This paper investigates the unsteady mixed convection flow of an incompressible electrically conducting nanofluid. The Cattaneo- Christov double diffusion with heat and mass transfer are taken into account. The governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations by using suitable similarity transformations. These equations are solved analytically by a homotopy perturbation technique. The profiles of the velocity, the temperature, the induced magnetic field and the nanoparticles concentration are sketched with various parameters. The influences of these parameters are discussed in details
Keywords: Nanofluid; Mixed convection; Relaxation time; Magnetohydrodynamics
|A||unsteadiness parameter||T∞||ambient temperature|
|b||characteristic temperature||u||velocity component in the x-direction|
|C||nanoparticle concentration||Ue||x-velocity at the edge of the boundary layer|
|c||positive constant||velocity vector|
|Cw||nanoparticle concentration at the surface||v||velocity component in the y-direction|
|C∞||ambient nanoparticle concentration||x||distance along the plate|
|cf||volumetric volume expansion of the fluid||y||distance perpendicular to the plate|
|cp||volumetric volume expansion of the particle|
|d||characteristic nanoparticle concentration||Greek symbols|
|DB||Brownian diffusion coefficient||λ||positive constant|
|DT||thermophoretic diffusion coefficient||η||independent similarity variable|
|f||dimensionless stream function||ψ||stream function|
|Grx||local Grashof number||θ||dimensionless temperature|
|acceleration vector due to gravity||φ||dimensionless nanoparticle concentration|
|induced magnetic field vector||α||thermal diffusivity|
|He||x-magnetic field at the edge of the boundary layer||ρf||density of the fluid|
|ρP||density of the particle|
|H0||uniform induced magnetic field strength||(ρc)f||heat capacity of the fluid|
|h'||dimensionless induced magnetic field||(ρc)P||effective heat capacity of the nanoparticle material|
|hx||induced magnetic field in the x-direction|
|hy||induced magnetic field in the y-direction||μ||dynamic viscosity|
|mass flux vector||μe||magnetic diffusivity|
|KT||thermal conductivity||μ0||magnetic permeability|
|Le||Lewis number||ν||kinematic viscosity|
|M||magnetic parameter||τ||ratio between effective heat capacity of the nanoparticle material and heat capacity of the fluid|
|Nr||nanofluid buoyancy parameter|
|Nt||thermophoresis parameter||τC||relaxation time of mass flux|
|Pr||Prandtl number||τH||relaxation time of heat flux|
|heat flux vector||β||volumetric expansion coefficient of the fluid|
|Ri||Richardson number||β1||thermal relaxation parameter|
|Rex||local Reynolds number||β2||nanoparticle concentration relaxation parameter|
|t||time||γ||reciprocal of the magnetic Prandtl number|
|tw||temperature at the surface|
Nanofluids are defined as composition of solid and liquid materials. This solid (nanoparticles or nanofibers) with diameter 1-100 nm dispersed in the base fluid . As the nanoparticles are so small, they can flow smoothly without clogging through micro channels. Choi and Eastman  were the first to suggest the term of nanofluid to define designed colloids that consist of nanoparticles suspended in the base fluid. Nanofluids have many advantages like they have more stability. Also, they have suitable viscosity, properties of spreading and dispersion on solid surface. Masuda et al.  observed that to enhance the heat transfer, nanofluids should consist of nanoparticles with 5% vol. Choi et al.  indicated that adding a small amount (less than 1% by volume) of nanoparticles to a liquid rises the thermal conductivity of it up to approximately two times. For instance, a small amount of adding copper to ethylene glycol enhances the thermal conductivity of the liquid by 40% . This phenomenon enables us to use nanofluids in advanced nuclear systems . Researchers have defined some mechanisms based on the thermal enhancements properties of nanofluids. Buongiorno  discussed seven possible slip mechanisms and he indicated that Brownian diffusion, and thermophoresis are the two effective mechanisms. One of the significant factors which affects the nanofluid’s flow is nanoparticle’s charges. Nanofluids content on negatively charged particles generates an electric field that affects the velocity profile . Also, the size and type of nanoparticles play an important role in natural convection heat transfer enhancement . The nanometer sized materials have unrivaled properties. Therefore, nanofluids are used in different industry and engineering applications. Some of these applications [10-12] are food productions, microelectronics, microfluidics, transportation and manufacturing. Also, they are used in power generation in nuclear reactor coolant, fuel cells, hybrid-powered engines, space technology, defence and ships. In addition, medical applications: gold nanoparticle probes that detect DNA, cancer treatment. Mnyusiwella  mentioned some nanotechnology dangers for environmental health.
