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Unsteady magnetohydrodynamic mixed convection flow of an electrically conducting nanofluid in a stagnation region of a rotating sphere is studied numerically in the present article. Slip and convective boundary conditions are imposed to surface of the sphere and the thermal radiation effects are taken into account. The nanofluid is simulated using Buongiorno’s nanofluid model and the nanofluid particle fraction on the boundary is considered to be passively rather than actively controlled. Non-similar solutions are applied on the governing equations and the MATLAB function bvp4c is used to solve the resulting system. Effects of the key-parameters such as slip parameter, Biot number, radiation parameter, rotation parameter, Lewis number and Brownian motion parameter on the fluid flow, temperature and nanoparticle volume fraction characteristics are examined. Details of the numerical solution and a comprehensive discussion with the physical meaning for the obtained results are performed. The results indicated that the increase in slip parameter enhances the velocity profiles, while it decreases the temperature distributions. Also, the increase in either slip parameter or Biot number causes an improvement in the rate of heat transfer.

Nanofluids are new types of fluids consist of a base fluid and suspended nanoparticles (1 - 100 nm). These fluids have a higher thermal conductivity and a single-phase heat transfer coefficient compared with their base fluids [

On the other hand, the study of convective boundary layer flow over rotating bodies of revolution has several engineering applications such as fiber coating, design of rotating machinery, re-entry missile, projectile motion etc. [

Ahmed and Mahdy [

flow of an incompressible, laminar nanofluid over a rotating sphere with a constant angular velocity Ω in the stagnation region. It is assumed that there is a uniform magnetic field with strength B in the flow domain. A two-phase model (Buongiorno’s nanofluid model) is considered to simulate the nanofluid. The ambient velocity, temperature and nanoparticles volume fraction are denoted by U ( x ) , T ∞ and ϕ ∞ , respectively, while the noral fluxes of the nanoparticles vanish at the surface of the sphere. Also, a uniform slip and convective boundary conditions are imposed to the wall of the sphere. The thermal radiation is taken into account, but the Joule heating and viscous dissipation effects are neglected. All the fluid properties are constant except the nanofluid density which is approximated using a well-known Boussinesq approximation. Under all these assumptions, the continuity, x-component momentum, y-component momentum, energy and nanoparticles volume fractions equations are expressed as:

∂ ( x u ) ∂ x + ∂ ( x w ) ∂ z = 0 (1)

∂ u ∂ t + u ∂ u ∂ x + w ∂ u ∂ z − v 2 x = U ∂ U ∂ x + ν ∂ 2 u ∂ z 2 + [ ( 1 − ϕ ∞ ) ρ f ∞ β T ( T − T ∞ ) − ( ρ P − ρ f ∞ ) ( ϕ − ϕ ∞ ) ] g x r − σ B 0 2 ρ f ∞ ( u − U ) (2)

∂ v ∂ t + u ∂ v ∂ x + w ∂ v ∂ z + u v x = ν ∂ 2 v ∂ z 2 − σ B 0 2 ρ f v (3)

∂ T ∂ t + u ∂ T ∂ x + w ∂ T ∂ z = κ ( ρ c ) f ∂ 2 T ∂ z 2 + ( ρ c ) P ( ρ c ) f [ D B ( ∂ ϕ ∂ z ∂ T ∂ z ) + D T T ∞ ( ∂ T ∂ z ) 2 ] − 1 ( ρ c ) f ∂ q ∂ z (4)

∂ ϕ ∂ t + u ∂ ϕ ∂ x + w ∂ ϕ ∂ z = D B ∂ 2 ϕ ∂ z 2 + D T T ∞ ∂ 2 T ∂ z 2 (5)

In Equations (1)-(4), u, v and w are the velocity components in x, y and z-direction, respectively, ϕ is the nanoparticle volume fraction, ρ is the density, ν is the kinematic viscosity, T is the temperature, β_{T} is the thermal expansion, g is the gravity acceleration, σ is the electrical conductivity, B_{0} is the magnetic strength, k is the thermal conductivity, D_{B} is Brownian diffusion coefficient, D_{T} is thermophoretic diffusion, C is the specific heat, q is the radiative heat flux and the subscripts f, p and ∞ refer to base fluid, nanoparticle and free stream condition, respectively.

The radiative flux q can be expressed using the Rossel and approximation [

q = − 4 σ 1 3 k * ∂ T 4 ∂ z (6)

where, σ 1 is the Stefan-Boltzmann constant and k * is the absorption coefficient. Expanding T 4 using Taylor series about T ∞ and neglecting the higher terms, we obtain:

T 4 = − 3 T ∞ 4 + 4 T ∞ 3 T (7)

Substituting Equations (6) and (7) in Equation (4), we get:

∂ T ∂ t + u ∂ T ∂ x + w ∂ T ∂ z = ( κ ( ρ c ) f + 16 T ∞ 3 σ 1 3 k * ( ρ c ) f ) ∂ 2 T ∂ z 2 + ( ρ c ) P ( ρ c ) f [ D B ( ∂ ϕ ∂ z ∂ T ∂ z ) + D T T ∞ ( ∂ T ∂ z ) 2 ] (8)

