MHD Effects on Mixed Convective Nanofluid Flow with Viscous Dissipation in Surrounding Porous Medium

In existence of concerning magnetic field, heat together with mass transfer features on mixed convective copper-water nanofluid flow through inclined plate is investigated in surrounding porous medium together with viscous dissipation. A proper set of useful similarity transforms is considered as to transform the desired governing equations into a system as ordinary differential equations which are nonlinear. The transformed equations for nanofluid flow include interrelated boundary conditions which are resolved numerically applying Runge-Kutta integration process of sixth-order together with Nachtsheim and Swigert technique. The numerical consequences are compared together with literature which was published previously and acceptable comparisons are found. The influence of significant parameters like as magnetic parameter, angle for inclination, Eckert number, fluid suction parameter, nanoparticles volume fraction, Schmidt number and permeability parameter on concerning velocity, temperature along with concentration boundary layers remains examined and calculated. Numerical consequences are presented graphically. Moreover, the impact regarding these physical parameters for engineering significance in expressions of local skin friction coefficient in addition to local Nusselt together with Sherwood numbers is correspondingly examined.

Heat transfer enhancement concerning boundary layer fluid flow for different nanofluid flow passes through a vertical plate that has been studied with steady case via Rana and Bhargava [1] including with the effect such as temperature dependent heat source/sink. For different nanofluids, average of Nusselt number was found to diminish by them. The influence regarding viscous dissipation, chemical reaction together with Soret in existence of magnetic field into nanofluids flow has been considered pass through porous media through Yohannes and Shankar [2]. The Keller box process was used to solve governing equation of the concerning fluid flow, and numerical outcomes were presented for various parameters of convective heat together with mass transfer properties.
Magnetic field including thermal radiation effects for nanofluid flow has been analyzed along stretching surface through Khan et al. [3]. The flow field was discussed by them with dissimilar time steps and reported that average shear stress reductions with the development of magnetic field are observed. MHD boundary layer nanofluid flow regarding heat with mass transfer has been stated through porous media by Haile and Shankar [4] with considering thermal radiation including viscous dissipation with chemical reaction effects. They were considered copper (Cu)-water and Al 2 O 3 -water nanofluids and noted out that velocity field decreases with increase of magnetic field.
For considering the steady case, MHD mixed convective nanofluid flow through porous medium which has been deliberated past along a stretching sheet by Ferdows et al. [5]. They concluded that velocity together with temperature increases while concentration decreases gradually with increase of Eckert number. Heat transfer physical characteristics of flow field with three dissimilar categories of nanofluid pass through permeable stretching/shrinking surface has been considered and observed through porous medium via Pal et al. [6]. They found with the increment of suction/injection parameters as local Nusselt number rises for stretching sheet while decreasing for shrinking sheet.
Through inclined porous plate, magnetohydrodynamic mixed convective flow including Joule heating together with viscous dissipation on the field has been studied via Das et al. [7]. The velocity along with temperature of flow fields rise due to rise of particular magnetic field which was obtained by them. Within a porous medium, MHD mixed convection happening on fluid flow has been inspected and analyzed toward a vertical plate through Hari et al. [8] with the radiation, including heat generation with presence of chemical reaction effect. On increasing particular magnetic field, velocity profile overshoots adjacent the plate surface and then convergence to the boundary which was found by them.

