MHD Mixed Convective Heat Transfer of CuAl_{2}O_{3} Water Hybrid Nanofluid over a Stretching Wedge with Ohmic Heating
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The main purpose of the current article is present exploration scrutinizes a numerical simulation for the magnetomixed convection flows of hybrid nanofluid over a stretching wedge with Ohmic heating applying Tiwari and Das model. Proper dimensionless variables are applied the governing PDEs which are solved numerically using Matlab program. The obtained results of velocity and temperature profiles for several key parameters are portrayed and construed in detail. It has been observed that the amount of skin friction coefficient and the Nusselt number of the hybrid liquid nano improve and increase with the increase of the nanoparticle size fraction, magnetic field, mixed load, and wedge angle while it decreases when we increase the velocity ratio and increases with the increase of the Nusselt number.
mixed convection, MHD, hybrid nanofluid, ohmic heating
Mixed convection flow occurs when natural convection and forced convection technique work with each other to transfer heat. The study of the flow and heat transfer has great practical applications in the field of technology and science. Mixed convection flow over a stretching surface is one of these studies that play an important role in engineering, petroleum and agriculture industries like ion exchange columns, heattreated materials operating between rolls or on a conveyor belt, materials production by extrusion processes, distillation towers, solar power absorbers and subterranean chemical waste migration etc. [19] are some of the studies that concentrate on the mixed convective flow over a perpendicular stretching surface in various cases. The study of the magnetic properties and behavior of electrically conducting fluids is called "Magnetohydrodynamics". This word has come from magneto which means magnetic field, hydro which means water, and dynamics which mean movement. Saltwater, plasmas, liquid metals and electrolytes are examples of such magnetofluids. Alfvén [10] who is starting MHD. Mahdy and Nabwey [11] reported a numerical analysis of buoyancydriven nanoliquid flow of gyrotactic timemixed bioconvection heat transfer in stagnation domain of an impulsively revolving sphere due to convective boundary condition. Ahmed and Mahdy [12] delignated the case of mixed convection flow through an isothermal vertical wedge submersed in saturated porous medium using Buongiorno's nanofluid.
Magnetic ﬁelds are vastly utilized in modern metallurgical and metalworking processes like MHD power generators, the cooling of reactors, ﬂow meters and MHD pumps etc. Magnetohydrodynamic debates the induced magnetic features of the electrical conductive fluids. In general, its impact on the stream and heat transfer knows as Lorentz force and Joule heating. Actually, exerting the magnetic field provides the resistance in the stream course and reduce the fluid motion. MHD mixed convective ﬂow over a perpendicular stretching surface has been checked by Kumari and Nath [13] and Ishak et al. [14]. MHD mixed convection ﬂow in an upper sided liddriven cavity heated by a corner heater which was considered by Oztop et al. [15]. The number of studies associated with mixed convection flow over a wedge is very limited. The numerical results for mixed convection ﬂow over a wedge with variable surface temperature were given by Hossain et al. [16]. The MHD mixed convective mass and heat transfer over a permeable wedge in the existence of a chemical reaction was checked by Muhaimin et al. [17]. MHD mixed convective mass and heat transfer past a permeable wedge was studied by Kandasamy et al. [18] and took into the effects of chemical reaction, thermophoresis, and variable viscosity.
A fluid which contains nanoparticles like (oxides, metals, carbon nanotubes or carbides) suspended in a base fluid like (oil, ethylene glycol, and water) is defined as nanofluid. This technique improves the effective conductivity of the resulting fluid comparing to the base fluid Choi [19] considered the first one introduce nanofluids. This increasing on the nanofluid conductivity depends on various parameters such as volume concentration of nanoparticles, shape, size, and material of nanoparticles, working temperature, and the material of base fluid. Recently, Momin [20] and Suresh et al. [21] have made a new experimental work on reinforcing the thermal conductivity of the prime fluid. it observed a new type called hybrid nanofluid by adding two various types or more of nanoparticles in a base fluid which improve the rate of heat transfer and thermal conductivity of the prime fluid better than nanofluids. For example, Al_{2}O_{3} seems more chemical stagnation and stabilization than metallic nano particles, so metallic nanoparticles like zinc, aluminum and copper have high thermal conductivities. Hybrid nanoﬂuid have many applications in different engineering ﬁelds like micro ﬂuidics, nuclear safety, manufacturing, military, transportation, pharmaceutical, acoustics, naval structures, buildings or cooling of ﬂushmounted electronic heaters in supercomputers and modern electronic devices. So, due to these benefits of hybrid nanofluid, a lot of researchers are interested to use hybrid nanofluid for developing nanotechnology. Mourad et al. [22] presented a numerical contribution for heat transmit and magnetohydrodynamic in a fined cold wavywalled porous cavity with a hot elliptic inner cylinder occupied by hybrid Fe_{3}O_{4}MWCNT /water nanofluid. Alwawi et al. [23] concerned with natural convection of ethylene glycolbased nonNewtonian nanoliquid and it is affected by a magnetic field along a horizontal circular cylinder. Niihara [24] shown that the thermal and mechanical properties can be signiﬁcantly enhancement by adding nanoparticles to materials. The advantage of using the nanoﬂuid in the liquid block is greater than pure water this is which was explored by Mehdi Bahiraei et al. [25]. The heat capacity of a ﬂuid increased by single and hybrid nanoadditives as show in the works of many researchers [2628]. The aim of this article is discussing the effect of some parameters on Cu Al_{2}O_{3 }water hybrid nanofluid flow over a stretching wedge with ohmic heating in the existence of a magnetic field.
Table 1. Thermophysical properties of Copper and Alumina [29, 30]
Property 
Water 
Copper Cu 
Alumina Al_{2}O_{3} 
$\rho$ C_{p} k $\beta$ $\sigma$ $\mu$ 
997.1 4179 0.613 21x10^{5} 0.05 8.9x10^{4} 
8933 385 401 1.67x10^{5} 5.96x10^{7}  
3970 765 40 0.85x10^{5} 1x10^{10}  
Figure 1. Flow model
Consider the problem of steady 2D magnetohydrodynamic mixed convection flow of hybrid nanofluid CuAl_{2}O_{3} water and heat transfer over a stretching wedge in an animated viscous, incompressible and electrically conducting ﬂuid. As shown from Figure 1, the xaxis is been selected to be the along the wedge surface, and the normal to it is the yaxis. $U=U_{0} x^{m}$ is the velocity of the stream flow far away from the wedge. U_{w}=UR is the velocity of the stretching of the wedge, were $(R \geq 0). B(x)=B_{0} x^{(m1) / 2}$ is the magnetic field and applied in the ydirection. The induced magnetic ﬁeld is neglected due to the assuming that the magnetic Reynolds number is small. The temperature in the external side of the temperature boundary layer is $T_{\infty}$. The stretching wedge surface temperature is $T_{w}=T_{\infty}+b x^{2 m}. \beta=2 m /(m+1)$, is the angle parameter of the wedge and $\Omega=\beta \pi$ for an overall angle of the wedge. $\beta=0$ in the case of the wall is horizontal and $\beta=1$ in the case of the wall is perpendicular. Table 1 shows the properties of thermophysical of the nanoparticles and base fluid [29, 30]. By these assumptions, the equations governing can be written as:
Continuity equation
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
Momentum equation
$u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=U \frac{d U}{d x}+v_{h n f}\left(\frac{\partial^{2} u}{\partial y^{2}}\right)+g \beta_{h n f}\left(TT_{\infty}\right) \sin \frac{\Omega}{2}\frac{\sigma_{h n f} \,\,B^{2}}{\rho_{h n f}}(uU)$ (2)
Energy equation
$\left(\rho C_{p}\right)_{h n f}\left(u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}\right)=\rho_{h n f} u\left(U \frac{d U}{d x}+\frac{\sigma_{h n f}}{\rho_{h n f}} B_{0}{ }^{2} U\right)+\alpha_{h n f}\left(\frac{\partial^{2} T}{\partial y^{2}}\right)+\sigma_{h n f} B^{2} u^{2}$ (3)
with the boundary conditions
$\left.\begin{array}{c}y=0: u(x, y)=U_{w}, v(x, y)=0 \\ T=T_{w}(x)=T_{\infty}+b x^{2 m} \\ y \rightarrow \infty: u(x, y)=U(x), T(x, y)=T_{\infty},\end{array}\right\}$ (4)
where, u is the component of the velocity along xdirection and v is the component of the velocity along the ydirection. T is the temperature, g is the gravity ﬁeld, $\beta_{h n f}$ is the thermal expansion coefficient, $\left(\rho c_{p}\right)_{h n f}$ is the speciﬁc heat at a fixed pressure, $\mu_{h n f}$ is the viscosity of the hybrid nanoﬂuid, $\alpha_{h n f}$ is the thermal conductivity of the hybrid nanoﬂuid, $v_{h n f}=\mu_{h n f} / \rho{ }_{h n f}$ is the kinematics viscosity of the hybrid nanoﬂuid.
We can define some of the similarity transformations as:
$\left.\begin{array}{c}\psi(x, \eta)=\sqrt{U x v_{f}} f(\eta) \\ \eta=y \sqrt{\frac{U}{v_{f} x}}, \theta(x, \eta)=\frac{TT_{\infty}}{T_{w}T_{\infty}}\end{array}\right\}$ (5)
where, $\theta$ is the dimensionless temperature, $f^{\prime}=u / U$ is the dimensionless velocity.
By substituting from Eq. (5) into Eq. (1:3) we get:
$f^{\prime \prime \prime}=\frac{\sigma_{h n f}}{\sigma_{f}}(1\varphi)^{2.5} \mathrm{M}\left(f^{\prime}1\right)\frac{\rho_{h n f}}{\rho_{f}}(1\varphi)^{2.5}\left(\frac{m+1}{2} f f^{\prime \prime}+m\left(1f^{\prime 2}\right)+\frac{\beta_{h n f}}{\beta_{f}} \gamma\left(\sin \frac{\Omega}{2}\right) \theta\right)$ (6)
$\theta^{\prime \prime}=\operatorname{Pr}\left(\frac{k_{h n f}}{k_{f}} \frac{\left(\rho C_{p}\right)_{f}}{\left(\rho C_{p}\right)_{h n f}}\right)^{1}\left(\frac{\left(\rho C_{p}\right)_{h n f}}{\left(\rho C_{p}\right)_{f}}\left(\frac{m+1}{2} f \theta^{\prime}+2 m f^{\prime} \theta\right)E c\left(M \frac{\sigma_{h n f}}{\sigma_{f}} f^{\prime 2}\left(m \frac{\rho_{h n f}}{\rho_{f}}+\frac{\sigma_{h n f}}{\sigma_{f}} M\right) f^{\prime}\right)\right)$ (7)
The factors appeared are $\gamma=\frac{G r_{x}}{R e^{2} x}$, $\mathrm{M}=\frac{\sigma_{f} B_{0}^{2}}{\rho_{f} U_{0}}$, $\operatorname{Pr}=\frac{\mu_{f}\left(C_{p}\right)_{f}}{k_{f}}$, $R e_{x}=\frac{U x}{v_{f}}$, $E c=\frac{U^{2}}{\left(C_{p}\right)_{f}\left(T_{w}T_{\infty}\right)}$, $G r_{x}=\frac{g \beta_{0} f\left(T_{w}T_{\infty}\right) x^{3}}{v_{f}^{2}}$.
where, $\gamma$ is the mixed convection parameter that refers to the effect of the thermal buoyancy force on the field of the flow, Pr is the Prandtl number, Ec is Eckert number, M is the magnetic parameter, Gr_{x} is local Grashof number, Re_{x} is local Reynolds number.
The boundary conditions which defined in Eq. (4), after transformations become:
$\left.\begin{array}{c}f(0)=0, f^{\prime}(0)=R, f^{\prime}(\infty)=1 \\ \theta(0)=1, \theta(\infty)=0,\end{array}\right\}$ (8)
where, R is the velocity ratio, when R>1 the wedge stretching is faster, when R<1 the wedge stretching slower. The prime refers to the derivative with respect to $\eta$. The physical quantities of interest, local Nusselt number and local skin friction coeﬃcient are defined as:
