Hydromagnetic steady flow of Maxwell fluid over a bidirectional stretching surface with prescribed surface temperature and prescribed surface heat flux.

This paper investigates the steady hydromagnetic three-dimensional boundary layer flow of Maxwell fluid over a bidirectional stretching surface. Both cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF) are considered. Computations are made for the velocities and temperatures. Results are plotted and analyzed for PST and PHF cases. Convergence analysis is presented for the velocities and temperatures. Comparison of PST and PHF cases is given and examined.

Introduction

Interest of recent researchers in analysis of boundary layer flows over a continuously moving surface with prescribed surface temperature or heat flux has increased substantially during the last few decades. These flows have abundant applications in many metallurgical and industrial processes. Specific examples of such industrial and technological processes include wire-drawing, glass-fiber and paper production, the extrusion of polymer sheets, the cooling of a metallic plate in a cooling bath, drawing of plastic films etc. Such situations occur in the class of flow problems relevant to the polymer extrusion in which the flow is generated by stretching of plastic surface [1], [2]. In addition, internal heat generation/absorption has key role in the heat transfer from a heated sheet in several practical aspects. The heat generation/absorption effects are also important in the flow problems dealing with the dissociating fluids. Influences of heat generation/absorption may change the temperature distribution which corresponds to the particle deposition rate in electronic chips, nuclear reactors, semiconductor wafers etc. The idea of boundary layer flow over a moving surface was introduced by Sakiadis [3]. He discussed the boundary layer flow of viscous fluid over a solid surface. This analysis was extended by Crane [4] for a linearly stretched surface. He provided the closed form solutions of two-dimensional boundary layer flow of viscous fluid over a surface. Numerous literature now exists on the boundary layer flow with heat transfer and in the presence of heat generation/absorption effects (see [5]–[10] and many refs. therein).

A large number of industrial fluids like polymers, soaps, molten plastics, sugar solutions pulps, apple sauce, drilling muds etc. behave as the non-Newtonian fluids [11]. The Navier-Stokes equations cannot explore the properties of such materials. In the literature, different types of fluids models are developed according to the nature of fluids. The non-Newtonian fluids are mainly divided into three categories which are known as the differential, rate and integral types. The fluid considered here is called the Maxwell fluid. It is subclass of rate type fluids predicting the characteristics of relaxation time. The properties of polymeric fluids can be explored by Maxwell model for small relaxation time. Zierep and Fectecau [12] discussed the energetic balance for the Rayleigh-Stokes problem involving Maxwell fluid. Closed form solutions of unsteady flow of Maxwell fluid due to the sudden movement of the plate was described by Hayat et al. [13]. Fetecau et al. [14] provided the exact solutions for the unsteady flow of Maxwell fluid. Here they considered that the flow is generated due to the constantly accelerating plate. Flow of Maxwell fluid with fractional derivative model between two coaxial cylinders was also addressed by Fetecau et al. [15]. Here the inner cylinder is subjected to the time-dependent longitudinal shear stress generating the fluid motion. Helical unidirectional flows of Maxwell fluid due to shear stresses on the boundary have been studied by Jamil and Fetecau [16]. They provided the exact solution by Hankel transform method. Stability analysis for the flow of Maxwell fluid under soret-driven double-diffusive convection in a porous medium was examined by Wang and Tan [17]. Two-dimensional boundary layer flow of Maxwell fluid over a linearly stretching surface was analyzed by Hayat et al. [18]. Mukhopadhyay [19] presented an analysis for the unsteady flow of Maxwell fluid in a porous medium with suction/injection. Falkner-Skan flow of Maxwell fluid with mixed convection over a surface was analytically discussed by Hayat et al. [20].

The main theme of present analysis is to discuss the steady three-dimensional boundary layer flow of Maxwell fluid over a bidirectional stretching surface subject to prescribed surface temperature and prescribed surface heat flux. The effects of applied magnetic field are also included in this analysis. To our knowledge, not much is known about flows induced by a bidirectional stretching surface. Wang [21] discussed the three-dimensional flow of viscous fluid over a bidirectional stretching surface. Ariel [22] provided the exact and homotopy perturbation solution for ref. [21]. Liu and Andersson [23] discussed the heat transfer analysis over a bidirectional stretching surface with variable thermal conditions. Ahmed et al. [24] extended the analysis of ref. [23] for hydromagnetic flow in a porous medium. They presented the series solutions. Hayat et al. and Shehzad et al. [25], [26] studied the boundary layer flows of Maxwell and Jeffery fluids over a bidirectional stretching surface. The present analysis is arranged as follows. The next section contains the mathematical formulation of the problem. Sections three and four are for the homotopy solutions (HAM) [27]–[34], convergence study and discussion. Both cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF) are given due attention in the discussion section. The main observations of this research are listed in the last section. Further, the correct modelling for magnetohydrodynamic case of Maxwell fluid is given.

