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

Authors: Yan Zhang; Min Zhang; Shujuan Qi
Journal: Mathematical Problems in Engineering
 (2016)

---

## 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.


---

## Body

## 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 S, Hartmann number Ma, the Prandtl number Pr, the Schmidt number Sc, the thermocapillary number M
1, and the solutal capillary number M
2 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 Figure1. A variable magnetic field B
=
B
0
/
1
-
α
t
1
/
2 normal to the stretching sheet is applied, where α is a positive constant. And in the above model, we take concentration into consideration. A thin liquid film with uniform thickness h
(
t
) 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 by(1)
∂
u
∂
x
+
∂
v
∂
y
=
0
,
(2)
∂
u
∂
t
+
u
∂
u
∂
x
+
v
∂
u
∂
y
=
υ
∂
2
u
∂
y
2
-
σ
^
B
2
ρ
u
,
∂
T
∂
t
+
u
∂
T
∂
x
+
v
∂
T
∂
y
=
κ
∂
2
T
∂
y
2
,
∂
C
∂
t
+
u
∂
C
∂
x
+
v
∂
C
∂
y
=
D
∂
2
C
∂
y
2
.The boundary conditions are(3)
u
=
U
s
,
v
=
0
,
T
=
T
s
,
C
=
C
s
,
at
y
=
0
,
μ
∂
u
∂
y
=
∂
σ
∂
x
,
∂
T
∂
y
=
0
,
∂
C
∂
y
=
0
,
v
=
d
h
d
t
,
at
y
=
h
t
,where u and v are the velocity components of the fluid in the x- and y-directions, t is the time, υ is the kinematic viscosity, σ
^ is the electrical conductivity, B is the magnetic field, ρ is the density, T is the temperature, κ is the thermal diffusivity, C is the concentration, D is the mass diffusivity, μ is dynamic viscosity, and h
(
t
) is the uniform thickness of the liquid film. The dependence of surface tension on the temperature and concentration can be expressed as [40](4)
σ
=
σ
0
1
-
δ
T
T
-
T
0
-
δ
C
C
-
C
0
,where δ
T
=
-
1
/
σ
0
∂
σ
/
∂
T
C and δ
C
=
-
1
/
σ
0
∂
σ
/
∂
C
T. 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 U
s
x
,
t is assumed to be of the same form as that considered by Wang [17]:(5)
U
s
=
b
x
1
-
α
t
,where b is a positive constant and denotes the initial stretching rate. With unsteady stretching (i.e., α
≠
0), however, α
-
1 becomes the representative time scale of the resulting unsteady boundary layer problem. The adopted formulation of the velocity sheet U
s
x
,
t in (5) is valid only for times t
<
α
-
1 unless α
=
0. 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](6)
T
s
=
T
0
-
T
r
e
f
b
x
2
2
υ
1
-
α
t
-
3
/
2
,
C
s
=
C
0
-
C
r
e
f
b
x
2
2
υ
1
-
α
t
-
3
/
2
,where T
0 and C
0 are the temperature and the concentration at the slit, respectively, and T
r
e
f and C
r
e
f are the reference temperature and the reference concentration in the case of t
<
α
-
1, respectively. Equation (6) represents a situation in which the sheet temperature and concentration decrease from T
0 and C
0 at the slot in proportion to x
2. 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.Figure 1
Physics model with interface condition and coordinate system.


