Abstract
The objective of present research is to examine the thermal radiation effect in three-dimensional mixed convection flow of viscoelastic fluid. The boundary layer analysis has been discussed for flow by an exponentially stretching surface with convective conditions. The resulting partial differential equations are reduced into a system of nonlinear ordinary differential equations using appropriate transformations. The series solutions are developed through a modern technique known as the homotopy analysis method. The convergent expressions of velocity components and temperature are derived. The solutions obtained are dependent on seven sundry parameters including the viscoelastic parameter, mixed convection parameter, ratio parameter, temperature exponent, Prandtl number, Biot number and radiation parameter. A systematic study is performed to analyze the impacts of these influential parameters on the velocity and temperature, the skin friction coefficients and the local Nusselt number. It is observed that mixed convection parameter in momentum and thermal boundary layers has opposite role. Thermal boundary layer is found to decrease when ratio parameter, Prandtl number and temperature exponent are increased. Local Nusselt number is increasing function of viscoelastic parameter and Biot number. Radiation parameter on the Nusselt number has opposite effects when compared with viscoelastic parameter.
Introduction
Analysis of non-Newtonian fluids is an active area of research for the last few years. Such fluids represent many industrially important fluids including certain oils, shampoos, paints, blood at low shear rate, cosmetic products, polymers, body fluids, colloidal fluids, suspension fluids, pasta, ice cream, ice, mud, dough floor etc. In many fields such as food industry, drilling operations and bioengineering, the fluids, either synthetic or natural, are mixtures of different stuffs such as water, particle, oils, red cells and other long chain molecules. Such combination imparts strong rheological properties to the resulting liquids. The dynamic viscosity in non-Newtonian materials varies non-linearly with the shear rate; elasticity is felt through elongational effects and time-dependent effects. The fluids in these situations have been treated as viscoelastic fluids. Further, all the non-Newtonian fluids in nature cannot be predicted by single constitutive equation. Hence all the contributors in the field are using different models of non-Newtonian fluids in their theoretical and experimental studies (see [1]-[11] and several refs. therein). The boundary layer flows of non-Newtonian fluids in the presence of heat transfer have special importance because of practical engineering applications such as food processing and oil recovery. Especially the stretching flows in this direction are prominent in polymer extrusion, glass fiber and paper production, plastic films, metal extrusion and many others. After the pioneering works of Sakiadis [12] and Crane [13], numerous works have been presented for two-dimensional boundary layer flow of viscous and non-Newtonian fluids over a surface subject to linear and power law stretching velocities (see some recent studies [14]-[21]). It has been noted by Gupta and Gupta [22] that stretching mechanism in all realistic situations is not linear. For instance the stretching is not linear in plastic and paper production industries. Besides these the flow and heat transfer by an exponentially stretching surface has been studied by Magyari and Keller [23]. In this attempt the two-dimensional flow of an incompressible viscous fluid is considered. The solutions of laminar boundary layer equations describing heat and flow in a quiescent fluid driven by an exponentially permeable stretching surface are numerically analyzed by Elbashbashy [24]. Al- Odat et al. [25] numerically discussed the thermal boundary layer on an exponentially stretching surface with an exponential temperature distribution. Here magnetohydrodynamic flow is addressed. Nadeem and Lee [26] presented the steady boundary layer flow of nanofluid over an exponential stretching surface. Sajid and Hayat [27] examined the thermal radiation effect in the boundary layer flow and heat transfer of a viscous fluid. The flow is caused by an exponentially stretching sheet. The thermal radiation effect in steady hydromagnetic mixed convection flow of viscous incompressible fluid past an exponentially stretching sheet is examined by El-Aziz and Nabil [28]. Pal [29] carried out an analysis to describe mixed convection heat transfer in the boundary layer flow on an exponentially stretching continuous surface with an exponential temperature. Here analysis is given in the presence of magnetic field, viscous dissipation and internal heat generation/absorption. Khan and Sanajayand [30] investigated the heat and mass transfer effects of viscoelastic boundary layer flow over an exponentially stretching sheet in presence of viscous dissipation and chemical reaction. Bhattacharyya [31] numerically investigated the heat transfer boundary layer flow over an exponentially shrinking sheet. Shooting method is implemented here. Recently, Mukhopadhyay et al. [32] dealt with the boundary layer flow and heat transfer of a non-Newtonian fluid over an exponentially stretching permeable surface. Mustafa et al. [33] studied the boundary layer flow of nanofluid over an exponentially stretching sheet with convective boundary conditions. Flow and heat transfer for three-dimensional viscous flow over an exponentially stretching surface is discussed by Liu et al. [34]. Bhattacharyya et al. [35] studied the effects of thermal radiation in the flow of micropolar fluid past a porous shrinking sheet with heat transfer. The transient free convection interaction with thermal radiation of an absorbing emitting fluid along moving vertical permeable plate is discussed by Makinde [36]. Hayat et al. [37] considered a two-dimensional mixed convection boundary layer MHD stagnation point flow through a porous medium bounded by a stretching vertical plate with thermal radiation.
Literature survey indicates that the published studies about three-dimensional flow by an exponentially stretching surface are still scarce. To our knowledge, there is only one recent study by Liu et al. [34] which describes the three-dimensional boundary layer flow of a viscous fluid over an exponentially stretching surface. Thus motivation of present research is to venture further in the regime of three-dimensional mixed convection flow of viscoelastic fluid over an exponentially stretching surface with thermal radiation. The surface possess the convective type heat condition. No doubt the thermal radiation effects are significant in many environmental and scientific developments, for instance, in aeronautics, fire research, heating and cooling of channels, etc. It is found that radiative transport is often comparable and hence associated with that of convective heat transfer in several real-world applications. Therefore it is of great worth to the researchers to study combined radiative and convective flow and heat transfer aspects. Moreover, the skin friction coefficients for three-dimensional viscoelastic fluid have been computed which has not yet been available in the literature. This paper is structured into the following fashion. Section two consists of mathematical formulation and definitions of physical quantities of interest. Convergent series solutions of the involved nonlinear systems are developed in section three. The solutions in this section are developed by homotopy analysis method (HAM) [38]-[45]. Section four comprises discussion with respect to seven pertinent parameters involved in the solutions of velocity components and temperature. Section five syntheses the main observations.
Mathematical Modelling
We consider three dimensional mixed convection boundary layer flow of second grade fluid passing an exponentially stretching surface. The surface coincides with the plane 
and the flow is confined in the region 
The surface also possess the convective boundary condition. Influence of thermal radiation through Rosseland's approximation is taken into account. Flow configuration is given below in Fig. 1.
Figure 1. Geometry of Problem.
The governing boundary layer equations for steady three-dimensional flow of viscoelastic fluid can be put into the forms (see Nazar and Latip [11]):
 |
(1) |
 |
(2) |
 |
(3) |
 |
(4) |
where 
and 
are the velocity components in the 
and 
directions respectively, 
is the material fluid parameter, 
is the dynamic viscosity, 
is the kinematic viscosity, 
is the fluid temperature, 
is the fluid density, 
is the gravitational acceleration, 
is thermal expansion coefficient of temperature, 
is the specific heat, 
is the thermal conductivity and 
the radiative heat flux. Note that w-momentum equation vanishes by applying boundary layer assumptions (see Schlichting [46]).
By using the Rosseland approximation, the radiative heat flux 
is given by
 |
(5) |
Where 
is the Stefan-Boltzmann constant and 
the mean absorption coefficient. By using the Rosseland approximation, the present analysis is limited to optically thick fluids. If the temperature differences are sufficiently small then Eq. (5) can be linearized by expanding 
into the Taylor series about 
, which after neglecting higher order terms takes the form:
 |
(6) |
By using Eqs. (5) and (6), Eq. (4) reduces to
 |
(7) |
The boundary conditions can be expressed as
 |
 |
(8) |
where subscript w corresponds to the wall condition, 
is the thermal conductivity, 
is the hot fluid temperature, 
is the heat transfer coefficient and 
is the free stream temperature.
The velocities and temperature are taken in the following forms:
 |
(9) |
in which 
are the constants, 
is the reference length and 
is the temperature exponent.
