Nainaru Tarakaramu | Panyam Venkata Satya Narayana* | Dondu Harish Babu | Ganganapalli Sarojamma | Oluwole Daniel Makinde
© 2021 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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This work examines the effects of non-linear thermal radiation and Joule heating on MHD three-dimensional visco-elastic nanofluid flow due to a surface stretching in lateral directions. A coupled nonlinear differential system is generated from the boundary layer equations by using self-similarity variables and is then solved numerically by using most powerful shooting technique with Runge Kutta method of fourth order. The computational results for the flow variables are plotted graphically and are discussed in detail for various governing parameters that emerged in the analysis. It is observed that the momentum of the visco elastic nanofluid is better than that of a viscous fluid. Thicker thermal and concentration boundary layers are formed for increasing nonlinear thermal radiation and temperature ratio parameters. Also the results are in very good agreement with the outcomes available in the literature as a particular case. This model may play a significant role in the field of manufacturing and engineering applications.
magnetohydrodynamic, couple stress, nanofluid, joule heating, viscous dissipation, stretching sheet
Nowadays, many researchers have been attracted towards the study in MHD viscoelastic (biological solutions, colloids, asphalts, glues, tars, paints, and fluids contain melts of polymer) nanofluid flows in view of their diverse scientific applications. Thermal and concentration boundary layer flow of a viscoelastic fluid over a stretching sheet was presented numerically by Ashraf et al. [1], and Mohamed et al. [2]. Non- Newtonian nanofluid flow due to polymeric stretching sheet in the presence of dissipation and surface transpiration was devoted by Rana et al. [3]. They reported that their analysis finds applications in the manufacturing process of rheological nano-bio-polymers. Seth et al. [4] explored the viscoelastic nanofluid flow past a stretching sheet with thermal radiation and soret effects. The influence of Cattaneo-Christov double diffusion in a viscoelastic nanofluid flow has been discussed by Hayat et al. [5]. Recently, some of the researchers [6-12] investigated the viscoelastic nanofluid flow induced due to a sheet stretching.
The influence of Joule heating and non-linear radiation on magnetohydrodynamic nanofluid flow caused due to sheet stretching plays a significant role in the fields of manufacturing and engineering. He et al. [13] developed the fictitious domain method with distributed Lagrange multipliers to study the unsteady flow in a screw extruder. Tarakaramu and Satyanarayana [14] investigated the hydromagnetic nanofluid flow induced by a sheet stretching with chemical reaction. Kumar et al. [15] analysed the 3D flow of non -Newtonian nanofluid in the presence of radiation and Joule heating. Rehman et al. [16] presented the thermo-physical aspects of nanofluid flow induced by a cylindrical stretching surface. Babu and Narayana [17] explored the magnetohydrodynamic non-Newtonian fluid flow induced by sheet stretching with Joule heating. Several researchers [18-23] explored 3D convection flows caused due to a sheet stretching in the presence of Lorentz forces.
The couple stresses [24-26] are non-central forces exerted between particles in a fluid flow. These forces in nanofluid flows play a significant role in several industrial applications (food industry, waste heat recovery, air conditioning, refrigeration and automobile radiators) due to enhanced heat transfer. Also, these fluids are capable of describing different fluid characteristics. The influence of nonlinear thermal radiation on three-dimensional boundary layer flow of a couple stress nanofluid was explored by Hayat et al. [27]. MHD couple stress viscoelastic nanofluid flow induced by a continuously sheet stretching has been analysed by Turkyilmazoglu [28]. The effects of thermal radiation and thermodiffusion on couple stress fluid flow between two vertical parallel plates have been discussed by Kaladhar et al. [29], and Hayat et al. [30]. They presented the influence of heat transfer characteristics of a couple stress nanofluid on a magnetohydrodynamic three-dimensional flow due to bidirectional stretching. Beg et al. [31] presented the oscillatory flow of a non-Newtonian bio-fluid in a rotating channel with Lorentz forces. Hayat et al. [32] presented the three-dimensional magnetohydrodynamic flow of a couple stress nanofluid past a stretching surface. Kumar et al. [33] experimentally developed the Fe3O4 nanofluid flow through longitudinal strip inserts. Many authors [34-44] considered various mathematical nanofluid flow models to analyse the heat transfer aspects. Satyanarayana [45] adopted the lie group analysis to examine the nanofluid flow induced by the sheet stretching.
