Figures
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.
Citation: Shehzad SA, Alsaedi A, Hayat T (2013) Hydromagnetic Steady Flow of Maxwell Fluid over a Bidirectional Stretching Surface with Prescribed Surface Temperature and Prescribed Surface Heat Flux. PLoS ONE 8(7): e68139. https://doi.org/10.1371/journal.pone.0068139
Editor: Enrique Hernandez-Lemus, National Institute of Genomic Medicine, Mexico
Received: April 13, 2013; Accepted: May 24, 2013; Published: July 12, 2013
Copyright: © 2013 Shehzad et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: 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.
Competing interests: The authors have declared that no competing interests exist.
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
(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 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
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
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.
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.
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.
Author Contributions
Conceived and designed the experiments: SAS AA TH. Performed the experiments: SAS AA TH. Analyzed the data: SAS AA TH. Contributed reagents/materials/analysis tools: SAS AA TH. Wrote the paper: SAS AA TH.
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