There are three main types of convection; free, force, and mixed. A free (Natural or Buoyant) convection flow field is a continuous self flow resulted from temperature gradients. This type can be found in many physical phenomena. For example, the free convection occurs when food is placed inside freezer with no circulation assisted by fans . On the other hand, the forced convection is generated by an external source. When free and forced convections take place together, this status is known as the mixed convection. Cheng and Minkowycz  considered the problem of free convection through a vertical plate with a porous medium. Bejan and Khair  investigated the same problem with the addition of heat and mass transfer. Bejan  wrote a book on mixed convection heat and mass transfer. Das et al.  studied the MHD mixed convection flow in a vertical channel. They indicated that the magnetic field enhances the velocity and the temperature. Subhashini et al.  studied different effects of thermal and concentration diffusions on a mixed convection boundary layer flow across a permeable surface. Some of applications on mixed convection are electronic devices, heat exchanger, and nuclear reactors .
The heat transfer process has various applications like energy production, and power generation . Cattaneo  modified Fourier law of heat conduction by introducing the term of relaxation time. This term represents the finite velocity of heat propagation. Han et al.  made a comparison of Fourier’s law and the Cattaneo–Christov heat flux model. Nadeem et al.  used Cattaneo-Christov heat flux model instead of Fourier's law of heat conduction. Also, Hayat et al.  used the same model to study the steady two-dimensional MHD flow of an Oldroyd-B fluid over a stretching surface. Rubab and Mustafa  studied the MHD three-dimensional viscoelastic flow over stretching surface with the Cattaneo-Christov heat flux model. This model tends to clarify the characteristics of thermal relaxation time. Hayat et al.  used the Cattaneo-Christov double diffusion, which are the generalized Fourier's and Fick's laws, in studying the boundary-layer flow of viscoelastic nanofluids.
Magnetohydrodynamics (MHD) is the study of the magnetic properties of electrically conducting fluids. This branch has many applications in geophysics, astrophysics, sensors, and engineering. Chen  combined the boundary layer heat and mass transfer of an electrically conducting fluid in MHD natural convection from a vertical surface. For the importance of MHD, many works were made to study the flow characteristics over a stretching sheet under different conditions of MHD. Hayat et al.  studied MHD flow and heat transfer characteristics of the boundary layer flow over a permeable stretching sheet. Further, the induced magnetic field has many applications such as liquid metals and ionized gases . Kumari et al.  added the effect of the induced magnetic field to the problem of MHD boundary layer flow and heat transfer over a stretching sheet. Ali et al. investigated some works in this field which can be found in .
The purpose of the current work is to explore the unsteady boundary layer flow over a surface embedded in a nanofluid. Cattaneo- Christov double diffusion with heat and mass transfer is taken into account. The influence of the induced magnetic field is also investigated. Our problem can be clarified as follows: the physical description of the paper at hand is investigated in section 2. This section involves many items like; the mathematical formulation of the problem, the governing equations of motion, the related initial and appropriated boundary conditions. Finally, the suitable similarity transformations are also added in this section. Section 3 is devoted to introduce the method of solution. This technique depends mainly on the homotopy perturbation . The required distribution functions such as velocity, induced magnetic field, temperature, and nanoparticle concentration are given in this section. The influence of the various parameters on the above distribution functions are presented in section 4. Finally, the concluding remarks are introduced in section 5.
The two-dimensional, unsteady, laminar, incompressible mixed convective boundary layer flow over a semi-infinite vertical plate is taken into account. The fluid is considered as a Newtonian and electrically conducting nanofluid flow. The Cartesian coordinates (x,y) are considered. The x-axis is considered as the coordinate measured along the boundary layer surface. The y-axis is the normal coordinate to that surface. It is assumed that the flow outside the boundary layer moves with a stream velocity Ue that is parallel to the flat plate. As considered by , we may choose an unsteady induced magnetic field strength as . This field is normal to the flat surface where H0 is the initial strength of the induced magnetic field. The flat plate is considered to be electrically non-conducting so that no surface current sheet occurs. In addition, a magnetic field He is applied at the outer edge of the boundary layer which is parallel to the flat plate. Moreover, at the surface, Tw and Cw represent the temperature and nanoparticle concentration, respectively. Meanwhile, far away from the surface, the temperature and nanoparticle concentration are taken as T∞ and C∞ , respectively. It is worthwhile to note that:Ue , He,Tw and Cw are functions of x and t. The acceleration vector due to the gravity taken as , is taken into account. A schematic diagram of the physical model is shown in Figure 1.