Md. N. Uddin et al. Journal of Applied Mathematics and Physics
On boundary layer flow, MHD effects toward exponentially shrinking sheet have been analyzed by Jain and Choudhary [9]. The numerical outcomes were observed graphically and then explored through them. For MHD Williamson nanofluid flow, effects regarding chemical reaction, melting and radiation through porous medium have been examined and discussed with specific physical parameters by Krishnamurthya et al. [10]. Considering Brinkman nature nanofluid in unsteady magnetohydrodynamic flow towards on vertical plate has been discussed into porous medium via Ali et al. [11]. Fluid velocity diminishes with the enhancement of such nanoparticle volume fraction which is noticed by them.
Through porous media, MHD flow regarding nanofluid with characteristic of heat including mass transfer has been reported along stretching sheet via Reddy and Chamkha [12]. The velocity reduces while temperature together with concentration enhances as per magnetic parameter increases which were found by them. For unsteady case, MHD along with free convective nanofluid flow pass through a flat plate has been stated with radiation absorption by Prasad et al. [13] and influence of regarding several physical factors on flow field was studied by them.
Double diffusive magnetohydrodynamic nanofluids flow together with effects of thermal radiation including viscous-Ohmic dissipation has been discussed along nonlinear stretching/shrinking sheet via Pal and Mandal [14]. In existence of magnetohydrodynamics, unsteady natural convective regarding nanofluid flow through above a porous vertical plate has been studied for three dissimilar categories of nanofluids by Geetha et al. [15]. The numerical outcomes were analyzed and then presented for numerous physical factors which are concerned by them. A mathematical relation for two dimensional flow of magnetic Maxwell nanofluid subject to heat sink/source influenced by a stretched cylinder has been attained by Irfan et al. [16]. As stated aforementioned, it remains more practical to include MHD effects to study the impact regarding momentum including heat with mass transfer flow, and to the best of author's knowledge; no investigation is considered as MHD effects on mixed convective through nanofluid flow along viscous force in surrounding porous medium.

Mathematical Model
The two dimensional incompressible nanofluid flows which is steady and viscous is considered for analysis. The nanofluid flow is passing inclined porous plate in surrounding porous medium. In addition to consider magnetic field B 0 , this is introduced normally to direction of nanofluid flow. For two dimensional coordinate system which is shown as below in Figure 1, (x, y) is (direction along of flow, direction normal of flow) and (u, v) is (velocity components of x, velocity components of y). It is supposed that U ∞ denotes free stream velocity of flow field; g denotes the gravity by virtue of acceleration and α denotes angle to vertical porous plate. Moreover, the temperature T w at the wall is larger than ambient temperature T ∞ while concentration C w at the wall is larger than ambient concentration C ∞ .
Consequently under the above flow field consideration, conservation law for mass is obeyed automatically as given below: The common fluid as water is considered such as base fluid while copper (Cu) is considered such as nanoparticles for flow field together with both are locally thermal equilibrium. The thermophysical properties which are used for nanofluid are specified into Table 1.
where for the nanofluid, v nf is the kinematic viscosity, (β t ) nf and (β c ) nf are the coefficient for thermal and concentration expansions, g is the acceleration owing to gravity, α is the angle for inclination, σ nf is the electrical thermal conductivity, ρ nf is the density, nf α is the thermal diffusivity and ( ) p nf C ρ is the heat capacitance. Furthermore, k pp is permeability regarding porous medium whereas D is mass diffusivity.
The operational dynamic viscosity for nanofluid was given via Brinkman [19] while relation between physical quantities regarding nanoparticle and base fluid which were familiarized by Abu-Nada [20] as: where, bf µ is considered for dynamic viscosity while ϕ is considered for nanoparticle solid volume fraction. The subscripts in aforementioned equations bf and np symbolize namely base fluid and nanoparticle respectively. , where permeability of porous plate is as v w (x) which is for suction (<0) or blowing (>0) whereas the subscripts in aforementioned equations w and ∞ are denoted wall and boundary layer edges respectively.
The well-known stream function is convenient to consider as equation of continuity is satisfied identically through following relations: In the above, ψ is the stream function. To reduce complexity of nanofluid flow field, the succeeding dimensionless transformations which were introduced via Cebeci et al. [21] are used: In the above, η is the similarity transform. The components for velocity of Equation (8) using Equation (9) can be re-written such as wherever the prime of aforementioned equation indicates as differentiation with respect to η.
Using similarity transformations in Equations (2) and (4) including the accompanying boundary conditions Equations (6) and (7), the transformed equations in which momentum, energy together with concentration equations are as follows: and corresponding boundary conditions are supposed as below: 1, 0 and 0 at f s In the aforementioned Equation (14), the coefficient for wall mass transfer is ( ) However, local Sherwood number is transformed into the succeeding form as: Re 0

Procedures of Numerical Solutions Using Nachtsheim and Swigert Technique
The system regarding nonlinear boundary value problem which was represented by Equations (11)-(13), together with corresponding boundary conditions Equations (14) and (15) is solved using Nachtsheim and Swigert [22] technique which is used to find unspecific initial conditions. The transformed boundary value system is re-transformed to initial value system and solved numerically employing sixth order Runge-Kutta initial value solver. The useful numerical methods were described by Nachtsheim and Swigert [22] and sixth order Runge-Kutta initial value solver by Al-Shimmary [23].