The main objective of present work is to analyse the magnetohydrodynamic 3D flow of a couple stress fluid in the presence of Joule heating and non-linear thermal radiation. The governing boundary layer equations are converted to a system of coupled ODEs by using the similarity variables. The transformed system can be solved numerically by RKF scheme with shooting method. Expressions for various values of parameters on the flow field and other aspects are discussed graphically and numerically.
nsidered in this mathematical model. A constant Lorentz force is applied on the fluid normal to the flow direction. Choose a Cartesian coordinates system (x,y,z) in which x- and y- axes are along the lateral directions of the stretchable surface and z- axis is normal to it. The fluid flow occurs for z>0 as displayed in Figure 1. The stretching components of velocities along x and y directions are respectively defined as A=ax and B=by. Joule heating, dissipation and nonlinear thermal radiations are considered in the energy equation. Using the above assumptions, the equations of continuity, momentum, energy and species concentration are as follows
Figure 1. Physical model and configuration
$\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}=0$ (1)
$A \frac{\partial A}{\partial x}+B \frac{\partial A}{\partial y}+C \frac{\partial A}{\partial z}=v \frac{\partial^{2} A}{\partial z^{2}}-v^{\prime} \frac{\partial^{4} A}{\partial z^{4}}-\frac{\sigma H_{0}^{2}}{\rho_{f}} A$ (2)
$A \frac{\partial B}{\partial x}+B \frac{\partial B}{\partial y}+C \frac{\partial B}{\partial z}=v \frac{\partial^{2} B}{\partial z^{2}}-v^{\prime} \frac{\partial^{4} B}{\partial z^{4}}-\frac{\sigma B_{0}^{2}}{\rho_{f}} B$ (3)
$\left.\begin{array}{l}A \frac{\partial \Theta}{\partial x}+B \frac{\partial \Theta}{\partial y}+C \frac{\partial \Theta}{\partial z}=\alpha_{m} \frac{\partial^{2} \Theta}{\partial z^{2}}-\frac{1}{\left(\rho c_{p}\right)_{f}} \frac{\partial q_{r}}{\partial z} \\ +\xi\left(D_{0}\left(\frac{\partial \Psi}{\partial z} \frac{\partial \Theta}{\partial z}\right)+\frac{D_{\theta}}{\Theta_{\infty}}\left(\frac{\partial \Theta}{\partial z}\right)^{2}\right)+\frac{\sigma H_{0}^{2}}{\left(\rho c_{p}\right)_{f}}\left(A^{2}+B^{2}\right) \\ +\frac{\mu}{\left(\rho c_{p}\right)_{f}}\left(\left(\frac{\partial^{2} A}{\partial z^{2}}\right)^{2}+\left(\frac{\partial^{2} B}{\partial z^{2}}\right)^{2}\right) \\ +\frac{2 \mu}{\left(\rho c_{p}\right)_{f}}\left(\left(\frac{\partial A}{\partial z}\right)^{2}+\left(\frac{\partial B}{\partial z}\right)^{2}\right)\end{array}\right\}$ (4)
$A \frac{\partial \Psi}{\partial x}+B \frac{\partial \Psi}{\partial y}+C \frac{\partial \Psi}{\partial z}=D_{0}\left(\frac{\partial^{2} \Psi}{\partial z^{2}}\right)+\frac{D_{\Theta}}{\Theta_{\infty}}\left(\frac{\partial^{2} \Theta}{\partial z^{2}}\right)$ (5)
The relevant boundary conditions (B.Cs) for this model are
According to the Rosseland’s approximation (Brewster [46]), the non-linear radiative heat flux qr is defined as:
