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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.

Magnetic fluids which are types of particular nanofluids took advantage of magnetic material goods of nanoparticles inside as for example liquid rotary seals functioning with no maintenance along with tremendously low leakage in very extensive variety of applications. Magnetohydrodynamics (MHD) nanofluid flows have broadly uses of MHD generators, tunable optical fiber, optical grating, optical modulators and switches, polymer, petroleum technologies and then metallurgical industries.

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 [

Magnetic field including thermal radiation effects for nanofluid flow has been analyzed along stretching surface through Khan et 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. [

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. [

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 [

Therefore, in the light of above literatures, the purpose of the current work is to observe MHD effects on mixed convective nanofluid flow along with viscous dissipation in surrounding porous medium. The governing equations for considered nanofluid flow are converted into a combination as nonlinear ordinary differential equations via introducing similarity variables and solved numerically. Moreover, for cogency of numerical results, a comparison is prepared with the literature which is published and comparatively satisfactory comparison is achieved. The terms of engineering interests such as wall shear stress including rate of heat transfer together with rate of mass transfer are shown into tabular form. The influence for various physical features as magnetic parameter, angle of inclination, Eckert number and fluid suction parameter are presented on the field of flow and analyzed thereafter.

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 _{∞} 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:

∂ u ∂ x + ∂ v ∂ y = 0 (1)

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

Particles | Thermo Physical Properties | ||||
---|---|---|---|---|---|

ρ (kg/m^{3}) | C_{p} (J/kg K) | k (W/m K) | β × 10^{-5} (1/K) | σ (S/m) | |

Water (H_{2}O) | 997.1 | 4179 | 0.613 | 21 | 5.5 × 10^{−6} |

Copper (Cu) | 8933 | 385 | 401 | 1.67 | 59.6 × 10^{−6} |

Using the present physical features, the modified dimensional boundary layer equations of Pal and Mandal [

u ∂ u ∂ x + v ∂ u ∂ y = ν n f ∂ 2 u ∂ y 2 + g ρ n f { ( ρ β t ) n f ( T − T ∞ ) + ( ρ β c ) n f ( C − C ∞ ) } cos ( α ) − ( σ n f B 0 2 ρ n f + ν n f k p p ) ( u − U ∞ ) (2)

u ∂ T ∂ x + v ∂ T ∂ y = α n f ∂ 2 T ∂ y 2 + 1 ( ρ C p ) n f [ μ n f { u ∂ 2 u ∂ y 2 + ( ∂ u ∂ y ) 2 } + σ n f B 0 2 ( u − U ∞ ) 2 ] (3)

u ∂ C ∂ x + v ∂ C ∂ y = D ∂ 2 C ∂ y 2 (4)

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, α n f is the thermal diffusivity and ( ρ C p ) n f 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 [

μ n f = μ b f ( 1 − ϕ ) 2.5 , ρ n f = ( 1 − ϕ ) ρ b f + ϕ ρ n p , ( ρ β t ) n f = ( 1 − ϕ ) ( ρ β t ) b f + ϕ ( ρ β t ) n p ( ρ β c ) n f = ( 1 − ϕ ) ( ρ β c ) b f + ϕ ( ρ β c ) n p , ( ρ C p ) n f = ( 1 − ϕ ) ( ρ C p ) b f + ϕ ( ρ C p ) n p ν n f = μ n f ρ n f , K n f K b f = ( K n p + 2 K b f ) − 2 ϕ ( K b f − K n p ) ( K n p + 2 K b f ) + ϕ ( K b f − K n p ) , α n f = K n f ( ρ C p ) n f , σ n f σ b f = 1 + 3 ( σ n p σ b f − 1 ) ϕ ( σ n p σ b f + 2 ) − ( σ n p σ b f − 1 ) ϕ (5)

where, μ b f 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.

The accompanying boundary conditions for the existing nanofluid flow field are as follows:

u = 0 , v = ± v w ( x ) , T = T w and C = C w at y = 0 (6)

u = U ∞ , T = T ∞ and C = C ∞ at y → ∞ (7)

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:

u = ∂ ψ ∂ y and v = − ∂ ψ ∂ x (8)

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. [

η = y U ∞ ν b f x , ψ = ν b f x U ∞ f ( η ) , θ ( η ) = T − T ∞ T w − T ∞ and s ( η ) = C − C ∞ C w − C ∞ (9)

In the above, η is the similarity transform. The components for velocity of Equation (8) using Equation (9) can be re-written such as

u = U ∞ f ′ ( η ) and v = 1 2 ν b f U ∞ x [ η f ′ ( η ) − f ( η ) ] (10)

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:

f ‴ + ϕ 1 { 1 2 ϕ 2 f f ″ + ( ϕ 3 R i t θ + ϕ 4 R i c s ) cos ( α ) } + ( ϕ 1 ϕ 5 M + K ) ( 1 − f ′ ) = 0 (11)

θ ″ + 1 2 ( Pr ϕ 6 ) { ϕ 7 f θ ′ + 2 ϕ 1 E c ( f ′ f ‴ + f ″ 2 ) + 2 ϕ 6 M E c ( f ′ − 1 ) 2 } = 0 (12)

s ″ + 1 2 S c f s ′ = 0 (13)

and corresponding boundary conditions are supposed as below:

f = f w , f ′ = 0 , θ = 1 , s = 1 , at η = 0 (14)

f ′ → 1 , θ → 0 and s → 0 at η → ∞ (15)