Comparison of Numerical Results
For the accurateness of numerical results, comparisons are prepared considering the special effects regarding velocity ratio parameter λ on velocity and temperature profiles. The results for Bachok et al. [18] which was published is compared with verify of Bachok et al. [18] work by authors for copper (Cu)-water nanofluid flow along with Prnadtl number as Pr = 6.2, nanoparticle volume fraction as ϕ = 0.1 and velocity ratio parameter as λ = −0.5 as shown into Figure 2.
It is detected that, verified corresponding numerical results of first solution are found excellent agreement. This favorable acceptable comparison indication is to improve numerical approach of Nachtsheim and Swigert [22] for present work. As well, the obtained numerical results for present work will report graphically into next subsequent section and analyze thereafter.

Results and Discussion of Significant Parameters
The  Figure 3(a) and Figure   3(b) and Figure 4(a) and Figure 4(b). The fluid velocity for nanofluid flow rises because of rise of concerning magnetic field as given away in Figure 3(a) and alike result was established through Das et al. [7]. Because, as magnetic field strength increases; the Lorentz force regarding magnetic field creates boundary layer for nanofluid flow as thinner. On free stream velocity, magnetic lines of forces move pass through the plate. The fluid which is decelerated by the viscous force, receives an impulsion from magnetic field which counteracts a viscous effects. As results, velocity of considered nanofluid flow rises and converges to the boundary together with rising magnetic field parameter. Additionally as detected in Figure 3(b), the thickness of thermal boundary layer of considered nanofluid flow rises owing to rise magnetic field because functional magnetic field which has a tendency to heat the fluid owing to electromagnetic work reduces heat transfer to the wall. Furthermore, skin friction coefficient namely C f is established to increases with increase magnetic field strength as given away in Figure   4(a). The reason for this is that, functioning magnetic field regarding nanofluid flow tends to improve flow motion and thus to improve surface friction force. In existence of growing magnetic field strength, the temperature gradient decreases at the wall which in turn leads to a reduction in rate of heat transfer. Subsequently, Nu x namely local Nusselt number which is schemed in Figure 4(b) decreases together with rise magnetic field strength.
The influence for α (α = 0˚, 30˚ and 60˚) which is angle for inclination for the copper (Cu) and water nanofluid flow fields is revealed in Figure 5 Figure 7(a). It is also observed from Figure 7(b) that with increase of ϕ, local Nusselt number Nu x increases for as much the temperature gradient at the wall increased and as a result Nu x increases. On the other hand, the effect of the permeability parameter K (K = 0, 3 and 6) on the momentum boundary layer as well as local skin friction coefficient C f are observed in Figure 8(a) and Figure 8(b). On the basis of the velocity variation in Figure 8(a), it is seen that the velocity of the flow field within the velocity boundary layer increases near the plate while decreases in the next and finally converges to the boundary condition as the permeability of the porous medium increases. However, Figure 8(b) shows the influence of permeability of the porous medium on the local skin friction coefficient against the streamwise distance x. In Figure 8(b), it is found that the entire local skin friction coefficient increases with the increase of permeability of the porous medium parameter.

Conclusions
The effect of magnetohydrodynamic on mixed convective nanofluid flow with viscous dissipation has been examined in surrounding porous medium. Consequently, the concerning governing equations regarding nanofluid flow are converted into nonlinear boundary layer equations by proper similarity transformations and solved numerically through sixth-order Runge-Kutta method with Nachtsheim and Swigert [22] approaches. For validity of numerical results, comparisons of numerical results remain prepared with Bachok et al. [18] and a comparatively acceptable comparison is reached. The concerning numerical results mainly calculated for influence of magnetic parameter M, angle for inclination α, Eckert number Ec, fluid suction parameter f w , nanoparticles volume fraction ϕ, Schmidt number Sc and permeability parameter K on flow field.
The following results on velocity, temperature together with concentration of nanofluid flow fields including local skin friction coefficient C f , local Nusselt Md. N. Uddin et al.