Flow Model

Consider three-dimensional magnetohydrodynamic (MHD) boundary layer flow of an incompressible Maxwell fluid. The flow is induced by bidirectional stretching surface (at with PST and PHF. Steady flow of an incompressible Maxwell fluid is considered for Flow analysis is carried out in the presence of heat generation/absorption parameter. The fluid is electrically conducting in the presence of applied magnetic field with constant strength No electric field contribution is taken into account. Induced magnetic field effects are ignored through large magnetic Reynolds number consideration. The geometry of considered flow is shown in Fig. 1. The conservation of mass, momentum and energy for steady flow in presence of magnetic field and heat source/sink can be expressed as

Figure 1

Physical model.

in which depicts the density, the current density, the magnetic field in the direction, the specific heat, the thermal conductivity and the heat generation/absorption parameter with (heat generation) and (heat absorption). is a unit vector parallel to the axis). The definition of for present flow consideration iswhere denotes the fluid velocity and the electrical conductivity. The Lorentz force thus reduces to

Expressions of Cauchy and extra stress tensors in Maxwell fluid are [11]: where is the Covariant differentiation and is the relaxation time. The first Rivilin Ericksen tensor is defined as

where * indicates the matrix transpose and the velocity field here is taken as

The definition of is [11]

Following the procedure of ref. [11] at pages 221–223 and using above equations, we have the following scalar expressions

After employing the boundary layer assumptions [35], the above equations in the absence of pressure gradient yield

The associated boundary conditions are defined as follows.

For temperature, the boundary conditions are specified as [23], [24]:

Type i

Prescribed surface temperature (PST)

Type ii

Prescribed surface heat flux (PHF)

Here is the thermal conductivity of the fluid, the constant temperature outside the thermal boundary layer, and the positive constants. The power indices and determine how the temperature or the heat flux varies in the plane.

Following [23], [24] similarity variables for the velocity field are introduced asand the temperature similarity variables take different forms depending on the boundary conditions being considered. These are equation (15) is automatically satisfied and Eqs. (16)–(21) take the following forms:

where is the Deborah number, the magnetic parameter, the ratio of stretching rates, the Prandtl number, the thermal diffusivity and the internal heat parameter.

Homotopy Analysis Solutions

In this section, we solve the problem consisting of Eqs. (24)–(27) with boundary conditions in Eq. (28) by HAM. For that the initial guesses and auxiliary linear operators are taken as follows: subject to the propertieswhere are the arbitrary constants.

At zeroth order, the problems satisfy

In above expressions, shows the embedding parameter, and the non-zero auxiliary parameters and and the nonlinear operators. When and then we obtain

It should be pointed out that when increases from to then and vary from to and Using Taylors' expansion we write where the parameters and have a key role in the convergence of series solutions. The values of parameters are chosen in such a manner that Eqs. converge at Hence Eqs. give

The general solutions are arranged as follows in which the special solutions are denoted by and

Convergence of Series Solutions and Discussion

It is well known fact that the homotopy analysis method has a great freedom to choose the auxiliary parameters and for adjusting and controlling the convergence of series solutions. To determine the appropriate convergence interval of the constructed series solutions, the curves at -order of approximations are sketched. Figs. 2 and 3 clearly show that the range of admissible values of and are and

Figure 2

curves for the functions and when and

Figure 3

curve for the function when and

The results are displayed graphically to see the effects of and on the prescribed surface temperature and prescribed surface heat flux. We denote temperature variation for PST case by and for PHF situation by in the Figs. 4–17. Figs. 4 and 5 illustrate the variations of Deborah number on and From these Figs., we have seen that both and are increased with an increase in Deborah number is based on the relaxation time. When Deborah number increases, the relaxation time increases. This increase in relaxation time causes an increase in and Comparison of Figs. 4 and 5 shows that has similar effects on and Figs. 6 and 7 are plotted to see the effects of magnetic parameter on and Clearly the thermal boundary layer thicknesses are increased for larger values of magnetic parameter. In fact the magnetic parameter involves the Lorentz force. Larger values of magnetic parameter correspond to the stronger Lorentz force. This stronger Lorentz force give rise to the thermal boundary layer thicknesses. Figs. 8 and 9 illustrate the variations of on and From these Figs. it is noticed that both and are reduced when we increased the values of Also the thermal boundary layer becomes thinner for higher values of This reduction in thermal boundary layer for larger values of is due to the entertainment of cooler to ambient fluid. The power indices and control the non-uniformity of the surface temperature in the prescribed surface temperature situation. Figs. 10 and 11 depict that and are decreasing functions of Also we noted that reduces rapidly as comparison to Effect of on and are seen in the Figs. 12 and 13. The values of and are reduced when values of are increased. It is concluded that the non-uniformity of the sheet temperature has prominent effect on the temperature fields for the reduction in temperature and thinner thermal boundary layer. Comparison of Figs. 12 and 13 illustrates that the variations in are more pronounced when compared to the variations in Also we examined that at the wall reduced rapidly when the values of are larger. Figs. 14 and 15 depict the variations of heat generation/absorption parameter on and Both and are increased by increasing values of heat generation/absorption parameter. Physically an increase in heat generation/absorption parameter produced more heat due to which the temperature of fluid increases. This increase in temperature gives rise to and The effects of Prandtl number on and are analyzed in the Figs. 16 and 17. These Figs. clearly show that and their related thermal boundary layer thicknesses are reduced for the larger values of Prandtl number Obviously the Prandtl number depends upon the thermal diffusivity. Larger values of Prandtl number give smaller thermal diffusivity and consequently the values of and decrease.