Skin friction coefficient and Nusselt number are
$C_{f}=\frac{\mu_{h n f}\left(\frac{\partial u}{\partial y}\right)_{y=0}}{\rho_{f} U_{w}^{2}}$
$N u_{x}=\frac{x k_{h n f}\left(\frac{\partial T}{\partial y}\right)_{y=0}}{k_{f}\left(T_{w}T_{\infty}\right)}$
Hence
$R e_{x}^{1 / 2} C_{f}=\frac{f^{\prime \prime}(0)}{(1\varphi)^{2.5}}$
$R e_{x}^{1 / 2} N u_{x}=\frac{k_{h n f}}{k_{f}} \theta^{\prime}(0)$ (9)
The equation of density in nanofluid is:
$\rho_{n f}=(1\varphi) \rho_{f}+\varphi \rho s \rho_{n f}=(1\varphi) \rho_{f}+\varphi \rho_{s}$
And in Hybrid nanofluid is:
$\rho_{h n f}=\left(1\left(\varphi_{s 1}+\varphi_{s 2}\right)\right) \rho_{f}+\varphi_{s 1} \rho_{s 1}+\varphi_{s 2} \rho_{s 2}$
The effective dynamic viscosity of the nanofluid according to the Brinkman model [31] is given by
$\mu_{n f}=\frac{\mu_{f}}{(1\varphi)^{2.5}}$
where, $\mu_{f}$ is the viscosity of the fluid fraction, so the effective dynamic viscosity of hybrid nanofluid is
$\mu_{h n f}=\frac{\mu_{b f}}{\left(1\left(\varphi_{s 1}+\varphi_{s 2}\right)\right)^{2.5}}$
The equation of heat capacity in nanofluid is
$\left(\rho c_{p}\right)_{n f}=(1\varphi)\left(\rho c_{p}\right)_{f}+\varphi\left(\rho c_{p}\right)_{s}$
The equation of heat capacity in hybrid nanofluid is:
$\left(\rho_{C p}\right)_{h n f}=(1\varphi)\left(\rho c_{p}\right)_{f}+\varphi_{s 1}\left(\rho c_{p}\right)_{s 1}+\varphi_{s 2}\left(\rho c_{p}\right)_{s 2}$
The equation of thermal conductivity in nanofluid for spherical nanoparticles and according to the MaxwellGarnetts model [28] is defined as:
$\frac{k_{n f}}{k_{f}}=\frac{\left(k_{s}+2 k_{f}\right)2 \varphi\left(k_{f}k_{s}\right)}{\left(k_{s}+2 k_{f}\right)+\varphi\left(k_{f}k_{s}\right)}$
From this, the thermal diffusivity of nanofluid defined as:
$\alpha_{n f}=\frac{k_{n f}}{\left(\rho C_{p}\right)_{n f}}$
The equation of thermal conductivity in hybrid nanofluid if thermal conductivity is defined as according to the MaxwellGarnetts model should give by
$\frac{k_{h n f}}{k_{f}}=\frac{\frac{\varphi_{s 1} k_{s 1}+\varphi_{s 2} k_{s 2}}{\varphi}+2 k_{f}+2\left(\varphi_{s 1} k_{s 1}+\varphi_{s 2} k_{s 2}\right)2 \varphi k_{f}}{\frac{\varphi_{s 1} k_{s 1}+\varphi_{s 2} k_{s 2}}{\varphi}+2 k_{f}\left(\varphi_{s 1} k_{s 1}+\varphi_{s 2} k_{s 2}\right)+\varphi k_{f}}$
So, the thermal diffusivity of hybrid nanofluid given by
$\alpha_{h n f}=\frac{k_{h n f}}{\left(\rho C_{p}\right)_{h n f}}$
The thermal expansion coefficient of the nanofluid can be determined by:
$(\rho \beta)_{n f}=\varphi(\rho \beta)_{s}+(1\varphi)(\rho \beta)_{f}$
where, $\beta_{f}$ is the coefficient of thermal expansion of the fluid and $\beta_{s}$ for the solid fractions.
For a hybrid nanofluid, the thermal expansion can be defined as follows:
$(\rho \beta)_{h n f}=\varphi_{s 1}(\rho \beta)_{s 1}+\varphi_{s 2}(\rho \beta)_{s 2}+(1\varphi)(\rho \beta)_{f}$
The effective electrical conductivity of the nanofluid was given by Maxwell [32] as:
$\frac{\sigma_{n f}}{\sigma_{f}}=1+\frac{3\left(\frac{\sigma_{s}}{\sigma_{f}}1\right) \varphi}{\left(\frac{\sigma_{s}}{\sigma_{f}}+2\right)\left(\frac{\sigma_{s}}{\sigma_{f}}1\right) \varphi}$
where, $\varphi=\varphi_{s 1}+\varphi_{s 2}$, and for hybrid nanofluid is given by
$\frac{\sigma_{h n f}}{\sigma_{f}}=1+\frac{3\left(\frac{\varphi_{s 1} \sigma_{s 1}+\varphi_{s 2} \sigma_{s 2}}{\sigma_{b f}}\varphi\right)}{\left(\frac{\varphi_{s 1} \sigma_{s 1}+\varphi_{s 2} \sigma_{s 2}}{\varphi \sigma_{f}}+2\right)\left(\frac{\varphi_{s 1} \sigma_{s 1}+\varphi_{s 2} \sigma_{s 2}}{\sigma_{f}}\varphi\right)}$
Table 2. Comparison of numerical values of $f^{\prime \prime}(0)$ and $\theta^{\prime}(0)$
M 
R 
$f^{\prime \prime}(0)$ 

Su et al. [33] 
Present study 

0.0
3.0 
0.1 2 0.1 2 
1.024909003 1.122862620 1.991260941 2.144131581 
1.0249090030 1.122862620 1.9912609410 2.1441330609 
M 
R 
$\theta^{\prime}(0)$ 

Su et al.[33] 
Present study 

0.0
3.0 
0.1 2.0 0.1 2.0 
0.626473534 0.878136319 0.786144878 0.407961083 
0.626473534 0.878136342 0.786144878 0.407966998 
Table 3. Numerical values of skinfriction coeﬃcient and Nusselt number for different parameters when Pr=1
$\varphi$ 
M 
R 
$\gamma$ 
$\beta$ 
$R e_{x}^{1 / 2} C_{f}$ 
$R e_{x}^{1 / 2} N u_{x}$ 
1% 
2.0 
0.1 
1.0 
1.0 
2.00055 
2.39129 
5% 




2.36538 
2.58523 
10% 




2.90624 
2.83484 
5% 
0 



1.74447 
2.42940 

1.0 



2.08025 
2.51840 

2.0 



2.58523 
2.58523 

2.0 
0.1 


2.58523 
2.58523 


0.5 


1.45502 
3.36365 


2.0 


2.82672 
5.58771 


0.1 
2.0 

1.66258 
2.43215 



1.0 

1.90339 
2.48678 



0 

2.13736 
2.53762 



1.0 

2.36538 
2.58524 



2.0 

2.58814 
2.63010 



1.0 
0 
1.60072 
1.03198 




2/3 
2.12664 
2.00331 




1.0 
2.36538 
2.58524 
From Table 3 we notice that the quantity of skin friction coefﬁcient of hybrid nanoﬂuid is improved with increasing of nanoparticle volume fraction, magnetic field, mixed convection, and the angle of the wedge whereas decreases when we increase the ratio of the velocity. Also observed the Nusselt number increase with increasing nanoparticle volume fraction, magnetic field, mixed convection, the angle of the wedge, and the ratio of the velocity.The numerical solutions are obtained for many values of physical parameters. Skin friction and Nusselt number coefficients. The ranges of the magnitude of the nanoparticles volume fraction $\varphi$, mixed convection $\gamma$, the angle of the wedge $\beta$ the magnetic field M, and the ratio of the velocity R, used in this study are $\varphi=[0,1]$, $\gamma=[2,2]$, $\beta=[0,1]$, $M=[0,2]$, $R=[0.1,2]$ respectively. Table 2 shows the obtained outcomes of our study in the case of $\left(\varphi_{s 1}=0\right.$ and $\left.\varphi_{s 2}=0\right)$ and notice that our outcomes are excellent agreement with the last published results of Su et al. [33].
Figure 2 shows the effect of $\gamma$ on $f^{\prime}$ in the case of existence and absence the magnetic field. It shows that $f^{\prime}$ increases as $\gamma$ increases. It can be seen that related boundary layer thickness and the ﬂuid velocity is improved due to increment in values of $\gamma$. This is because of the fact that an improvement in $\gamma$ means an improvement in the buoyancy force through which the ﬂuid velocity is improved. The effect of $\gamma$ on $\theta$ is depicted in Figure 3 in the case of existence or absence the magnetic field. It refers that the proﬁle $\theta$ decreases with increasing in the values of $\gamma$, so the thickness of the thermal boundary layer is decreased. The mixed convection does not exist in the energy equation explicitly; this explains the low reliance of the heat transfer at the surface. Figure 4 shows the impact of m on the velocity profile $f^{\prime}$. It is observed that an increase in the values m leads to that the thickness of the ﬂuid layer and $f^{\prime}$ are decreased.
Figure 5 shows the impact of m on the temperature profile. It is observed that an increase in m leads to a decrease in the temperature profile. Because the angle parameter of the wedge is reliant over the gradient of the pressure, which its values can be negative or positive.
Figure 2. Velocity distributions versus M and $\gamma$
Figure 3. Temperature distributions versus M and $\gamma$
Figure 4. Velocity distributions versus m
Figure 5. Temperature distribution versus m
Figure 6. Velocity distribution versus $\varphi$
Figure 7. Temperature distributions versus $\varphi$
Figures 4 and 5 show that the thickness of the velocity and temperature boundary layer in the Cuwater case is less than that in Al_{2}O_{3}water. Figure 6 shows the effect of volume fraction on the velocity profile as R=0.1 or R=2. It is observed that an increase in the volume fractions leads to that a thickness of the velocity boundary layer is decreased when R=0.1 or R=2. Physically, a varied velocity line is generated by the occurrence of divergent collisions between adjacent particles during ﬂuid diffusion. Moreover, it is found that an increase in the concentration of the nanoparticle improved the resistance among the neighboring layers of moving ﬂuid, which leads to reduce the velocity proﬁle.
Further, Figure 7 Shows the effect of volume fraction on the temperature profile as R=0.1 or R=2. It is observed that an increase in the volume fractions leads to an increase in the temperature profile. This is due to that the exchange of thermal energy in the layer of ﬂuid is enhanced by the dispersion of nanoparticles, that is neighboring to the surface and next neighboringﬂuid layers dispersion of nanoparticles. So, results expect that the utilize of nanoparticles can aid to enhance the heat distribution in heat transfer particular equipment and can store energy through the chemical process.
The problem of heat transfer by MHD mixed convection of CuAl_{2}O_{3}water hybrid anoﬂuid past over a stretching wedge with Ohmic heating is studied. The governing equations were converted into a similar form. The obtained equations were solved numerically by using the Matlab bvp4c method. The numerical results very agree with the last work. The effects of the different parameters namely magnetic parameter, mixed convection, the angle of the wedge parameter, the ratio of the velocity, the nanoparticles volume fraction, the velocity, temperature proﬁles as well as the local Nusselt number and the local skin friction coefﬁcients are discussed and shown graphically. From the obtained outcomes of the problem, it was noticed that a rising of the values of the nanoparticles volume fraction, magnetic ﬁeld, mixed convection and the angle of the wedge parameters resulted in increases in the skinfriction coefﬁcient, local Nusselt numbers whereas the skinfriction coefﬁcient decrease and local Nusselt numbers increase as the ratio of the velocity is an increase.
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