$q_{r}=-\frac{4 \sigma^{*}}{3 k^{*}} \frac{\partial \Theta^{4}}{\partial z}=-\frac{16 \sigma^{*}}{3 k^{*}} \Theta^{3} \frac{\partial \Theta}{\partial z}$ (7)
On differentiation we get
$\frac{\partial q_{r}}{\partial z}=-\frac{16 \sigma^{*}}{3 k^{*}} \frac{\partial}{\partial z}\left(\Theta^{3} \frac{\partial \Theta}{\partial z}\right)$ (8)
In view of Eq. (8), Eq. (4), can be written as
$\left.\begin{array}{l}A \frac{\partial \Theta}{\partial x}+B \frac{\partial \Theta}{\partial y}+C \frac{\partial \Theta}{\partial z}=\alpha_{m} \frac{\partial^{2} \Theta}{\partial z^{2}} \\ +\frac{1}{\left(\rho c_{p}\right)_{f}}\left(\frac{16 \sigma^{*}}{3 k^{*}} \frac{\partial}{\partial z}\left(\Theta^{3} \frac{\partial \Theta}{\partial z}\right)\right) \\ +\xi\left(D_{0}\left(\frac{\partial \Theta}{\partial z} \frac{\partial \Theta}{\partial z}\right)+\frac{D_{\theta}}{\Theta_{\infty}}\left(\frac{\partial \Theta}{\partial z}\right)^{2}\right) \\ +\frac{\sigma H_{0}^{2}}{\left(\rho c_{p}\right)_{f}}\left(A^{2}+B^{2}\right) \\ +\frac{2 \mu}{\left(\rho c_{p}\right)_{f}}\left(\left(\left(\frac{\partial A}{\partial z}\right)^{2}+\left(\frac{\partial B}{\partial z}\right)^{2}\right)\right) \\ +\frac{\mu}{\left(\rho c_{p}\right)_{f}}\left(\left(\frac{\partial^{2} A}{\partial z^{2}}\right)^{2}+\left(\frac{\partial^{2} B}{\partial z^{2}}\right)^{2}\right)\end{array}\right\}$ (9)
The suitable similar transformations for this model are
$A=\operatorname{axf}^{\prime}(\eta), B=\operatorname{ayg}^{\prime}(\eta), C=-\sqrt{a v}(f(\eta)+g(\eta)$
$\theta(\eta)=\frac{\Theta-\Theta_{\infty}}{\Theta_{w}-\Theta_{\infty}} \phi(\eta)=\frac{\Psi-\Psi_{\infty}}{\Psi_{w}-\Psi_{\infty}} \eta=\left(\frac{a}{c}\right)^{1 / 2} z$ (10)
Using the above Eq. (10), we can recast the Eqns. (2)-(5) and (9) as
$f^{\prime \prime \prime}-K f^{v}-\left(f^{\prime}\right)^{2}-M^{2} f^{\prime}+f^{\prime \prime}(f+g)=0$ (11)
$g^{\prime \prime \prime}-K g^{v}-\left(g^{\prime}\right)^{2}-M^{2} g^{\prime}+g^{\prime \prime}(f+g)=0$ (12)
$\left.\begin{array}{l}\operatorname{Pr}\left((f+g) \theta^{\prime}-f^{\prime} \theta+N_{b} \theta \phi^{\prime}+N_{t}\left(\theta^{\prime}\right)^{2}\right) \\ +M^{2}\left(E c_{x}\left(f^{\prime}\right)^{2}+E c_{y}\left(g^{\prime}\right)^{2}\right) \\ +\left(E c_{x}\left(f^{\prime \prime}\right)^{2}+E c_{y}\left(g^{\prime \prime}\right)^{2}\right) \\ +K\left(E c_{x}\left(f^{\prime \prime \prime}\right)^{2}+E c_{y}\left(g^{\prime \prime \prime}\right)^{2}\right) \\ +\left(\left(1+R_{d}\left(1+\left(\theta_{w}-1\right) \theta\right)\right)^{3} \theta^{\prime}\right)^{\prime}=0\end{array}\right\}$ (13)
$\phi^{\prime \prime}+L e \operatorname{Pr}(f+g) \phi^{\prime}+\left(\frac{N t}{N_{b}}\right) \theta^{\prime \prime}=0$ (14)
The corresponding boundary conditions are given by
$\left.\begin{array}{l}f=0 g=0 f^{\prime}=1, \\ g^{\prime}=\lambda \quad \theta=1 \quad N_{b} \phi^{\prime}+N_{t} \theta^{\prime}=0 \\ a t \eta=0 \\ f^{\prime} \rightarrow 0 g^{\prime} \rightarrow 0, \\ \theta \rightarrow 0 \quad \phi \rightarrow 0 \quad \text { as\eta } \rightarrow \infty\end{array}\right\}$ (15)