The continuity equation
Gauss's law of magnetism (one of Maxwell's equations)
The momentum equation
The coupling between the fluid velocity, magnetic field and nanofluid on the conservation of momentum yields
where is the magnetohydrodynamic pressure, P0 is the pressure of the fluid, and is the magnetic pressure.
Magnetic induction equation gives
The energy equation yields 
where satisfies the following heat flux equation
where the relaxation time of heat flux is the initial relaxation time of heat.
The nanoparticle concentration equation yields 
where satisfies the following mass flux equation
where the relaxation time of mass flux is the initial relaxation time of mass.
In 1904, Ludwig Prandtl introduced an approximate form of the Navier-Stokes equations. He considered the no-slip condition at the surface and the frictional effects occur only in a thin region near the surface called boundary layer. Outside the boundary layer, the flow is inviscid flow. The boundary layer approximations can be written as follows:
By taking the 2-dimensional Cartesian coordinates with and . Under the Oberbeck-Boussinesq and the standard boundary layer approximations, the system of equations that govern the motion can be formulated as follows
The continuity equation
Gauss's law of magnetism
The momentum equation
According to the boundary layer approximations the y-momentum equation reduced to , and the x-momentum equation can be written as follows
Magnetic induction equation
By taking the influence of the Brownian motion and thermophoresis into consideration, Eqs. (5) & (7) become as follows
The energy equation
where the term is the heat convection, the term is the heat conduction, the term is the thermal energy transport due to Brownian diffusion, the term is the energy transport due to thermophoretic effect .
The nanoparticle concentration equation
where the term indicates that nanoparticles can move homogeneously within the fluid, the term is due to Brownian diffusion, the term is due to thermophoresis effect .
The appropriate initial conditions may be taken as 
The appropriate boundary conditions for t≥0 are 
where c and λ are constants and both have dimension (t-1) such that ( c>0 and λ≥0 ,λt<1 ), d is a constant with dimension (L-1), and b is a constant with dimension (TL-1) , such that "b>0 and d>0" represents the assisting flow (the case of heating plate), "b<0 and d<0" is corresponding to the opposing flow (the case of cooling plate), and "b=0 and d=0" represents the forced convection limit which means the absence of the free convection.
The system of partial differential Eqs.(11) – (14) is converted into a system of four coupled ordinary differential equations by using the following similarity transforms
To satisfy the continuity Eq.(9), we may consider a stream function ψ such that and . Also, hx and hy are defined as the previous forms to satisfy Eq.(10).
Substitute from Eq.(17) into Eqs.(11-14), we get the following system of four coupled ordinary differential equations
The momentum equation
The magnetic induction
The energy equation
The nanoparticle concentration equation
where the non-dimensional parameters are defined as follows
Also, the initial and boundary conditions of equations (15) and (16) take the following forms The appropriate initial conditions become
The appropriate boundary conditions for t ≥ 0 become
Now, the governing equations of motion are converted to the nonlinear ordinary differential equations (18- 21). Their solutions should satisfy the conditions (23) and (24). To relax the mathematical manipulation, we will use the homotopy perturbation technique . This technique is a combination of the traditional perturbation methods and homotopy techniques. Through this technique, there is no need for a small parameter. According to the homotopy technique, a homotopy imbedding parameter p ∈ [0,1] is considered. Therefore, Eqs. (18-21) may be rewritten as
At this stage, any function M may be written as
where M stands for f , h , θ, and φ.
The above equation represents the approximation solutions for the functions f , h , θ, and φ in terms of the power series of the homotopy parameter p.
The initial and boundary conditions which satisfy the above system can be written as
To obtain a good solution series for Eq.(29). We solved it till second order. The solutions for the various orders are lengthy but straight forward, away from the detail, these solutions may be written as follows:
Zero order solution
First order solution
Second order solution
where ξ1 and ξ2 are two functions of η which are given in the Appendix.
The coefficients (l1→l68) , (s1→s52) , and (r1→ r94) are given in the Appendix.
For the complete solution corresponding to p→1 in Eq. (29), the analytical perturbed solutions for the velocity, the induced magnetic field, the temperature, and the nanoparticle concentration are written as
To get a good convergence, we choose the length of the semi-infinite plate is limited to 2, i.e. η∞ = 2.
This section is devoted to discuss the influence of the various physical parameters on the velocity, the induced magnetic field, the temperature, and the nanoparticle concentration. These parameters are the unsteadiness parameter (A) , magnetic parameter (M) , reciprocal magnetic Prandtl number (γ) , thermal relaxation parameter(β1), nanoparticle relaxation parameter(β2), Prandtl number (Pr), Lewis number (Le),Brownian parameter (Nb), and thermophoresis parameter (Nt). Moreover, Richardson number (Ri), which represents the ratio of the buoyancy term to the shear stress term, its values are taken according to the type of convection. For mixed convection case, we took 0.1 < Ri < 10 . Meanwhile, at Ri < 0.1 the natural convection is negligible, and the forced convection is negligible at Ri > 10. Furthermore, Ri > 0 means that Tw > T∞ (the assisting flow), Ri < 0 means that Tw < T∞ (the opposing flow), and Ri = 0 is the case of the forced convection.
