β Deborah number
η Similarity variable
ν Kinematic viscosity (m2s−1)
θ Dimensionless temperature
φ Stream function
λ1 Internal heat generation/absorption parameter
λR Relaxation time (t)
C Dimensional constant (t−1)
f Similarity variable
K Thermal conductivity
M Magnetic field (Te)
Nux Nusselt number
p Density of the fluid (kg m3)
Pr Prandtl number
Q Hear source/sink parameter
S Suctions/injection parameter(ms−1)
T Fluid’s temperature (K)
T∞ Ambient temperature (K)
Tw Wall temperature (k)
U, V Velocity components (ms−1)
V1, V2 Wall velocity
x,y Cartesian coordinates (I)
The flow problem of a hydromagnetic fluid with the effects of porous media and heat transfer is one of the most important problems in the field of engineering and applied sciences. Recently, several mathematicians and applied researchers have proposed that the cooling rate be critical for products in order to improve their quality. For instance, the heat transfer is very important in extracting metal from ores. As a result, a number of field researchers from all of the developed countries recently studied various fluid mechanics problems in different flow configurations, including suction/blowing, magnetohydrodynamics, internal heat generation / absorption, rotation effects, permeability of the porous medium, simultaneous effects of energy and concentration process, viscous dissipations, Joule and Newtonian heating processes, etc. For example, hydromagnetic natural convection flow over a moving surface was studied by Fetecau et al. . They presented the radiative heat transfer solution considering the effect of slip boundary conditions. Closed form solutions have been obtained by utilizing the Laplace transform and it was found that the slip parameter has significant effects on the solutions. A note on a Sisko fluid with radiation heat transfer was given by Mehmood and Fetecau . Poiseuille flow was considered in an asymmetric channel and the effect of nonlinear wall temperature was investigated. The expressions for stream functions, axial velocities, and pressure were computed analytically. The micropolar fluid problem was studied by Sheikoleslami et al.  with heat transfer effects in a channel. The series solution was obtained by the homotopy perturbation method (HPM) and the effects of chemical reactions were investigated. Moreover, the effects of physical quantities such as Reynolds number, Peclet number, and micro rotation was analyzed and discussed through graphs. In another study, the same author  used the Differential Transform Method (DTM) for computations considering a micropolar fluid with high mass transfer filled in a porous channel. Ellahi et al.  analyzed the effects of heat transfer using third grade fluid in a channel. They obtained an analytical solution for axial velocity and temperature distribution for incompressible viscoelastic third grade fluid. Ellahi presented  the flow of nanofluid in a circular pipe. The effects of magnetohydrodynamic and variable viscosity are investigated in the solutions. MHD flow of a third grade fluid was investigated by Adesanya and Falade . They explored the heat transfer rate of entropy in two parallel plates. The hydromagnetic slip flow over a shrinking wall of non-Newtonian fluid was computed by Turkyilmazoglu . The dual and triple solutions were obtained. Raza et al.  presented the rotating flow of nanofluid with the effects of hydromagnetic and slip parameters. Freidoonimehr et al.  studied unsteady convective flow in a vertical permeable stretching surface with MHD effects. Rashidi et al.  used a permeable vertical sheet for the MHD free convective flow of non-Newtonian fluid. They presented the effects of radiation and buoyancy in their proposed model. The same author approached the similar procedure to study the mixed convective heat transfer for magnetohydrodynamic viscous fluid with thermal radiation and porous medium . Rashidi et al.  gave an analytical solution for MHD viscoelastic fluid flow over a stretching surface with continuous motion. For this purpose they applied the homotopy analysis method (HAM). Zahid et al.  investigated the dual effects of viscous dissipation and thermal radiation on the stagnation point flow induced by an exponentially stretching wall. Awais et al.  studied the combined effects of Newtonian heating, thermal diffusion and diffusion thermos on an axisymmetric non-Newtonian fluid flow. In the present analysis, we have explored the solution for the internal heat generation/absorption phenomenon of the non-Newtonian Upper-convected-Maxwell fluid flow. The governing equations are first modeled and then solved analytically and numerically. In this context, the constitutive equations for velocity and temperature profiles are solved analytically by applying the homotopy analysis method (HAM).
