Abstract
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.
 |
(1) |
 |
(2) |
 |
(3) |
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 is
 |
(4) |
where 
denotes the fluid velocity and 
the electrical conductivity. The Lorentz force thus reduces to
 |
(5) |
Expressions of Cauchy 
and extra stress 
tensors in Maxwell fluid are [11]:
 |
(6) |
 |
(7) |
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
 |
(8) |
The definition of 
is [11]
 |
(9) |
Following the procedure of ref. [11] at pages 221–223 and using above equations, we have the following scalar expressions
 |
(10) |
 |
(11) |
 |
(12) |
 |
(13) |
 |
(14) |
After employing the boundary layer assumptions [35], the above equations in the absence of pressure gradient yield
 |
(15) |
 |
(16) |
 |
(17) |
 |
(18) |
The associated boundary conditions are defined as follows.
 |
(19) |
For temperature, the boundary conditions are specified as [23], [24]:
Type i
Prescribed surface temperature (PST)
 |
(20) |
Type ii
Prescribed surface heat flux (PHF)
 |
(21) |
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 as
 |
(22) |
and the temperature similarity variables take different forms depending on the boundary conditions being considered. These are
 |
(23) |
equation (15) is automatically satisfied and Eqs. (16)–(21) take the following forms:
 |
(24) |
 |
(25) |
 |
(26) |
 |
(27) |
 |
(28) |
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:
 |
(29) |
 |
(30) |
subject to the properties
 |
(31) |
where 
are the arbitrary constants.
At zeroth order, the problems satisfy
 |
(32) |
 |
(33) |
 |
(34) |
 |
(35) |
 |
(36) |
 |
(37) |
 |
(38) |
 |
(39) |
 |
(40) |
In above expressions, 
shows the embedding parameter, 
and 
the non-zero auxiliary parameters and 
and 
the nonlinear operators. When 
and 
then we obtain
 |
(41) |
It should be pointed out that when 
increases from 
to 
then 
and 
vary from 
to 
and 
Using Taylors' expansion we write
 |
(42) |
 |
(43) |
 |
(44) |
 |
(45) |
 |
(46) |
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
 |
(47) |
 |
(48) |
 |
(49) |
 |
(50) |
The general solutions are arranged as follows
 |
(51) |
 |
(52) |
 |
(53) |
 |
(54) |
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.
Funding Statement
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah under grant no. 10-130/1433HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support. The funder had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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