### 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:(7)
u
=
b
x
1
-
α
t
f
′
η
,
v
=
-
υ
b
1
-
α
t
1
/
2
f
η
β
,
T
=
T
0
-
T
r
e
f
b
x
2
2
υ
1
-
α
t
-
3
/
2
θ
η
,
C
=
C
0
-
C
r
e
f
b
x
2
2
υ
1
-
α
t
-
3
/
2
ϕ
η
,where the similarity variable η is given by Wang [19](8)
η
=
b
υ
1
/
2
1
-
α
t
-
1
/
2
y
β
-
1
,and β is yet an unknown constant denoting the dimensionless film thickness, defined by(9)
β
=
b
υ
1
/
2
1
-
α
t
-
1
/
2
h
t
.Substituting (7)–(9) into (1)–(3), (1)–(3) can be transformed into the following nonlinear ordinary differential equations:(10)
f
′
′
′
+
γ
f
f
′
′
-
f
′
2
-
1
2
η
S
f
′
′
-
S
+
M
a
f
′
=
0
,
P
r
-
1
θ
′
′
+
γ
f
θ
′
-
3
2
S
θ
-
1
2
S
θ
′
η
-
2
f
′
θ
=
0
,
S
c
-
1
ϕ
′
′
+
γ
f
ϕ
′
-
3
2
S
ϕ
-
1
2
S
ϕ
′
η
-
2
f
′
ϕ
=
0
,subject to the boundary conditions(11)
f
0
=
0
,
f
′
0
=
1
,
θ
0
=
1
,
ϕ
0
=
1
,
at
η
=
0
,
(12)
f
1
=
S
2
,
f
′
′
1
=
M
1
θ
1
+
M
2
ϕ
1
,
θ
′
1
=
0
,
ϕ
′
1
=
0
,
at
η
=
1
,where M
a
=
σ
^
B
0
/
(
ρ
b
) is Hartmann number, S
=
α
/
b is the dimensionless unsteadiness parameter, P
r
=
υ
/
κ is the Prandtl number, S
c
=
υ
/
D is the Schmidt number, γ
=
β
2 is the dimensionless film thickness, and the thermocapillary number M
1 and the solutal capillary number M
2 are defined as M
1
=
δ
T
σ
0
T
r
e
f
β
/
μ
b
υ and M
2
=
δ
C
σ
0
C
r
e
f
β
/
μ
b
υ, respectively.The skin friction coefficientC
f and the Nusselt number N
u
x are defined as(13)
C
f
=
τ
w
ρ
U
s
2
/
2
,
N
u
x
=
q
w
x
k
T
r
e
f
,where the wall skin friction τ
w and the sheet heat flux q
w are(14)
τ
w
=
μ
∂
u
∂
y
y
=
0
,
q
w
=
-
k
∂
T
∂
y
y
=
0
.Substituting (14) into (13), then we get(15)
C
f
=
2
β
f
′
′
0
R
e
x
-
1
/
2
,
N
u
x
=
1
2
β
1
-
α
t
θ
′
0
R
e
x
3
/
2
,where R
e
x
=
U
s
x
/
υ 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 ε; 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:(16)
f
η
=
ε
g
ξ
+
η
+
α
1
2
η
2
=
ε
g
ξ
+
ε
1
/
3
ξ
+
α
1
2
ε
2
/
3
ξ
2
,
θ
η
=
ε
2
/
3
h
ξ
+
1
+
α
2
η
=
ε
2
/
3
h
ξ
+
1
+
α
2
ε
1
/
3
ξ
,
ϕ
η
=
ε
2
/
3
ϖ
ξ
+
1
+
α
3
η
=
ε
2
/
3
ϖ
ξ
+
1
+
α
3
ε
1
/
3
ξ
,where ξ
=
ε
-
1
/
3
η and ε is an artificial small parameter. We assume f
′
′
0
=
α
1, θ
′
0
=
α
2, and ϕ
′
0
=
α
3; here α
1, α
2, and α
3 are constants; the boundary conditions become (17)
g
0
=
0
,
g
′
0
=
0
,
g
′
′
0
=
0
,
h
0
=
0
,
h
′
0
=
0
,
ϖ
0
=
0
,
ϖ
′
0
=
0
.Equation (10) reduces to(18)
g
′
′
′
ξ
+
γ
ε
g
ξ
+
ε
1
/
3
ξ
+
α
1
2
ε
2
/
3
ξ
2
ε
1
/
3
g
′
′
+
α
1
-
1
2
S
ε
1
/
3
ξ
ε
1
/
3
g
′
′
+
α
1
-
ε
2
/
3
g
′
+
1
+
α
1
ε
1
/
3
ξ
2
-
S
+
M
a
ε
2
/
3
g
′
+
1
+
α
1
ε
1
/
3
ξ
=
0
,
h
′
′
ξ
P
r
+
γ
ε
g
0
ξ
+
ε
1
/
3
ξ
+
α
1
2
ε
2
/
3
ξ
2
ε
1
/
3
h
′
ξ
+
α
2
-
2
ε
2
/
3
g
′
ξ
+
1
+
α
1
ε
1
/
3
ξ
ε
2
/
3
h
ξ
+
1
+
α
2
ε
1
/
3
ξ
-
1
2
S
ε
1
/
3
ξ
ε
1
/
3
h
′
ξ
+
α
2
-
3
S
2
ε
2
/
3
h
ξ
+
1
+
α
2
ε
1
/
3
ξ
=
0
,
ϖ
′
′
ξ
S
c
+
γ
ε
g
0
ξ
+
ε
1
/
3
ξ
+
α
1
2
ε
2
/