This technique has already been used for the solution of various problems [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Ghadikolaei et al.  studied nonlinear thermal radiation effect on magneto Casson nanofluid flow with joule heating effect over an inclined porous stretching sheet. Hosseinzadeh et al.  investigated a numerical investigation on ethylene glycoltitanium dioxide nanofluid convective flow over a stretching sheet in presence of heat generation/absorption. Rahmati et al.  studied dimultaneous investigations the effects of non-Newtonian nanofluid flow in different volume fractions of solid nanoparticles with slip and no-slip boundary conditions. Ghadikolaei et al.  carried out analytical and numerical solution of non-Newtonian second-grade fluid flow on a stretching sheet. Sheikholeslami et al.  studied heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid; an experimental procedure. Guo et al.  studied numerical study of nanofluids’ thermal and hydraulic characteristics considering Brownian motion effect in a micro fin heat sink. Amini et al.  investigated thermal conductivity of highly porous metal foams using experimental and image based finite element analysis. Tian et al.  studied heat conduction investigation of functionally graded material plates with variable gradient parameters under exponential heat source load [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67].
Furthermore, for the sake of clarity the proposed method is also compared with ND-Solve method  and ADM [70, 71]. The effect of modeled parameters such as internal heat generation/absorption, porosity, magnetic parameter, suctions/injection, and Deborah number on the solutions has been shown graphically and discussed. Furthermore, the present result was also compared with published work. A table has been constructed in order to represent the numerical values of a local Nusselt number of different involved physical quantities.
2 Statement of the problem
The system deals with two dimensional steady flow of an Upper-Convected-Maxwell fluid over pa shrinking surface subject to a constant transverse magnetic field B0. A porous channel is used to show the effects of suction/junction. The fluid saturates the porous medium fo y > 0 and the flow occupies the positive region y-axis. The porous medium saturates for > 0 . The shrinking velocity of the wall is V1 = −Hw1 where and the suction/blowing parameter is represented by V2. It is pointed out here that V2 < 0 represents suction phenomena while V2 > 0 corresponds to a blowing situation. It is also assumed that the temperature of the wall and free stream conditions are θw and θ∞, respectively. The heat generation or absorption q is also taken into account. In view of these assumptions, the governing basic flow equations are given in the following [1, 2, 3, 4, 5]:(1)(2)(3)
with the following boundary conditions(4)(5)
u and v are the velocity components along the x and y axes, respectively. Here V1 = −cx is the shrinking velocity of the wall where c > 0 and V2 is the suction/blowing. Furthermore, (x, y) represents the coordinate system, λR is relaxation time, σ is electrical conductivity, B0 is magnetic field strength, ρf is fluid density, M p is permeability of porous media, T is fluid temperature, α is thermal diffusivity, and cp is the specific heat. Eqs. (2) and (3) can be transformed into a set of nonlinear ordinary differential equations by introducing the following similarity variables. ,(6)
The transformed ordinary differential equations are(7)(8)
The boundary conditions (4) and (5) become(9)(10)
Here De = λRc represents the Deborah number in terms of relaxation time, is the magnetic parameter, S = −V2/(vc)1/2 is the suction/blowing parameter where S > 0 means wall mass suction and S < 0 means wall mass injection,is the porosity parameter, represents the Prandtl number, is the internal heat generation/absorption parameter, and prime represents differentiation with respect to η.
The local Nusselt number Nux is the physical quantity of interest for the readers. It is defined as(11)
where qw (the wall heat flux) is defined as(12)
In dimensionless form we can write the above expression as(13)
3 HAM solution
In order to solve Equation (7) and (8) under the boundary conditions (8) and (10), we utilize the homotopy analysis method (HAM) with the following procedure. The solutions having the auxiliary parameters ħ regulate and control the convergence of the solutions. The initial guesses are selected as follows:
We select the initial approximation ssuch that the boundary conditionsaare satisfied as follows:(14)
The linear operators are introduced as(15)
With the following properties:(16)
where ci(i = 1 − 6) are arbitrary constants in general solution.
The nonlinear operators, according to (7) and (8), are defined as:(17)
The auxiliary function becomes(18)
The symbolic software Mathematica is employed to solve ith order deformation equations:(19)
where is auxiliary non-zero parameter and(20)
are the involved parameters in HAM theory (for more infor
To control and speed the convergence of the approximation series by the help of the so-called h-curve, it is significant to choose a proper value of auxiliary parameter. The calculation are carried out on a personal computer with 4GB RAM and 2.70GHz CPu. The h-curves of f (0) and θ(0) obtained by the 18th order of HAM solution are shown in Figure 1 which take approximately less than a minute in the exicution. To obtain the optimal values of auxiliary parameters, the averaged residual errors are defined as(21)(22)
In order to survey the accuracy of the present method, the residual errors for the 20th order of HAM solutions of (21) and (22) are illustrated in Figures 2 and 3, and listed numerically in Table 1. An efficient numerical (ND-Solve) method is also applied to solve the transformed equation (7) and (8) correspond ding to the boundary conditions given in equation (9) and (10) and compared with a HAM solution graphically and numerically as shown in Figures 4-5 and Tables 1-2, respectively.
4 Results and discussion
The transformed equation (7) and (8) subject to the boundary conditions (9) and (10) are solved analytically
by the homotopy analysis method. An efficient numerical method called ND-Solve method is also used for the sake of comparison. In this section, numerical values are assigned to the physical parameters involved in the velocity, temperature
and local Nusselt number. The paper examined the effects of governing parameters on the transient velocity profile, temperature profile and local Nusselt number. For this purpose the SRM approach has been applied for various values of flow controlling parameters De = 0.1, Mp = 0.3, λH = 0.2, M = 0.2, R = 0.3 and Pr = 0.5, to obtain a clear insight into the physics of the problem. Therefore, all the graphs and tables correspond to the values above and the rest will be mentioned.
Figures 6 and 7 show the influence of Deborah number De on the velocities f and f ' respectively. It is obvious from these figures that the boundary layer thickness decreases for larger values of De, with an increase in η. Physically, viscous effects increase for the larger Deborah numbers. These retard the flow in the entire domain and consequently the momentum boundary layer will be thinner. Since the Deborah number defines the difference between the solids and liquid (or fluids), the material shows fluid like behavior for a small Deborah number and for larger values of Deborah number the material behaves like a viscoelastic solid such as rubber, jelly, polymers etc. From the present analysis it is quite obvious that the velocity field decelerating for larger numbers.
Figure 8 and 9 depict the effects of porosity parameter Mp on f and f ' respectively. It is observed that the velocity profile increases with increasing porosity parameter. Small variation is observed for f '. Furthermore, under the influence of constant magnetic field M = 0.3, the momentum boundary layer decreases due to the Lorentz force.
Figures 10 and 11 present the effects of suction/injection parameter s on f and f '. It is noticed that the velocity fields f and f ' satisfy the far field boundary conditions (9) asymptotically, thus verifying the analytical and numerical results obtained.