Figure 4

Influence of on when and

Figure 17

Influence of on when and

Figure 5

Influence of on when and

Figure 6

Influence of on when and

Figure 7

Influence of on when and

Figure 8

Influence of on when and

Figure 9

Influence of on when and

Figure 10

Influence of on when and

Figure 11

Influence of on when and

Figure 12

Influence of on when and

Figure 13

Influence of on when and

Figure 14

Influence of on when and

Figure 15

Influence of on when and

Figure 16

Influence of on when and

Table 1 has been prepared to analyze the convergent values of the velocities, and We have seen that our solutions for velocities converge from 16th order of approximations whereas one needs 25th order of deformations for and Hence we need less deformations for the velocities in comparison to temperatures for a convergent solution. Table 2 provides the values of temperature gradient for different values of and when and One can see that our solutions has an excellent agreement with the previous results in a limiting case [20], [21]. Further, it is observed that the temperature gradient at surface becomes positive and reduces for and and negative for and Table 3 presents the numerical values of and for different values of and when and From this Table we noted that our series solutions have very good agreement with the previous results available in the literature.

Table 1

Convergence analysis of series solutions by numerical data for different order of deformations when and

Order of deformations	f′′(0)	g′′(0)	θ′(0)	φ′(0)
1	−1.345900	−0.592325	−0.92800	0.55000
10	−1.341759	−0.600119	−0.84012	0.50038
16	−1.341761	−0.600122	−0.83823	0.50111
25	−1.341761	−0.600122	−0.83775	0.50128
30	−1.341761	−0.600122	−0.83775	0.50128
35	−1.341761	−0.600122	−0.83775	0.50128
40	−1.341761	−0.600122	−0.83775	0.50128

Table 2

Temperature gradient at surface for different values of and with and

		r = s = 0	r = −2, s = 0	r = 2, s = 0	r = 0, s = −2	r = 0, s = 2
[23]	α = 0.25	−0.665933	0.554512	−1.364890	−0.413111	−0.883125
[24]		−0.665927	0.554573	−1.364890	−0.413101	−0.883123
Present		−0.66593	0.55457	−1.36489	−0.41310	−0.88312
[23]	α = 0.50	−0.735334	0.308578	−1.395356	−0.263381	−1.106491
[24]		−0.735333	0.308590	−1.395357	−0.263376	−1.106500
Present		−0.73533	0.30858	−1.39536	−0.26338	−1.10649
[23]	α = 0.75	−0.796472	0.135471	−1.425038	−0.126679	−1.292003
[24]		−0.796470	0.135470	−1.425037	−0.126679	−1.292010
Present		−0.79472	0.13547	−1.42504	−0.12667	−1.29200

Table 3

Temperature gradient at surface and for different values of and when and

		−θ′(0) for PST			φ(0) for PHF
		B = −0.2	B = 0.0	B = 0.2	B = −0.2	B = 0.0	B = 0.2
[23]	Pr = 1.0	1.348064	1.255781	1.148932	0.741805	0.796317	0.870355
[24]		1.348064	1.255780	1.148934	0.741808	0.796318	0.870372
Present		1.34806	1.25578	1.14893	0.74180	0.79632	0.87037
[23]	Pr = 5.0	3.330392	3.170979	3.002380	0.300265	0.315360	0.333069
[24]		3.330394	3.170981	3.002384	0.300265	0.315363	0.333071
Present		3.33039	3.17098	3.00238	0.30028	0.31537	0.33308
[23]	Pr = 10.0	4.812149	4.597141	4.371512	0.207807	0.217527	0.228754
[24]		4.812151	4.597143	4.371516	0.207809	0.217529	0.228756
Present		4.81215	4.59714	4.37152	0.20781	0.21753	0.22876

Concluding Remarks

In this study, the three-dimensional MHD flow of Maxwell fluid generated by bidirectional stretching surface is investigated for two cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The effects of applied magnetic field are also taken into account. Interesting observations of this study can be mentioned below:

Effects of Deborah number on and are similar in a qualitative manner.

Both and are increasing functions of magnetic parameter

Increase in ratio parameter reduces the temperatures and their boundary layer thicknesses.

Temperature for case decreases rapidly in comparison to case when larger values of and are employed.

An increase in heat generation/absorption parameter enhances the temperatures and

Our series solutions have an excellent agreement with the previous results in limiting cases.