The skin friction coefficients and Nusselt number are given as
$\left.\begin{array}{l}R e_{x}^{1 / 2} C_{f x}=f^{\prime \prime}(0)-K f^{i v}(0), \\ R e_{y}^{1 / 2} C_{f y}=g^{\prime \prime}(0)-K g^{i v}(0) \\ R e_{x}^{-1 / 2} N u_{x}=-\left(1+R_{d} \theta_{w}^{3}\right) \theta^{\prime}(0) \\ R e_{x}^{-1 / 2} S h_{x}=-\theta^{\prime \prime}(0)\left(\frac{N t}{N_{b}}\right)\end{array}\right\}$ （16）
where, $R e_{x}=A_{w} x / v$ read as local Reynolds number.
The converted Eqns. (11)-(14) with corresponding boundary conditions (15) have been calculated numerically by Runge-Kutta-Fehlberg scheme along with well-known shooting technique. The graphical results are displayed from Figures 2-10 for distinct values of the physical parameters on velocity, temperature and concentration distributions as well as the skin friction coefficient and Nusselt number.
The influence of stretching ratio parameter λ is illustrated in Figures 2(a)-2(d). It is noticed that the transverse velocity rises with larger values of λ as the stretching velocity in the y direction exceeds that of its counterpart in the x-direction and it is obvious that the x-component of velocity diminishes. θ(η) and ϕ(η) profiles follow the trend of axial velocity for the same set of values of λ and therefore the thermal and solutal boundary layers are thinner. Also, λ=0, corresponds to 2D flow analysis and it is pertinent to emphasise that the temperature boundary layer thickness in the case of 2D flow is greater than that of the corresponding case of 3D flow. Moreover, the velocity along x-axis for couple stress nanofluid is more comparing to the water nanofluid. These results are useful in high heat transfer experiments in industrial applications.
(a)
(b)
(c)
(d)
Figure 2. Influence of λ on (a) f’(η) (b) g’(η) (c) θ(η) (d) ϕ(η)
Figure 3 signify the influence of Prandtl number Pr on temperature distributions. It is noticed that the thermal boundary layers thickness enhances for smaller Pr values. Physically, the thermal diffusivity of the fluid is higher for smaller Pr and hence thicker thermal boundary layers occur. It is also observed that the temperature distribution in the case of a non-Newtonian fluid is fewer than that of viscous fluid.
Figure 4(a)-(b) exhibit the influence of Eckert number Ecx and Ecy along x and y-directions respectively on ϕ(η), θ(η). From these figures it is noticed that both ϕ(η) and θ(η) augmented with an increase in Ec along x and y-direction. This is due to the frictional force effect in the fluid layers. Also, noticed that the influence of Ecx on ϕ(η) is more than that of Ecy.
Figures 5(a)-5(b) display the impact of Rd (thermal radiations parameter) on temperature θ(η) and concentration ϕ(η) profiles, respectively. It is pointed out that thickness of the thermal concentration boundary layers increases for enhanced values of Rd due to the fact that increasing radiation release more thermal energy in the fluid.