In the aforementioned Equation (14), the coefficient for wall mass transfer is

f w = − v w ( x ) x ν b f U ∞ (16)

such as f w > 0 for suction and f w < 0 for injection or blowing while physical parameters are defined as under:

G r t = g ( β t ) b f ( T w − T ∞ ) x 3 ν b f 2 , G r c = g ( β c ) b f ( C w − C ∞ ) x 3 ν b f 2 , Re x = x U ∞ ν b f , R i t = G r t ( Re x ) 2 , R i c = G r c ( Re x ) 2 , M = σ b f B 0 2 x ρ b f U ∞ , K = x ν b f k p p U ∞ , Pr = ν b f ( ρ C p ) b f K b f , E c = U ∞ 2 ( C p ) b f ( T w − T ∞ ) , S c = ν b f D , ϕ 1 = ( 1 − ϕ ) 2.5 , ϕ 2 = ( 1 − ϕ ) + ϕ ρ n p ρ b f , ϕ 3 = ( 1 − ϕ ) + ϕ ( ρ β t ) n p ( ρ β t ) b f , ϕ 4 = ( 1 − ϕ ) + ϕ ( ρ β c ) n p ( ρ β c ) b f , ϕ 5 = σ n f σ b f , ϕ 6 = K n f K b f and ϕ 7 = ( 1 − ϕ ) + ϕ ( ρ C p ) n p ( ρ C p ) b f (17)

In the aforementioned Equations (11)-(13), Gr_{t} is the local thermal Grashof number, Gr_{c} is the local mass Grashof number, Re_{x} the local Reynolds number, Ri_{t} is represented as the local Richardson number of thermal, Ri_{c} is denoted as the local Richardson number of mass, K is indicated as the parameter of permeability, M is designated as magnetic parameter, Ec is denoted as Eckert number, Sc is indicated as the Schmidt number whereas ϕ i ( i = 1 , 2 , ⋯ , 7 ) are constants.

The parameters which are the engineering interest are skin-friction coefficient, local Nusselt together with Sherwood numbers. The nondimensional form of local skin-friction coefficient C f is

C f = 2 ϕ 1 ( Re x ) − 1 2 f ″ ( 0 ) (18)

Furthermore, using the thermophysical property of nanofluid, the local Nusselt number is converted through the resulting form

N u x = − ϕ 6 ( Re x ) 1 2 θ ′ ( 0 ) (19)

However, local Sherwood number is transformed into the succeeding form as:

S h = − ( Re x ) 1 2 s ′ ( 0 ) (20)

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 [

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. [

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 [

The comprehensive numerical results are computed for water (H_{2}O)-copper (Cu) nanofluid flow including dissimilar values of nondimensional parameters that described flow characteristics of the nanofluid. The influences of namely 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 are analyzed to describe flow characteristics. The values for nondimensional parameters R_{it} = 1, R_{ic} = 1, M = 0.5, K = 0.5, α = 30˚, Pr = 6.2, Ec = 0.5, Sc = 0.60, f_{w} = 1.5, ϕ = 0.03 and U_{∞}/ν = 1.0 are considered except otherwise specified. The selective results are shown graphically with velocity, temperature together with concentration flow fields. Moreover, the interest of engineering terms as local skin friction coefficient C_{f} and local Nusselt number Nu_{x} together with local Sherwood number Sh are presented graphically.

The impacts for magnetic parameter namely M (M = 0, 1 and 2) on the copper (Cu) and water (H_{2}O) nanofluid flow fields are shown in _{f} is established to increases with increase magnetic field strength as given away in _{x} namely local Nusselt number which is schemed in

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

for α = 90˚. The gravitational effects is maximum at a vertical position whereas minimum at horizontal position. Consequently, momentum boundary layer nanofluid flow decreases to increase of α. As a consequence this is observed into _{f} of nanofluid declines which is detected in

The variation of velocity, temperature together with concentration of water-copper nanofluid flow for various values regarding fluid suction parameter f_{w} (f_{w} = 1.5, 3.0 and 4.5) are given away in _{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

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 [_{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 number Nu_{x} together with local Sherwood number Sh is deduced from the present analysis:

➢ The skin friction coefficient increases but local Nusselt number decreases due to increasing values for magnetic parameter.

➢ The local skin friction coefficient shows diminishing behavior for rising values angle of inclination.

➢ The local Sherwood number of nanofluid flow displays raising behavior for raising values of fluid suction parameter.

➢ The local Nusselt number of nanofluid flow shows raising behavior for raising values of nanoparticles volume fraction.

➢ The skin friction coefficient of nanofluid flow increases for increasing value of permeability parameter.

The authors declare no conflicts of interest regarding the publication of this paper.

Uddin, Md.N., Alim, Md.A. and Rahman, Md.M. (2019) MHD Effects on Mixed Convective Nanofluid Flow with Viscous Dissipation in Surrounding Porous Medium. Journal of Applied Mathematics and Physics, 7, 968-982. https://doi.org/10.4236/jamp.2019.74065