Figure 12 displays the influence of magnetic field on the velocity profiles f with s = 1, De = 0.1, Mp = 0.3, M = 0.2 and R = 0.3. Due to the inflection of the vertical magnetic field to the electrically conducting fluid, the
Lorentz force is produced. This force has the tendency to slow down the flow and as a result the velocity profile decreases. Figures 13-?? show the effect of different parameters on the temperature profiles. The effect of Prandtl number on the temperature distribution is shown in Figure 13. Based on the Prandtl number’s definition Pr = , this parameter is defined as the ratio between the momentum diffusion to thermal diffusion. Thus, with the increase of Prandtl number the thermal diffusion decreases and so the thermal boundary layer becomes thinner as seen in Figure 13. It physically means that the flow with a large Prandtl number dissipates the heat faster to the fluid as the temperature gradient gets steeper and hence increases the heat transfer coefficient between the surface and the fluid. Figure 14 depicts the variation of Deborah number on the
temperature distribution. Increasing the Deborah number De increases the temperature inside the fluid. Also, it is observed that for larger values of Deborah number the boundary layer thickness decreases with increasing η.
The influence of magnetic parameter Mon the temperature distribution is shown in Figure 15. It is observed that the temperature distribution increases with increasing values of magnetic parameter M. Physically, the Lorentz force due to the transverse magnetic field has the property of relaxing the fluid velocity and temperature distributions. Accordingly, the velocity and temperature boundary layer thickness decreases as the magnetic parameter increases.
The effect of the porous permeability parameter Mp on the fluid temperature distribution is shown in Figure 16.
The result shows that as the porous permeability parameter increases, there is a corresponding increase in the fluid temperature due to increased diffusion of heat within the flow channel.
The effect of heat source (λH > 0) and heat sink (λH < 0) parameters are shown in Figures 17 and 18 respectively. It is observed that a heat source enhances the thermal conductivity and increases the fluid temperature as shown in Figure 13. Figure 14 incorporates the effects of the heat sink parameter. As expected, a heat sink provides a decrease in the fluid temperature.
Table 3 shows different values of skin friction parameter −f (0) for several values of permeability parameter M p. We see from this table that the magnitude of shear stress in the boundary is smaller for injection in comparison to the case of suction. These results are in accordance with the physical situation because the injection of the fluid amounts to an increase of fluid velocity, resulting in a decrease of the frictional force. The permeability parameter introduces additional shear stress on the boundary. Note that the increase of permeability parameter leads to the increase of skin friction parameter in all the cases of suction, blowing, and impermeability of the surface.
In order to discuss the results of local Nusselt number against different physical quantities including Deborah number, Prandtl number, and internal heat generation/absorption quantity, we have prepared Table 4. It is evident from this table that the local Nusselt number decreases due to an increase in Deborah number and quantity oif internal heat generation/absorption, whereas the local Nusselt number increases due to an increase in Prandtl number.
In this research paper, the semi-analytical/numerical technique known as HAM has been implemented to solve the transformed nonlinear differential equations describing the MHD flow of an Upper-Convected-Maxwell fluid with the influence of the internal heat generation/absorption. The dynamics of the magneto-hydrodynamic fluid flow in porous medium over a porous wall are investigated. The present semi-numerical/analytical simulations agree closely with the previous studies for some special cases. HAM has been shown to be a very strong and efficient technique in finding analytical solutions for nonlinear differential equations. HAM is shown to illustrate excellent convergence and accuracy and is currently being employed to extend the present study to mixed convective heat transfer simulations. The convergence of the series solution is established. Furthermore, the present method is also compared with an efficient numerical technique so called ND-Solve method. The effects of different physical key parameters which effect fluid motion such as Deborah number, magnetic parameter, suction/injection parameter, heat generation/absorption parameter, and porosity parameter are plotted and discussed. It is observed that with the increasing Deborah number the velocity decreases and the temperature inside the fluid increases. The results show that the velocity and temperature distribution increases with a porous medium. It is also investigated that the magnetic parameter has a decelerating effect on velocity while the temperature profiles increases in the entire domain. Due to increase in Prandtle number the temperature profile increases. It is observed that the heat source enhance the thermal conductivity and increases the fluid temperature while the heat sink provides a decrease in the temperature fluids.
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About the article
Published Online: 2018-12-31
Citation Information: Open Physics, Volume 16, Issue 1, Pages 917–928, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0113.
© 2018 Z. Khan et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0