Figure 2 show the relation between the velocity f′ and the different physical parameters. In general, the velocity starts at its lowest value at the surface, f′(0) = 0, then it increases till it approaches its free stream value that satisfying the far field boundary condition, f′(2) = 1. Figure 2(a) sketches the rising in velocity f′ with the increasing of A. Also, the increasing in magnetic parameter M accelerates the velocity f′. Physically, when the induced magnetic field is normal to the surface, the Lorentz force acts in the upwards direction to enhance the flow and increase the fluid velocity. Figure 2(b) illustrates the effect of the relaxation parameters β1 and β2 on the velocity. The velocity f′ reduces with the increasing of β1. Meanwhile velocity f′ enhances with the increasing of β2. From Figure 2(c), the velocity f′ slightly decreases as Prandtl number Pr increases. In fact Prandtl number Pr is defined as the ratio between the momentum (viscous) diffusivity and the thermal diffusivity. This means growing in Pr enhances the rate of viscous diffusion which in turns decreases the velocity. Whilst, a reduction of f′ is indicated as the reciprocal magnetic Prandtl number γ increases. Figure 2(d) indicates that f′ reduces as Lewis number Le increases till Le ≈ 3.5, and then it starts to increase with the increasing of Le. Figure 2(e) shows that f′ decreases as Richardson number Ri increases.
Figure 3(a) indicates that the induced magnetic field h′ reduces with the increase in unsteadiness parameter A till A ≈ 0.2, and then h′ starts to increase as A increases. Figure 3(b) shows that h′ increases with the growing in the reciprocal magnetic Prandtl number γ. This is because of γ is defined as the ratio between the magnetic diffusivity and viscous diffusivity. So, the increasing in γ enhances the magnetic diffusivity which in turns enhances h′. Figure 3(b) indicates that h′ increases with the increase in Richardson number Ri.
Figure 4(a), plots the variation of temperature θ due to the unsteadiness parameter A. This figure indicates that for small values of A, the temperature decreases as long as A increases till the critical value Ac = 0.55, andthen the temperature slightly increases near the surface with the increasing of A. Moreover, the curves approach of the higher values of η . In Figures 4(b) and (c), there exists a certain point (η≈ 1.6) called crossing over point in which temperature profile has a conflicting behavior before and after that point. It should be noted that the increasing in the relaxation parameters &beta1 and β2, the Prandtl number Pr, and Lewis number Le leads to a decrease in the temperature before that point and an increase in the temperature after that point. Figure 4(d) shows that the increasing in Nb, the temperature θ decreases till the certain point (η ≈1.5), and then it starts to increase. Also, it is depicted that with the growing in Nt, the temperature θ decreases up to a point (η ≈ 1), and then increases. The temperature θ is slightly rises as Richardson number Ri increases, this can be shown in Figure 4(e).
From Figures 5 (a) and (b), it is illustrated the decreasing in nanoparticle concentration φ with the increasing of each of A and β1. Physically, as A increases, the mass transfer rate reduces from the fluid to the plate. This action caused a decrease in nanoparticle concentration. Also, the nanoparticle concentration φ decreases with the increase in β2 till a certain point (η ≈ 1.6) and then it starts to increase. Figures 5 (c) shows that φ decreases with the increasing of Pr and Le. This is because that the growing of Le reduces the Brownian diffusion coefficient that leads the flow to decrease the nanoparticle concentration. Figures 5(d) indicates that φ decreases with the increase of Nt, but φ enhances with the increasing values of Nb. Figures 5 (e) elucidates the reduction in nanoparticle concentration φ that resulted from the growing of Ri.
An analytical study is investigated for an unsteady boundary layer flow of nanofluid over a vertical plate. The boundary layer is affected by an induced magnetic field. The model of Cattaneo- Christov double diffusion with heat and mass transfer is taken into account. The governing partial differential equations are transformed to ordinary differential equations by using suitable similarity transformations. The resulted system is solved analytically by the homotopy perturbation method. The obtained functions such as the velocity, the induced magnetic field, the temperature, and the nanoparticle concentration are plotted graphically. The influences of the various parameters, in these functions, are sketched. The important results are summarized as follow: