Series Solutions for Marangoni Convection on a Vertical Surface

The problem of steady, laminar, thermal Marangoni convection flow of Newtonian fluid over a flat surface is investigated. The boundary layer equations for the momentum and energy equations are transformed with the similarity solutions to ODEs to obtain the analytical approximate solutions. The analysis assumes that the temperature variation is a power law function of the location. The approximate solutions to the similarity equations are obtained by exponential series. The effects of the power law exponent and Prandtl number on the velocity and temperature profiles are presented.


Introduction
For many metallurgical and materials processing applications in space, it has been evidenced that if a free liquid surface or liquid-liquid interface is present, a surface tension gradient of this free surface will cause liquid movement at this free surface which alternatively drives fluid movements in the corresponding phase.This phenomenon is referred to as capillary motion.When a free liquid surface is present, the surface tension variation resulting from the temperature or concentration gradient along the surface can also induce motion in the fluid called solutal capillary, and thermocapillary motion, respectively.
The study of liquid movement resulting from thermocapillarity or so-called Marangoni convection is very important for a liquid system either in microgravity or in normal gravity 1 .Under normal gravity, liquid movement is mainly driven by buoyancy force because of the temperature-dependent density while the liquid is exposed to a temperature gradient field.As the size of the liquid system decreases especially having the size decrease in the direction of gravity, the buoyancy effect begins to diminish and the Marangoni effect will then dominate the system as the main driving force for liquid interface movement.In the absence of gravity, Marangoni convection always plays a main role in the determination of the fluid movement because of varying liquid surface tension in a temperature gradient field governing equations for two-dimensional Navier-Stokes and energy equations describing thermocapillary flows in a liquid layer of infinite extent are considered.The layer is bounded by a horizontal rigid plate from one side and opened from the other one.The rigid boundary is considered as thermally insulated.The physical properties of the liquid are assumed to be constant except the surface tension.This investigation are based on the balance laws of mass, momentum and energy.In the domain x > 0, y > 0, these can be written in the form 2 : where the x and y axes are taken parallel and perpendicular to the surface, u and v are the parallel and normal velocity components to the surface, respectively, and α f denotes the thermal diffusivity, K μ/ρ is the kinematic viscosity and density ratio of the ambient fluid.Marangoni effect is incorporated as a boundary condition relating the temperature field to the velocity.The boundary conditions at the surface at y 0 are as y → ∞, where σ T dσ/dT, A denotes the temperature gradient coefficient, m is a parameter relating to the power law exponent.The case m 0 refers to a linear profile, m 1 to the quadratic one.The minimum value of m is −1 which corresponds to no temperature variation on the surface and no Marangoni induced flow. Introducing For 2.3 by the similarity temperature function Θ with the corresponding boundary conditions we get where Pr μ/ ρα f is the Prandtl number.For the dimensionless stream function f η and the temperature field Θ η , the system 2.8 , 2.10 is derived and the primes denote the differentiation with respect to η.Now, the velocity components can be expressed by similarity function f as follows where It should be noted that u and v are proportional to x 2m 1 /3 and x m−1 /3 , respectively.It means, that for m −1/2 the velocity component u is a constant on the upper surface of the boundary layer.If m 1 then η 3 m 1 ρσ T A/μ 2 y.In the case of m > 1, v is proportional to x m−1 /3 and is strictly monotone increasing to infinity as x tends to infinity, which is not accepted in physics.Therefore, we restrict our investigations for the interval −1 < m ≤ 1.
We note that the special case m 1 do admit explicit solution.In 21, 22 the solution to 2.8 , 2.9 is given by and easy computation shows that is the solution to 2.10 , 2.11 .Due to the inherent complexity of such flows, to give exact analytical solutions of˜Marangoni flows are almost impossible.Exact analytical solutions were given by Magyari and Chamkha for thermosolutal Marangoni convection when the wall temperature and concentration variations are quadratic functions of the location 6 .
Our goal is to present approximate exponential series solution to the nonlinear boundary value problem 2.8 , 2.9 , moreover to 2.10 , 2.11 .Several values of the power law exponent and Prandtl number are considered.The influences of the effects of these parameters are illustrated.

Exponential Series Solution
First, our aim is to determine the approximate local solution of f η to 2.8 , 2.9 .We replace the condition at infinity by one at η 0. Therefore, 2.8 , 2.9 is converted into an initial value problem of 2.8 with initial conditions

3.1
In view of the third of the boundary conditions 2.9 , let us take the solution of the initial value problem 2.8 , 3.1 in the form where α > 0, A 0 3/ m 2 , A i i 1, 2, . . .are coefficients and α > 0 and d are constants.Conditions in 2.9 yield the following equations:

6 Mathematical Problems in Engineering
It may be remarked that the classic Briot-Bouquet theorem 23 guarantees the existence of formal solutions 3.2 to 2.8 , 3.1 , the value of A 0 , and also the convergence of formal solutions.
Let us introduce the new variable Z such as Z de −αη .

3.4
It is evident that the third boundary condition in 2.9 is automatically satisfied.From differential equation 2.8 with 3.2 we get

3.5
Equating the coefficients of like powers of Z one can obtain the expressions for coefficients A 2 , A 3 , . . .with m and A 1 :

3.6
From system 3.3 with the choice of A 1 1 the parameter values of d and α can be numerically determined.By these parameters the complete series solution 3.2 is reached.The calcutaled values of d, α and f 0 are shown in Table 1, and the variation of f 0 with m on Figure 1.The series forms for f η and f η are given below for some special values of the exponent m m −0.
3.7 It can be seen that for the case m 1 the obtained solution coincides with the exact solution 2.13 .The effect of the exponent m on the velocity profiles f η is illustrated in Figure 2. The values of f 0 ζ decrease as m is changing from negative values to positive ones.Applying the series solution for f the second-order linear differential equation 2.10 for Θ can be solved similarly, which presents the temperature distribution.Here we define Θ η as the series with coefficients B i i 0, 1, 2, . . .and hence the individual coefficients will be determined from differential equation 2.10 with 3.2 as follows . . .Remark that these coefficients as expressions of B 0 can be calculated only for noninteger values of the low Prandtl numbers.In 2.11 the second boundary condition is automatically satisfied, and from the first condition coefficient B 0 is to be determined, that is, from the equation

3.11
It may be noted that the Prandtl number Pr 298 corresponds to the power transformer oil.We point out that for the case m 1 the solution Θ η coincides with the exact solution 2.14 .
The effects of the power law exponent m and the Prandtl number are exhibited in Figures 3-14, where Pr 0.27 corresponds to the mercury and Pr 0.7 corresponds to the air.

Conclusion
In this paper the incompressible flow and heat transfer over a flat impermeable plate has been investigated.The resulting governing equations have been transformed into a system of nonlinear ordinary differential equations by applying suitable similarity transformation.For these equations approximate exponential series solutions are determined and the effects of the power exponent and the Prandtl number are illustrated in Figures 1-14, and it is observed that the values of f decrease as power exponent m increases see Figures 1 and 2 , moreover the boundary layer thickness increases as m or Pr increases.The temperature profiles are exhibited in Figures 3-8, it is observed that for low Prandtl number the temperature Θ decreases as Pr increases and for high Prandtl numbers the influence of Pr is opposite.

Figures 3 -
5 illustrate the influence of the Prandtl number on the temperature Θ for m 1.It can be observed in Figure 3 that for low Prandtl numbers 0.27 ≤ Pr ≤ 1.00001 the maximum value of Θ decreases as Pr increases and for a high Prandtl numbers 2.5 ≤ Pr ≤ 7.00001 and 70 ≤ Pr ≤ 298 the maximum value of Θ increases as Pr increases.In all three cases the boundary layer thickness increases as Pr increases.Figures 6-8 depict the effect power exponent m for fixed values of Pr.It can be observed in Figures 6 and 7 that the boundary layer thickness increases as m increases and the maximum value of Θ decreases as m increases for Pr 0.27, 2.2, while for high Prandtl number Pr 298 the reverse effect of m on the

Figure 9 :
Figure 9: The effect of the Prandtl number on Θ for m 1 0.27 ≤ Pr ≤ 1.00001 .

Figure 10 :
Figure 10: The effect of the Prandtl number on Θ for m 1 2.5 ≤ Pr ≤ 7.00001 .