3
ξ
2
ε
1
/
3
ϖ
′
ξ
+
α
3
-
2
ε
2
/
3
g
′
ξ
+
1
+
α
1
ε
1
/
3
ξ
ε
2
/
3
ϖ
ξ
+
1
+
α
3
ε
1
/
3
ξ
-
1
2
S
ε
1
/
3
ξ
ε
1
/
3
ϖ
′
ξ
+
α
3
-
3
S
2
ε
2
/
3
ϖ
ξ
+
1
+
α
3
ε
1
/
3
ξ
=
0
.And we can assume(19)
g
ξ
=
g
0
ξ
+
ε
1
/
3
g
1
ξ
+
ε
2
/
3
g
2
ξ
+
ε
g
3
ξ
+
ε
4
/
3
g
4
ξ
+
⋯
,
h
ξ
=
h
0
ξ
+
ε
1
/
3
h
1
ξ
+
ε
2
/
3
h
2
ξ
+
ε
h
3
ξ
+
ε
4
/
3
h
4
ξ
+
⋯
,
ϖ
ξ
=
ϖ
0
ξ
+
ε
1
/
3
ϖ
1
ξ
+
ε
2
/
3
ϖ
2
ξ
+
ε
ϖ
3
ξ
+
ε
4
/
3
ϖ
4
ξ
+
⋯with the boundary conditions (20)
g
n
0
=
0
,
g
n
′
0
=
0
,
g
n
′
′
0
=
0
,
g
n
′
′
′
0
=
0
,
n
=
0,1
,
2
,
…
,
h
n
0
=
0
,
h
n
′
0
=
0
,
h
n
′
′
0
=
0
,
n
=
0,1
,
2
,
…
,
ϖ
n
0
=
0
,
ϖ
n
′
0
=
0
,
ϖ
n
′
′
0
=
0
,
n
=
0,1
,
2
,
…
.Let S
=
0.8 and M
a
=
1, and we can obtain(21)
f
η
=
η
+
1
2
α
1
η
2
+
7
15
γ
η
3
+
2
15
α
1
γ
η
4
+
1
120
α
1
2
γ
+
91
1500
γ
2
η
5
+
1
60
α
1
γ
2
η
6
+
23
25200
α
1
2
γ
2
+
77
15000
γ
3
η
7
+
589
504000
α
1
γ
3
-
77
40320
α
1
3
γ
2
η
8
+
383
9072000
α
1
2
γ
3
+
3871
16200000
γ
4
η
9
+
1
432000
α
1
3
γ
3
+
209
7560000
α
1
γ
4
η
10
+
⋯
,
f
′
η
=
1
+
α
1
η
+
7
45
γ
η
2
+
1
30
α
1
γ
η
3
+
1
600
α
1
2
γ
+
91
4500
γ
2
η
4
+
1
360
α
1
γ
2
η
5
+
23
176400
α
1
2
γ
2
+
11
15000
γ
3
η
6
+
589
4672000
α
1
γ
3
-
77
322560
α
1
3
γ
2
η
7
+
383
81648000
α
1
2
γ
3
+
3871
145800000
γ
4
η
8
+
1
4320000
α
1
3
γ
3
+
209
75600000
α
1
γ
4
η
9
+
⋯
,
f
′
′
η
=
α
1
+
7
90
γ
η
+
1
90
α
1
γ
η
2
+
1
2400
α
1
2
γ
+
91
18000
γ
2
η
3
+
1
1800
α
1
γ
2
η
4
+
23
1058400
α
1
2
γ
2
+
11
90000
γ
3
η
5
+
589
34804000
α
1
γ
3
-
77
2257920
α
1
3
γ
2
η
6
+
383
653184000
α
1
2
γ
3
+
3871
116400000
γ
4
η
7
+
1
38880000
α
1
3
γ
3
+
209
680400000
α
1
γ
4
η
8
+
⋯
,
θ
η
=
α
2
η
+
8
5
γ
η
2
+
13
30
γ
α
2
+
1
3
α
1
γ
η
3
+
1
2
γ
2
+
1
8
γ
α
2
α
1
η
4
+
147
1000
γ
2
α
2
+
47
300
α
1
γ
2
η
5
+
1
120
γ
2
α
1
2
+
599
4500
γ
3
+
1
24
γ
2
α
1
α
2
η
6
+
989
21000
α
1
γ
3
+
2533
9000
γ
3
α
2
+
1
560
γ
2
α
2
α
1
2
η
7
+
⋯
,
θ
′
η
=
α
2
+
4
5
γ
η
+
13
90
γ
α
2
+
1
9
α
1
γ
η
2
+
1
8
γ
2
+
1
32
γ
α
1
α
2
η
3
+
147
5000
γ
2
α
2
+
47
1500
α
1
γ
2
η
4
+
1
720
γ
2
α
1
2
+
599
27000
γ
3
+
1
144
γ
2
α
1
α
2
η
5
+
989
147000
α
1
γ
3
+
2533
63000
γ
3
α
2
+
1
3920
γ
2
α
2
α
1
2
η
6
+
⋯
,
ϕ
η
=
1
+
α
3
η
+
16
5
γ
η
2
+
13
15
γ
α
3
+
2
3
α
1
γ
η
3
+
23
15
γ
2
+
1
4
γ
α
1
α
3
η
4
+
133
375
γ
2
α
3
+
13
25
γ
2
α
1
η
5
+
1171
2250
γ
3
+
1
36
γ
2
α
1
2
+
47
450
γ
2
α
1
α
3
η
6
+
1
280
γ
2
α
3
α
1
2
+
1424
7875
γ
3
α
1
+
1961
22500
γ
3
α
3
η
7
+
⋯
,
ϕ
′
η
=
α
3
+
8
5
γ
η
+
13
45
γ
α
3
+
2
9
α
1
γ
η
2
+
23
60
γ
2
+
1
16
γ
α
1
α
3
η
3
+
133
1875
γ
2
α
3
+
13
125
γ
2
α
1
η
4
+
1171
13500
γ
3
+
1
216
γ
2
α
1
2
+
47
2700
γ
2
α
1
α
3
η
5
+
1
1960
γ
2
α
3
α
1
2
+
1424
22125
γ
3
α
1
+
1961
157500
γ
3
α
3
η
6
+
⋯
.Substituting (21) into (12), we can get four equations involving the four variables such as α
1, α
2, α
3, and γ. Analytical solutions of (10)–(12) in the form of series can be obtained with this method.


## 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 Figure1. A variable magnetic field B
=
B
0
/
1
-
α
t
1
/
2 normal to the stretching sheet is applied, where α is a positive constant. And in the above model, we take concentration into consideration. A thin liquid film with uniform thickness h
(
t
) 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 by(1)
∂
u
∂
x
+
∂
v
∂
y
=
0
,
(2)
∂
u
∂
t
+
u
∂
u
∂
x
+
v
∂
u
∂
y
=
υ
∂
2
u
∂
y
2
-
σ
^
B
2
ρ
u
,
∂
T
∂
t
+
u
∂
T
∂
x
+
v
∂
T
∂
y
=
κ
∂
2
T
∂
y
2
,
∂
C
∂
t
+
u
∂
C
∂
x
+
v
∂
C
∂
y
=
D
∂
2
C
∂
y
2
.The boundary conditions are(3)
u
=
U
s
,
v
=
0
,
T
=
T
s
,
C
=
C
s
,
at
y
=
0
,
μ
∂
u
∂
y
=
∂
σ
∂
x
,
∂
T
∂
y
=
0
,
∂
C
∂
y
=
0
,
v
=
d
h
d
t
,
at
y
=
h
t
,where u and v are the velocity components of the fluid in the x- and y-directions, t is the time, υ is the kinematic viscosity, σ
^ is the electrical conductivity, B is the magnetic field, ρ is the density, T is the temperature, κ is the thermal diffusivity, C is the concentration, D is the mass diffusivity, μ is dynamic viscosity, and h
(
t
) is the uniform thickness of the liquid film. The dependence of surface tension on the temperature and concentration can be expressed as [40](4)
σ
=
σ
0
1
-
δ
T
T
-
T
0
-
δ
C
C
-
C
0
,where δ
T
=
-
1
/
σ
0
∂
σ
/
∂
T
C and δ
C
=
-
1
/
σ
0
∂
σ
/
∂
C
T. 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 U
s
x
,
t is assumed to be of the same form as that considered by Wang [17]:(5)
U
s
=
b
x
1
-
α
t
,where b is a positive constant and denotes the initial stretching rate. With unsteady stretching (i.e., α
≠
0), however, α
-
1 becomes the representative time scale of the resulting unsteady boundary layer problem. The adopted formulation of the velocity sheet U
s
x
,
t in (5) is valid only for times t
<
α
-
1 unless α
=
0. 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](6)
T
s
=
T
0
-
T
r
e
f
b
x
2
2
υ
1
-
α
t
-
3
/
2
,
C
s
=
C
0
-
C
r
e
f
b
x
2
2
υ
1
-
α
t
-
3
/
2
,where T
0 and C
0 are the temperature and the concentration at the slit, respectively, and T
r
e
f and C
r
e
f are the reference temperature and the reference concentration in the case of t
<
α
-
1, respectively. Equation (6) represents a situation in which the sheet temperature and concentration decrease from T
0 and C
0 at the slot in proportion to x
2. 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.Figure 1
Physics model with interface condition and coordinate system.


## 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:(7)
u
=
b
x
1
-
α
t
f
′
η
,
v
=
-
υ
b
1
-
α
t
1
/
2
f
η
β
,
T
=
T
0
-
T
r
e
f
b
x
2
2
υ
1
-
α
t
-
3
/
2
θ
η
,
C
=
C
0
-
C
r
e
f
b
x
2
2
υ
1
-
α
t
-
3
/
2
ϕ
η
,where the similarity variable η is given by Wang [19](8)
η
=
b
υ
1
/
2
1
-
α
t
-
1
/
2
y
β
-
1
,and β is yet an unknown constant denoting the dimensionless film thickness, defined by(9)
β
=
b
υ
1
/
2
1
-
α
t
-
1
/
2
h
t
.Substituting (7)–(9) into (1)–(3), (1)–(3) can be transformed into the following nonlinear ordinary differential equations:(10)
f
′
′
′
+
γ
f
f
′
′
-
f
′
2
-
1
2
η
S
f
′
′
-
S
+
M
a
f
′
=
0
,
P
r
-
1
θ
′
′
+
γ
f
θ
′
-
3
2
S
θ
-
1
2
S
θ
′
η
-
2
f
′
θ
=
0
,
S
c
-
1
ϕ
′
′
+
γ
f
ϕ
′
-
3
2
S
ϕ
-
1
2
S
ϕ
′
η
-
2
f
′
ϕ
=
0
,subject to the boundary conditions(11)
f
0
=
0
,
f
′
0
=
1
,
θ
0
=
1
,
ϕ
0
=
1
,
at
η
=
0
,
(12)
f
1
=
S
2
,
f
′
′
1
=
M
1
θ
1
+
M
2
ϕ
1
,
θ
′
1
=
0
,
ϕ
′
1
=
0
,
at
η
=
1
,where M
a
=
σ
^
B
0
/
(
ρ
b
) is Hartmann number, S
=
α
/
b is the dimensionless unsteadiness parameter, P
r
=
υ
/
κ is the Prandtl number, S
c
=
υ
/
D is the Schmidt number, γ
=
β
2 is the dimensionless film thickness, and the thermocapillary number M
1 and the solutal capillary number M
2 are defined as M
1
=
δ
T
σ
0
T
r
e
f
β
/
μ
b
υ and M
2
=
δ
C
σ
0
C
r
e
f
β
/
μ
b
υ, respectively.The skin friction coefficientC
f and the Nusselt number N
u
x are defined as(13)
C
f
=
τ
w
ρ
U
s
2
/
2
,
N
u
x
=
q
w
x
k
T
r
e
f
,where the wall skin friction τ
w and the sheet heat flux q
w are(14)
τ
w
=
μ
∂
u
∂
y
y
=
0
,
q
w
=
-
k
∂
T
∂
y
y
=
0
.Substituting (14) into (13), then we get(15)
C
f
=
2
β
f
′
′
0
R
e
x
-
1
/
2
,
N
u
x
=
1
2
β
1
-
α
t
θ
′
0
R
e
x
3
/
2
,where R
e
x
=
U
s
x
/
υ 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 ε; 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:(16)
f
η
=
ε
g
ξ
+
η
+
α
1
2
η
2
=
ε
g
ξ
+
ε
1
/
3
ξ
+
α
1
2
ε
2
/
3
ξ
2
,
θ
η
=
ε
2
/
3
h
ξ
+
1
+
α
2
η
=
ε
2
/
3
h
ξ
+
1
+
α
2
ε
1
/
3
ξ
,
ϕ
η
=
ε
2
/
3
ϖ
ξ
+
1
+
α
3
η
=
ε
2
/
3
ϖ
ξ
+
1
+
α
3
ε
1
/
3
ξ
,where ξ
=
ε
-
1
/
3
η and ε is an artificial small parameter. We assume f
′
′
0
=
α
1, θ
′
0
=
α
2, and ϕ
′
0
=
α
3; here α
1, α
2, and α
3 are constants; the boundary conditions become (17)
g
0
=
0
,
g
′
0
=
0
,
g
′
′
0
=
0
,
h
0
=
0
,
h
′
0
=
0
,
ϖ
0
=
0
,
ϖ
′
0
=
0
.Equation (10) reduces to(18)
g
′
′
′
ξ
+
γ
ε
g
ξ
+
ε
1
/
3
ξ
+
α
1
2
ε
2
/
3
ξ
2
ε
1
/
3
g
′
′
+
α
1
-
1
2
S
ε
1
/
3
ξ
ε
1
/
3
g
′
′
+
α
1
-
ε
2
/
3
g
′
+
1
+
α
1
ε
1
/
3
ξ
2
-
S
+
M
a
ε
2
/
3
g
′
+
1
+
α
1
ε
1
/
3
ξ
=
0
,
h
′
′
ξ
P
r
+
γ
ε
g
0
ξ
+
ε
1
/
3
ξ
+
α
1
2
ε
2
/
3
ξ
2
ε
1
/
3
h
′
ξ
+
α
2
-
2
ε
2
/
3
g
′
ξ
+
1
+
α
1
ε
1
/
3
ξ
ε
2
/
3
h
ξ
+
1
+
α
2
ε
1
/
3
ξ
-
1
2
S
ε
1
/
3
ξ
ε
1
/
3
h
′
ξ
+
α
2
-
3
S
2
ε
2
/
3
h
ξ
+
1
+
α
2
ε
1
/
3
ξ
=
0
,
ϖ
′
′
ξ
S
c
+
γ
ε
g
0
ξ
+
ε
1
/
3
ξ
+
α
1
2
ε
2
/
3
ξ
2
ε
1
/
3
ϖ
′
ξ
+
α
3
-
2
ε
2
/
3
g
′
ξ
+
1
+
α
1
ε
1
/
3
ξ
ε
2
/
3
ϖ
ξ
+
1
+
α
3
ε
1
/
3
ξ
-
1
2
S
ε
1
/
3
ξ
ε
1
/
3
ϖ
′
ξ
+
α
3
-
3
S
2
ε
2
/
3
ϖ
ξ
+
1
+
α
3
ε
1
/
3
ξ
=
0
.And we can assume(19)
g
ξ
=
g
0
ξ
+
ε
1
/
3
g
1
ξ
+
ε
2
/
3
g
2
ξ
+
ε
g
3
ξ
+
ε
4
/
3
g
4
ξ
+
⋯
,
h
ξ
=
h
0
ξ
+
ε
1
/
3
h
1
ξ
+
ε
2
/
3
h
2
ξ
+
ε
h
3
ξ
+
ε
4
/
3
h
4
ξ
+
⋯
,
ϖ
ξ
=
ϖ
0
ξ
+
ε
1
/
3
ϖ
1
ξ
+
ε
2
/
3
ϖ
2
ξ
+
ε
ϖ
3
ξ
+
ε
4
/
3
ϖ
4
ξ
+
⋯with the boundary conditions (20)
g
n
0
=
0
,
g
n
′
0
=
0
,
g
n
′
′
0
=
0
,
g
n
′
′
′
0
=
0
,
n
=
0,1
,
2
,
…
,
h
n
0
=
0
,
h
n
′
0
=
0
,
h
n
′
′
0
=
0
,
n
=
0,1
,
2
,
…
,
ϖ
n
0
=
0
,
ϖ
n
′
0
=
0
,
ϖ
n
′
′
0
=
0
,
n
=
0,1
,
2
,
…
.Let S
=
0.8 and M
a
=
1, and we can obtain(21)
f
η
=
η
+
1
2
α
1
η
2
+
7
15
γ
η
3
+
2
15
α
1
γ
η
4
+
1
120
α
1
2
γ
+
91
1500
γ
2
η
5
+
1
60
α
1
γ
2
η
6
+
23
25200
α
1
2
γ
2
+
77
15000
γ
3
η
7
+
589
504000
α
1
γ
3
-
77
40320
α
1
3
γ
2
η
8
+
383
9072000
α
1
2
γ
3
+
3871
16200000
γ
4
η
9
+
1
432000
α
1
3
γ
3
+
209
7560000
α
1
γ
4
η
10
+
⋯
,
f
′
η
=
1
+
α
1
η
+
7
45
γ
η
2
+
1
30
α
1
γ
η
3
+
1
600
α
1
2
γ
+
91
4500
γ
2
η
4
+
1
360
α
1
γ
2
η
5
+
23
176400
α
1
2
γ
2
+
11
15000
γ
3
η
6
+
589
4672000
α
1
γ
3
-
77
322560
α
1
3
γ
2
η
7
+
383
81648000
α
1
2
γ
3
+
3871
145800000
γ
4
η
8
+
1
4320000
α
1
3
γ
3
+
209
75600000
α
1
γ
4
η
9
+
⋯
,
f
′
′
η
=
α
1
+
7
90
γ
η
+
1
90
α
1
γ
η
2
+
1
2400
α
1
2
γ
+
91
18000
γ
2
η
3
+
1
1800
α
1
γ
2
η
4
+
23
1058400
α
1
2
γ
2
+
11
90000
γ
3
η
5
+
589
34804000
α
1
γ
3
-
77
2257920
α
1
3
γ
2
η
6
+
383
653184000
α
1
2
γ
3
+
3871
116400000
γ
4
η
7
+
1
38880000
α
1
3
γ
3
+
209
680400000
α
1
γ
4
η
8
+
⋯
,
θ
η
=
α
2
η
+
8
5
γ
η
2
+
13
30
γ
α
2
+
1
3
α
1
γ
η
3
+
1
2
γ
2
+
1
8
γ
α
2
α
1
η
4
+
147
1000
γ
2
α
2
+
47
300
α
1
γ
2
η
5
+
1
120
γ
2
α
1
2
+
599
4500
γ
3
+
1
24
γ
2
α
1
α
2
η
6
+
989
21000
α
1
γ
3
+
2533
9000
γ
3
α
2
+
1
560
γ
2
α
2
α
1
2
η
7
+
⋯
,
θ
′
η
=
α
2
+
4
5
γ
η
+
13
90
γ
α
2
+
1
9
α
1
γ
η
2
+
1
8
γ
2
+
1
32
γ
α
1
α
2
η
3
+
147
5000
γ
2
α
2
+
47
1500
α
1
γ
2
η
4
+
1
720
γ
2
α
1
2
+
599
27000
γ
3
+
1
144
γ
2
α
1
α
2
η
5
+
989
147000
α
1
γ
3
+
2533
63000
γ
3
α
2
+
1
3920
γ
2
α
2
α
1
2
η
6
+
⋯
,
ϕ
η
=
1
+
α
3
η
+
16
5
γ
η
2
+
13
15
γ
α
3
+
2
3
α
1
γ
η
3
+
23
15
γ
2
+
1
4
γ
α
1
α
3
η
4
+
133
375
γ
2
α
3
+
13
25
γ
2
α
1
η
5
+
1171
2250
γ
3
+
1
36
γ
2
α
1
2
+
47
450
γ
2
α
1
α
3
η
6
+
1
280
γ
2
α
3
α
1
2
+
1424
7875
γ
3
α
1
+
1961
22500
γ
3
α
3
η
7
+
⋯
,
ϕ
′
η
=
α
3
+
8
5
γ
η
+
13
45
γ
α
3
+
2
9
α
1
γ
η
2
+
23
60
γ
2
+
1
16
γ
α
1
α
3
η
3
+
133
1875
γ
2
α
3
+
13
125
γ
2
α
1
η
4
+
1171
13500
γ
3
+
1
216
γ
2
α
1
2
+
47
2700
γ
2
α
1
α
3
η
5
+
1
1960
γ
2
α
3
α
1
2
+
1424
22125
γ
3
α
1
+
1961
157500
γ
3
α
3
η
6
+
⋯
.Substituting (21) into (12), we can get four equations involving the four variables such as α
1, α
2, α
3, and γ. 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 Tables1–7, respectively. Based on Table 1, the values of dimensionless film thickness β and skin friction coefficient -
f
′
′
0 obtained are compared with the results by Wang [19] for the case M
a
=
0, and good agreement can be seen. From Table 2, it can be observed that increasing the value of unsteadiness parameter S will decrease the film thickness β, the skin friction coefficient -
f
′
′
0, the heat flux -
θ
′
0, and the diffusion flux -
ϕ
′
0 but increase the free velocity f
′
1, free temperature θ
(
1
), and free concentration ϕ
(
1
). 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 -
f
′
′
0.Table 1
Comparison of values of film thicknessβ and skin friction coefficient -
f
′
′
(
0
) with Ma = 0.
S
Wang [19]
Present results
β
-
f
′
′
(
0
)
β
-
f
′
′
(
0
)
0.8
2.151990
2.680940
2.152839
2.681920
1.0
1.543620
1.972385
1.543620
1.972385
1.2
1.127781
1.442631
1.127781
1.442625
1.4
0.821032
1.012784
0.821032
1.012780Table 2
Values ofβ, f
′
(
1
), θ
(
1
) and ϕ
(
1
), -
f
′
′
(
0
), -
θ
′
(
0
), and -
ϕ
′
(
0
) with Ma = 1.2, M
1
=
1, M
2
=
1, Pr = 1, and Sc = 1.8 and S is varied.
S
β
f
′
(
1
)
θ
(
1
)
ϕ
(
1
)
-
f
′
′
(
0
)
-
θ
′
(
0
)
-
ϕ
′
(
0
)
0.8
1.706139
0.245858
0.190409
0.079435
2.791406
2.738673
3.835584
1.0
1.340872
0.387092
0.249442
0.118379
2.180379
2.252363
3.160992
1.2
1.088400
0.541510
0.309461
0.161276
1.714381
1.897617
2.678301Table 3
Values ofβ, f
′
(
1
), θ
(
1
) and ϕ
(
1
), -
f
′
′
(
0
), -
θ
′
(
0
), and -
ϕ
′
(
0
) with S
=
0.8, M
1
=
1, M
2
=
1, Pr = 1, and Sc = 1.8 and Ma is varied.
Ma
β
f
′
(
1
)
θ
(
1
)
ϕ
(
1
)
-
f
′
′
(
0
)
-
θ
′
(
0
)
-
ϕ
′
(
0
)
0.5
1.955885
0.228035
0.131483
0.047797
2.776933
3.21537
4.463988
1.5
1.629020
0.252600
0.211365
0.092719
2.802801
2.590747
3.640895
3.0
1.370781
0.283080
0.299672
0.154063
2.866680
2.089107
2.983500Table 4
Values ofβ, f
′
(
1
), θ
(
1
) and ϕ
(
1
), -
f
′
′
(
0
), -
θ
′
(
0
), and -
ϕ
′
(
0
) with S = 0.8, Ma = 1, M
2
=
1, Pr = 1, and Sc = 1.8 and M
1 is varied.
M
1
β
f
′
(
1
)
θ
(
1
)
ϕ
(
1
)
-
f
′
′
(
0
)
-
θ
′
(
0
)
-
ϕ
′
(
0
)
1
1.765648
0.242329
0.178412
0.070413
2.783885
2.850602
3.984435
3
1.895449
0.296754
0.143799
0.054085
2.978572
3.080157
4.285635
5
1.987330
0.335046
0.123865
0.044928
3.120442
3.24009
4.498705
8
2.089998
0.380759
0.106127
0.036553
3.279882
3.415187
4.733832Table 5
Values ofβ, f
′
(
1
), θ
(
1
) and ϕ
(
1
), -
f
′
′
(
0
), -
θ
′
(
0
), and -
ϕ
′
(
0
) with S
=
0.8, Ma = 1, M
1
=
1, Pr = 1, and Sc = 1.8 and M
2 is varied.
