capillary number.(2)The velocity, temperature, and concentration all increase for the higher values of unsteadiness parameter.(3)Increasing Prandtl number can reduce the fluid temperature, and increasing Schmidt number causes a descent in the concentration.(4)Marangoni convection caused by surface-tension gradient creates a promotion in flow near the surface region of the film.
Nomenclature
| b: | Positive constant |
| : | Time (s) |
| : | -component of velocity (ms−1) |
| : | -component of velocity (ms−1) |
| : | Horizontal coordinates (m) |
| : | Vertical coordinate (m) |
| : | Velocity similarity variable |
| : | Mass diffusivity (m2s−1) |
| : | Magnetic field (T) |
| : | Initial magnetic field (T) |
| : | Hartmann number |
| : | Thermocapillary number |
| : | Solutal capillary number |
| : | Prandtl number |
| : | Dimensionless unsteadiness parameter |
| : | Stretching surface velocity |
| : | Temperature (K) |
| : | Temperature at slit (K) |
| : | Temperature of slit (K) |
| : | Reference temperature (K) |
| : | Concentration (kgm−3) |
| : | Concentration at slit (kgm−3) |
| : | Concentration of slit (kgm−3) |
| : | Reference concentration (kgm−3). |
| : | Constants |
| : | Dimensionless film thickness |
| : | Thermal diffusivity (m2s−1) |
| : | Electrical conductivity (s (cm)−1) |
| : | Similarity variable |
| : | Dynamic viscosity (Nsm−2) |
| : | Kinematic viscosity (m2s−1) |
| : | Dimensionless film thickness |
| : | Temperature similarity variable |
| : | Concentration similarity variable |
| : | Artificial small parameter |
| : | Density (kgm−3) |
| : | Surface tension (Nm−1). |
| : | Thermal quantity |
| : | Solutal quantity |
| : | At the slit |
| : | At the elastic sheet. |
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (nos. 21206009 and 21576023) and Funding Project for Academic Human Resources Development in Beijing University of Civil Engineering and Architecture (no. 21221214111).