Figure 3. Influence of Pr on θ(η)
(a)
(b)
Figure 4. Influence of (a) $E c_{x}$ on $\phi(\eta)$ (b) $_{E c_{y}}$ on $\theta(\eta)$
(a)
(b)
Figure 5. Influence of Rd on (a) θ(η) (b) ϕ(η)
Figure 6. Influence of θw on θ(η)
Figure 7. Influence K on ϕ(η)
Figure 6 illustrate the impact of temperature ratio parameter θw on θ(η) profile. Presence of θw is due to non-linear thermal radiation and its contribution is seen to enhance the thermal energy leading to higher temperatures. Figure 7 depict the couple stress parameter K on the concentration ϕ(η) distributions respectively. It is noticed that the reduction in temperature profile is observed with enhancing values of couple stress parameter K.
Figure 8 expose the variation of thermophoresis parameter Nt on θ(η) profile. It is clear that both the profiles θ(η) increase with higher values of Nt. Physically, increase in thermophoresis parameter Ntcauses the nanoparticles to move from hotter area to colder area and consequently the temperature and the thermal boundary layer thickness rise. The influence of magnetic field parameter M2 on $R e_{x}^{-1 / 2}$ against λ is exhibited in Figure 9. It is noticed that the $R e_{x}^{-1 / 2}$ rate of heat transfer enhance with increasing values of magnetic field parameter M2.
Figure 8. Influence K on ϕ(η)
Figure 9. Influence of M2on $R e_{x}^{-1 / 2} N u_{x}$
(a)
(b)
Figure 10. Influence of $K \cdot \mathrm{on} \cdot(\mathbf{a}) \mathrm{Re}_{x}^{-1 / 2} N u_{x}(\mathbf{b}) \operatorname{Re}_{x}^{1 / 2} C_{f x}$
Figures 10(a)-10(b) depict the effect K with respect to λ on the two components of skin friction coefficients and Nusselt number $R e_{x}^{-1 / 2} N u_{x}$. It is observed that the two components of coefficients of skin friction reduce with K. Also, observed that the skin friction coefficient in the presence of magnetic field (M2=0.2) is seen to be more than that of the nonmagnetic (M2=0) case. On the other hand, a reverse trend is observed in the case of rate of heat transfer $R e_{x}^{-1 / 2}$. In the absence of $E c_{x}$ and $E c_{y}$ the present model reduces to the flow model discussed by Hayat et al. [9] and they observed that the rate of heat transfer is seen to be lesser. Hence, it may be concluded that both Joule heating and viscous dissipation are vital in improving the rate of heat transfer of a couple stress nanofluid.
The validity of the current work outcomes of final values are compared with Wang [47] in Table 1. The skin friction coefficient values for λ=0 compared with those of Oyelakin et al. [48], Nadeem et al. [49], Gupta and Sharma [50], and Ahmad and Nazar [51], respectively in Table 2. The skin friction coefficient values for λ=1 compared with Nadeem et al. [49] in Table 3. It is observed that the current results are in good agreement with those existing results.
Table 1. Comparison of final values for various values of λ
|
λ |
Previous Study Wang [47] f(∞) |
Present study f(∞) |
Previous Study Wang [47] g(∞) |
Present study g(∞) |
|
0.00 |
1.000000 |
1.00000 |
0.000000 |
0.00000 |
|
0.25 |
0.907075 |
0.90707 |
0.257986 |
0.25798 |
|
0.50 |
0.842360 |
0.84236 |
0.451671 |
0.45167 |
|
0.75 |
0.792308 |
0.79230 |
0.612049 |
0.61212 |
|
1.00 |
0.751527 |
0.75152 |
0.751527 |
0.75148 |