The mathematical analysis of the problem is simplified by using the transformations (Liu et al. [34]):
 |
(10) |
Incompressibility condition is now clearly satisfied whereas Eqs. (2)–(7) give
 |
(11) |
 |
(12) |
 |
(13) |
 |
(14) |
 |
(15) |
in which
is the viscoelastic parameter, 
is the ratio parameter, 
is the Prandtl number, 
is the local Grashof number, 
is the radiation parameter, 
is the temperature exponent, 
is the Biot number, 
is the local Reynold number, 
is the mixed convection parameter and prime denotes the differentiation with respect to 
. These can be defined as
 |
(16) |
The skin-friction coefficients in the x and y directions are given by
 |
(17) |
where
 |
(18) |
By using Eq. (18) in Eq. (17) the non-dimensional forms of skin friction coefficients are as follows:
 |
(19) |
 |
(20) |
Further the local Nusselt number has the form
 |
(21) |
Series Solutions
The initial guesses and auxiliary linear operators in the desired HAM solutions are
 |
(22) |
 |
(23) |
subject to the properties
 |
(24) |
in which 
are the arbitrary constants, 
and 
are the linear operators and 
and 
are the initial guesses.
Following the idea in ref. [38] the zeroth order deformation problems are
 |
(25) |
 |
(26) |
 |
(27) |
 |
(28) |
For 
and 
one has
 |
(29) |
Note that when 
increases from 
to 
then 
and 
vary from 
and 
to 
and
So as the embedding parameter 
increases from 0 to 1, the solutions 
and 
of the zeroth order deformation equations deform from the initial guesses 
and 
to the exact solutions
and 
of the original nonlinear differential equations. Such kind of continuous variation is called deformation in topology and that is why the Eqs. (26-28) are called the zeroth order deformation equations. The values of the nonlinear operators are given below:
 |
(30) |
 |
(31) |
 |
(32) |
Here 
and 
are the non-zero auxiliary parameters and 
and 
the nonlinear operators. Taylor series expansion gives
 |
(33) |
 |
(34) |
 |
(35) |
where the convergence of above series strongly depends upon 
and 
Considering that 
and 
are chosen in such a manner that Eqs. (33)-(35) converge at 
then
 |
(36) |
 |
(37) |
 |
(38) |
The corresponding problems at mth order deformations satisfy
 |
(39) |
 |
(40) |
 |
(41) |
 |
(42) |
 |
(43) |
 |
(44) |
 |
(45) |
The mth order deformation problems have the solutions
 |
(46) |
 |
(47) |
 |
(48) |
where the special solutions are 
and 
.
Convergence Analysis
We recall that the series (36-38) contain the auxiliary parameters 
and 
. These parameters are useful to adjust and control the convergence of homotopic solutions. Hence the 
curves are sketched at 
order of approximations in order to determine the suitable ranges for 
and 
. Fig. 2 denotes that the range of admissible values of 
and 
are 
and 
Table 1 shows that the series solutions converge in the whole region of 
when 
and 
Figure 2.
Table 1. Convergence of series solutions for different order of approximations when 
and 
.
| Order of aproximations | 1 | 5 | 10 | 15 | 20 | 25 |
| -f′′(0) | 1.06111 | 1.02482 | 1.02609 | 1.02623 | 1.02618 | 1.02618 |
| -g′′ (0) | 0.544444 | 0.548057 | 0.548092 | 0.548043 | 0.548053 | 0.548053 |
| -θ′ (0) | 0.317778 | 0.305581 | 0.305729 | 0.305744 | 0.305738 | 0.305738 |
Discussion of Results
The effects of ratio parameter 
viscoelastic parameter 
mixed convection parameter 
Biot number 
and radiation parameter 
on the velocity component 
are shown in the Figs. 3-7. It is observed from Fig. 3 that velocity component 
and thermal boundary layer thickness are decreasing functions of ratio parameter 
This is due to the fact that with the increase of ratio parameter 
the x-component of velocity coefficient decreases which leads to a decrease in both the momentum boundary layer and velocity component 
Fig. 4 illustrates the influence of viscoelastic parameter 
on the velocity component 
It is clear that both the boundary layer and velocity component 
increase when the viscoelastic parameter increases. Influence of mixed convection parameter 
on the velocity component 
is analyzed in Fig. 5. Increase in mixed convection parameter 
shows an increase in velocity component 
. This is due to the fact that the buoyancy forces are much more effective rather than the viscous forces. Effects of Biot number 
and the radiation parameter 
on the velocity component 
can be predicted from Figs. 6 and 7. These Figs. depict that the influences of 
and 
on both the velocity component 
and thermal boundary layer thickness are similar i.e. there is increase in these quantities. Figs. 8 and 9 illustrate the variations of ratio parameter 
and viscoelastic parameter 
on the velocity component 
Variation of ratio parameter 
is analyzed in Fig. 8. Through comparative study with Fig. 3 it is noted that 
decreases while 
increases when 
increases. Physically, when 
increases from zero, the lateral surface starts moving in y-direction and thus the velocity component 
increases and the velocity component 
decreases. Fig. 9 is plotted to see the variation of viscoelastic parameter 
on the velocity component 
It is found that both the velocity component 
and momentum boundary layer thicknesses are increasing functions of 
. It is revealed from Figs. 4 and 9 that the effect of 
on both the velocities are qualitatively similar. Figs. 10-16 are sketched to see the effects of ratio parameter 
viscoelastic parameter 
, the temperature exponent 
Biot number 
mixed convection parameter 
Radiation parameter and Prandtl number 
on the temperature 
Fig. 10 is drawn to see the impact of ratio parameter 
on the temperature 
. It is noted that the temperature 
and also the thermal boundary layer thickness decrease with increasing
. Variation of the viscoelastic parameter 
on the temperature 
is shown in Fig. 11. Here both the temperature and thermal boundary layer thickness are decreasing functions of 
. Variation of mixed convection parameter 
is analyzed in Fig.12. It is seen that both the temperature 
and thermal boundary layer thickness are decreasing functions of mixed convection parameter 
Fig.13 presents the plots for the variation of Biot number 
Note that 
increases when 
increases. The thermal boundary layer thickness is also increasing function of 
. It is also noted that the fluid temperature is zero when the Biot number vanishes.
Influence of temperature exponent 
is displayed in Fig. 14. It is found that both the temperature 
and thermal boundary layer thickness decrease when A is increased. Also both the temperature 
and thermal boundary layer thickness are increasing functions of thermal radiation parameter 
(see Fig. 15). It is observed that an increase in 
has the ability to increase the thermal boundary layer. It is due to the fact that when the thermal radiation parameter increases, the mean absorption coefficient 
will be decreased which in turn increases the divergence of the radiative heat flux. Hence the rate of radiative heat transfer to the fluid is increased and consequently the fluid temperature increases. Fig. 16 is plotted to see the effects of 
on 
. It is noticed that both the temperature profile and thermal boundary layer thickness are decreasing functions of 
. In fact when 
increases then thermal diffusivity decreases. This indicates reduction in energy transfer ability and ultimate it results in the decrease of thermal boundary layer.
Figure 3. Influence of 
on the velocity 
.
Figure 7. Influence of R on the velocity 
.
Figure 4. Influence of K on the velocity 
.
Figure 5. Influence of 
on the velocity 
.
Figure 6. Influence of 
on the velocity .
Figure 8. Influence of 
on the velocity 
.
Figure 9. Influence of K on the velocity 
.
Figure 10. Influence of 
on the temperature 
.
Figure 11. Influence of K on the temperature 
.
Figure 12. Influence of 
on the temperature 
.
Figure 13. Influence of 
on the temperature 
.
Figure 14. Influence of A on the temperature 
.
Figure 15. Influence of R on the temperature 
.
Figure 16. Influence of 
on the temperature 
.
Table 1 presents the numerical values of 
and 
for different order of approximations when 
and 
It is seen that the values of 
and 
converge from 20th order of deformations whereas the values of 
converge from 25th order approximations. Further, it is observed that we have to compute less deformations for the velocities in comparison to temperature for convergent series solutions. Table 2 includes the values for comparison of existing solutions with the previous available solutions in a limiting case when 
and 
varies. This Table presents an excellent agreement with the previous available solutions. Table 3 is computed to see the influences of viscoelastic parameter 
and ratio parameter 
on skin friction coefficients in the x and y directions. It is noted that 
has quite opposite effect on skin friction coefficients while quite similar effect is seen within the increase of ratio parameter 
. Table 4 examines the impact of viscoelastic parameter 
, mixed convection parameter 
, ratio parameter 
, Biot number 
, radiation parameter 
, Prandtl number 
and temperature exponent 
on the local Nusselt number (rate of heat transfer at the wall). It is noted that the value of rate of heat transfer increases for larger viscoelastic parameter 
, mixed convection parameter 
, ratio parameter 
, Biot number 
, Prandtl number 
and temperature exponent 
while it decreases through an increase in radiation parameter R.