M
2
β
f
′
(
1
)
θ
(
1
)
ϕ
(
1
)
-
f
′
′
(
0
)
-
θ
′
(
0
)
-
ϕ
′
(
0
)
1
1.765648
0.242329
0.178412
0.070413
2.783885
2.850602
3.984435
3
1.823807
0.265936
0.162696
0.062564
2.870952
2.95405
4.120261
5
1.870447
0.284213
0.148046
0.056928
2.942314
3.03796
4.229606
8
1.927131
0.30837
0.135444
0.050747
3.028503
3.136665
4.360637Table 6
Values ofβ, f
′
(
1
), θ
(
1
) and ϕ
(
1
), -
f
′
′
(
0
), -
θ
′
(
0
), and -
ϕ
′
(
0
) with S
=
0.8, Ma = 1, M
1
=
1, M
2
=
1, and Sc = 1.8 and Pr is varied.
Pr
β
f
′
(
1
)
θ
(
1
)
ϕ
(
1
)
-
f
′
′
(
0
)
-
θ
′
(
0
)
-
ϕ
′
(
0
)
1
1.765648
0.242329
0.178412
0.070413
2.783885
2.850602
3.984435
2
1.70801
0.216269
0.065849
0.079270
2.701304
4.083229
3.851739
5
1.675911
0.203976
0.009869
0.084604
2.654055
6.569892
3.775766
10
1.671722
0.20159
0.000697
0.085353
2.648593
9.447705
3.766435Table 7
Values ofβ, f
′
(
1
), θ
(
1
) and ϕ
(
1
), -
f
′
′
(
0
), -
θ
′
(
0
), and -
ϕ
′
(
0
) with S
=
0.8, Ma = 1, M
1
=
1, M
2
=
1, and Pr = 1 and Sc is varied.
Sc
β
f
′
(
1
)
θ
(
1
)
ϕ
(
1
)
-
f
′
′
(
0
)
-
θ
′
(
0
)
-
ϕ
′
(
0
)
0.5
1.884207
0.289866
0.144404
0.313734
2.963305
3.062276
2.011923
1
1.810615
0.259498
0.163349
0.160799
2.852006
2.932347
2.933503
1.5
1.77745
0.245319
0.170494
0.093966
2.80298
2.874491
3.62599
1.8
1.765648
0.242329
0.178412
0.070413
2.783885
2.850602
3.984435From Tables4 and 5, it can be seen that M
1 and M
2 have the same effects on velocity, temperature, and concentration fields. Increasing the value of the thermocapillary number M
1 or solutal capillary number M
2 can cause a descent in the free temperature θ
(
1
) and free concentration ϕ
1 but a rise in the film thickness β, the skin friction coefficient -
f
′
′
0, the heat flux -
θ
′
0, and the free velocity f
′
1. Table 6 shows that the values of the free concentration ϕ
(
1
) and the heat flux -
θ
′
0 increase while the film thickness β, the skin friction coefficient -
f
′
′
0, the diffusion flux -
ϕ
′
(
0
), the free velocity f
′
0, and free temperature θ
1 decrease with an increment in the Prandtl number Pr. Table 7 shows the effect of Schmidt number S
c 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 Figures2–7. 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 S 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 M
a 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 M
a 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.Figure 2
(a) Velocity profiles for different values ofS with M
a
=
1.2, M
1
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8. (b) Temperature profiles for different values of S with M
a
=
1.2, M
1
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8. (c) Concentration profiles for different values of S with M
a
=
1.2, M
1
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8.
(a)
(b)
(c)Figure 3
(a) Velocity profiles for different values of Ma withS
=
0.8, M
1
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8. (b) Temperature profiles for different values of M
a with S
=
0.8, M
1
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8. (c) Concentration profiles for different values of M
a with S
=
0.8, M
1
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8.
(a)
(b)
(c)Figure 4
(a) Velocity profiles for different values ofM
1 with S
=
0.8, M
a
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8. (b) Temperature profiles for different values of M
1 with S
=
0.8, M
a
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8. (c) Concentration profiles for different values of M
1 with S
=
0.8, M
a
=
1, M
2
=
1, P
r
=
1, and S
c
=
1.8.
(a)
(b)
(c)Figure 5
(a) Velocity profiles for different values ofM
2 with S
=
0.8, M
a
=
1, M
1
=
1, P
r
=
1, and S
c
=
1.8. (b) Temperature profiles for different values of M
2 with S
=
0.8, M
a
=
1, M
1
=
1, P
r
=
1, and S
c
=
1.8. (c) Concentration profiles for different values of M
2 with S
=
0.8, M
a
=
1, M
1
=
1, P
r
=
1, and S
c
=
1.8.
(a)
(b)
(c)Figure 6
Temperature profiles for different values of Pr withS
=
0.8, M
a
=
1, M
1
=
1, M
2
=
1, and S
c
=
1.8.Figure 7
Concentration profiles for different values ofS
c with S
=
0.8, M
a
=
1, M
1
=
1, M
2
=
1, and P
r
=
1.Figures4(a) and 5(a) show the velocity profiles for different values of thermocapillary number M
1 and solutal capillary number M
2. 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 M
1 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 M
1 is more remarkable in comparison to the solutal capillary number M
2.The effect of Pr on temperature is presented in Figure6. 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 S 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.



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*Source: 8521580-2016-07-31.xml*