Table 2. Comparison of -f''(0) (Skin friction coefficient) for various values of M for λ=0
|
M |
Present study -f''(0) |
Oyelakin et al. [48] |
Nadeem et al. [49] |
Gupta and Sharma [50] |
Ahmad and Nazar [51] |
|
0.0 |
1.00000 |
1.0000 |
1.0004 |
1.0003 |
1.0042 |
|
10 |
3.31662 |
3.31662 |
3.3165 |
3.3165 |
3.3165 |
|
100 |
10.04987 |
10.04987 |
10.049 |
10.0498 |
10.049 |
Table 3. Comparison of -f''(0) (Skin friction coefficient) with various values of M for λ=1
|
M |
Present Study -f''(0) |
Nadeem et al. [49] -f''(0) |
|
0.0 |
1.1737 |
1.1737 |
|
10 |
3.3672 |
3.3667 |
|
100 |
10.0664 |
10.066 |
This article deals with the steady MHD 3D flow of a couple stress nanofluid caused due to a sheet stretching. The dimensionless equations are derived and solved computationally. The major conclusions in the current study are indicated here under:
|
(x, y) |
Cartesian coordinate’s |
|
A, B, C |
velocity components along x, y, z-axis |
|
$\Psi$w |
volume fraction of nanoparticle |
|
$c_{f}$ |
Skin friction coefficient |
|
$c_{p}$ |
Specific heat |
|
$c_{\infty}$ |
Uniform ambient concentration |
|
$D_{0}$ |
Brownian diffusion |
|
$D_{\Theta}$ |
Thermophoresis diffusion |
|
$E c_{x}$ |
Eckert number in the direction of x $=\frac{2 A_{w}^{2}}{\left(c_{p}\right)_{f}\left(\Theta_{w}-\Theta_{\infty}\right)}$ |
|
$E c_{y}$ |
Eckert number in the direction of y $=\frac{2 B_{w}^{2}}{\left(c_{p}\right)_{f}\left(\Theta_{w}-\Theta_{\infty}\right)}$ |
|
f |
Dimensionless stream function |
|
f' |
Dimensionless velocity |
|
k* |
Mean absorption coefficient |
|
k |
Thermal conductivity |
|
K |
Couple Stress Parameter $=a v^{\prime} / v^{2}$ |
|
Le |
Lewis number =$\alpha_{m} / D_{0}$ |
|
$M^{2}$ |
magnetic field parameter $=\frac{\sigma H_{0}^{2}}{a \rho}$ |
|
$N_{t}$ |
Thermophoresis parameter $=\frac{\mathrm{D}_{\Theta}(\rho c)_{p}}{\Theta_{\infty}(\rho c)_{f}}\left(\Theta_{w}-\Theta_{\infty}\right)$ |
|
$N_{b}$ |
Brownian motion coefficient $=\frac{\mathrm{D}_{0}(\rho c)_{p}}{v(\rho c)_{f}}\left(\Psi_{w}-\Psi_{\infty}\right)$ |
|
$\operatorname{Pr}$ |
Prandtl number $=v / \alpha_{m}$ |
|
$q_{r}$ |
radiative heat flux |
|
$\operatorname{Re}$ |
Reynolds number |
|
$R_{d}$ |
Radiation parameter $=\frac{16 \sigma^{*} \Theta_{\infty}^{2}}{3 k k^{*}}$ |
|
$\Theta_{\infty}$ |
fluid temperature far away from the surface |
|
$\Theta_{w}$ |
Constant fluid Temperature of the wall |
|
$A_{w}$ |
Stretching velocity |
|
$A_{\infty}$ |
Free stream velocity |
|
Greek symbols |
|
|
$\alpha_{m}$ |
Thermal diffusion |
|
$\mu$ |
Dynamic viscosity |
|
$\phi$ |
Dimensionless concentration |
|
$\lambda$ |
Ratio parameter $=b / a$ |
|
$v$ |
Kinematic viscosity |
|
$\sigma$ |
Electrical conductivity |
|
$\theta$ |
Dimensionless temperature |
|
$\alpha_{m}$ |
Base fluid thermal diffusivity $=k /\left(\rho c_{p}\right)_{f}$w |
|
$v^{\prime}$ |
Couple stress viscosity $=\mu / \rho$ |
|
$\xi$ |
Ratio of the nanoparticle to the fluid $(\rho c)_{p} /(\rho c)_{f}$ |
|
$(\rho c)_{f}$ |
Heat capacity of the fluid |
|
$(\rho c)_{p}$ |
Heat capacity of the nanoparticle to the fluid |
|
$\rho_{f}$ |
Fluid density |
|
$\rho$ |
Density |
|
$\sigma^{*}$ |
Boltzmann constant |
|
Subscripts |
|
|
$\infty$ |
condition at free stream |
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