Table 2. Comparative values of 
and 
for different values 
when 
.
| Liu et al. [34] | Present results | |||||
|
-f′′′ (0) | -g′′ (0) | f(∞)+g(∞) | -f′′ (0) | -g′′ (0) | f(∞)+g(∞) |
| 0.0 | 1.28180856 | 0 | 0.90564383 | 1.28181 | 0 | 0.90564 |
| 0.50 | 1.56988846 | 0.78494423 | 1.10918263 | 1.56989 | 0.78494 | 1.10918 |
| 1.00 | 1.81275105 | 1.81275105 | 1.28077378 | 1.81275 | 1.81275 | 1.28077 |
Table 3. Values of skin friction coefficients for different values of K and α when λ = γ = 0.5, R = 0.3, Pr = 1.2 and A = 0.2.
| K | α | -
|
-
|
| 0.0 | 0.5 | 4.95289 | 4.37363 |
| 0.2 | 5.16586 | 3.97055 | |
| 0.3 | 5.42622 | 3.96130 | |
| 0.3 | 0.0 | 3.72170 | 1.65409 |
| 0.2 | 4.30247 | 2.34617 | |
| 0.5 | 5.42622 | 3.96130 |
Table 2. Comparative values of 
and 
for different values 
when 
Table 4. Values of local Nusselt number 
for different values of the parameters 
, 
and 
.
| K |
|
|
|
R | Pr | A |
|
| 0.0 | 0.5 | 0.5 | 0.5 | 0.3 | 1.2 | 0.2 | 0.297492 |
| 0.3 | 0.308234 | ||||||
| 0.5 | 0.311853 | ||||||
| 0.2 | 0.0 | 0.303062 | |||||
| 0.3 | 0.304775 | ||||||
| 0.5 | 0.305738 | ||||||
| 0.2 | 0.5 | 0.0 | 0.282007 | ||||
| 0.3 | 0.297135 | ||||||
| 0.5 | 0.305738 | ||||||
| 0.1 | 0.0885730 | ||||||
| 0.3 | 0.216850 | ||||||
| 0.5 | 0.305738 | ||||||
| 0.2 | 0.5 | 0.5 | 0.5 | 0.0 | 0.329701 | ||
| 0.3 | 0.305738 | ||||||
| 0.5 | 0.292750 | ||||||
| 0.2 | 0.5 | 0.5 | 0.5 | 0.3 | 1.0 | 0.292152 | |
| 1.2 | 0.305738 | ||||||
| 1.5 | 0.321826 | ||||||
| 0.2 | 0.5 | 0.5 | 0.5 | 0.3 | 1.2 | 0.0 | 0.288530 |
| 0.2 | 0.305738 | ||||||
| 0.5 | 0.325492 |
Conclusions
Three-dimensional mixed convection flow of viscoelastic fluid over an exponentially stretching surface is analyzed in this study. The analysis is carried out in the presence of thermal radiation subject to convective boundary conditions. The main observations can be summarized as follows:
Influence of ratio parameter
on the velocities 
and 
is quite opposite. However the effect of viscoelastic parameter 
on the velocities 
and 
is qualitatively similar.Momentum boundary layer thickness increases for
when ratio parameter 
is large. Effect of 
on 
is opposite to that of 
Velocity component
is increasing function of mixed convection parameter 
However 
decreases with an increase of mixed convection parameter 
. The impact of Biot number 
and radiation parameter 
on 
and 
are qualitatively similar.Momentum boundary layer is an increasing function of mixed convection parameter
while thermal boundary layer is decreasing function of mixed convection parameter 
Increase in Prandtl number decreases the temperature
.Thermal boundary layer thickness decreases when ratio parameter
viscoelastic parameter 
, mixed convection parameter 
Prandtl number 
and temperature exponent 
are increased.Influence of viscoelastic parameter
on the x and y direction of skin friction coefficients is opposite.Both components of skin friction coefficient increase through an increase in ratio parameter
Local Nusselt number is an increasing function of Prandtl number
ratio parameter 
viscoelastic parameter 
, mixed convection parameter 
Biot number 
and temperature exponent 
while it decreases for radiation parameter
.
Supporting Information
Appendix.
(DOCX)
Acknowledgments
We are thankful to the reviewers for the useful comments.
Funding Statement
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant number 26-130-35 Hi Ci. 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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