The corresponding initial and boundary conditions are:

t < 0 , u ( x , z , t ) = v ( x , z , t ) = w ( x , z , t ) = 0 , T ( x , z , t ) = T ∞ , ϕ ( x , z , t ) = ϕ ∞ (9a)

t ≥ 0 , u ( x , 0 , t ) = N μ ∂ u ∂ z , v ( x , 0 , t ) = Ω x , w ( x , 0 , t ) = 0 , k ∂ T ∂ z = h f ( T − T f ) , D B ∂ ϕ ( x , 0 , t ) ∂ z + D T T ∞ ∂ T ( x , 0 , t ) ∂ z = 0 (9b)

t ≥ 0 , u ( x , ∞ , t ) = U ( x ) , v ( x , ∞ , t ) = 0 , T ( x , ∞ , t ) = T ∞ , C ( x , ∞ , t ) = C ∞ , ϕ ( x , ∞ , t ) = φ ∞ (9c)

where, is the dimensional slip parameter, μ is the dynamic viscosity, Ω is the angular velocity of the sphere and h_{f} is the convective heat transfer.

Introducing the following dimensionless variables:

ξ = 1 − exp ( − a t ) , η = ( 2 a ν ) 1 2 ξ − 1 2 z , a > 0 , U = a x , v ( x , z , t ) = Ω S ( ξ , η ) , w ( x , z , t ) = − ( 2 a ν ) 1 2 ξ 1 2 F ( ξ , η ) , λ = ( Ω a ) 2 , T ( x , z , t ) − T ∞ T f − T ∞ = θ ( ξ , η ) , ϕ ( x , z , t ) − ϕ ∞ ϕ ∞ = φ ( ξ , η ) (10)

Substituting Equation (10) into Equations (2), (3), (5) and (8), the following non-similar equations are obtained:

F ‴ + ξ F F ″ + 1 4 η ( 1 − ξ ) F ″ + 1 2 ξ ( 1 − F ′ 2 + λ S 2 ) + 1 2 M ξ ( 1 − F ′ ) + 1 2 ξ γ ( θ − N r φ ) = 1 2 ξ ( 1 − ξ ) ∂ F ′ ∂ ξ (11)

S ″ + ξ ( S ′ F − F ′ S ) + 1 4 η ( 1 − ξ ) S ′ − 1 2 M ξ S = 1 2 ξ ( 1 − ξ ) ∂ S ∂ ξ (12)

1 P r ( 1 + 4 3 R ) θ ″ + 1 4 η ( 1 − ξ ) θ ′ + ξ F θ ′ + + N b φ ′ θ ′ + N T θ ′ 2 = 1 2 ξ ( 1 − ξ ) ∂ θ ∂ ξ (13)

φ ″ + N T N b θ ″ + L e ξ F φ ′ + L e 4 ( 1 − ξ ) η φ ′ = L e 2 ξ ( 1 − ξ ) ∂ φ ∂ ξ (14)

where, G r R = g β T ( T f − T ∞ ) ( 1 − ϕ ∞ ) ρ f ∞ r 3 ν 2 is the Grashof number, γ = G r R R e R 2 is the buoyancy parameter, R e R = r 2 a ν is the Reynolds number, M = σ B 0 2 a ρ f ∞ is the magnetic field parameter , N r = ( ρ p − ρ f ∞ ) ϕ ∞ ( 1 − ϕ ∞ ) ρ f ∞ ( T f − T ∞ ) β T is the buoyancy ratio, λ = ( Ω a ) 2 is the rotation parameter, N b = ( ρ c ) p D B ϕ ∞ ( ρ c ) f ν is the Brownian motion parameter, N t = ( ρ c ) p D T ( T f − T ∞ ) ( ρ c ) f T ∞ ν is the thermophoreses parameter, P r = μ c p k is the Prandtl number, L e = ν D B is the Lewis number.

The boundary conditions are converted to:

F ′ ( ξ , 0 ) = δ F ″ ( ξ , 0 ) , S ( ξ , 0 ) = 1 , F ( ξ , 0 ) = 0 , θ ′ ( ξ , 0 ) = B i ( θ ( ξ , 0 ) − 1 ) , N b φ ′ ( ξ , 0 ) + N t θ ′ ( ξ , 0 ) = 0

F ′ ( ξ , ∞ ) = 1 , S ( ξ , ∞ ) = 0 , θ ( ξ , ∞ ) = 0 , φ ( ξ , ∞ ) = 0 (15)

where, δ = N μ [ 2 a υ ξ ] 1 2 is the slip parameter and B i = h f k [ ν ξ 2 a ] 1 2 is the Biot number.

In this section the MATLAB function bvp4c is applied to solve Equations (11)-(14) with the boundary conditions (15). The first step in this technique is the converting the system (11)-(14) to an ordinary differential equations system, so all the derivatives with respect to ξ in this system are dropped down, thus:

F ‴ + ξ F F ″ + 1 4 η ( 1 − ξ ) F ″ + 1 2 ξ ( 1 − F ′ 2 + λ S 2 ) + 1 2 M ξ ( 1 − F ′ ) + 1 2 ξ γ ( θ − N r r φ ) = 0 (16)

S ″ + ξ ( S ′ F − F ′ S ) + 1 4 η ( 1 − ξ ) S ′ − 1 2 M ξ S = 0 (17)

1 P r ( 1 + 4 3 R ) θ ″ + 1 4 η ( 1 − ξ ) θ ′ + ξ F θ ′ + N b φ ′ θ ′ + N T θ ′ 2 = 0 (18)

φ ″ + N T N b θ ″ + L e ξ F φ ′ + L e 4 ( 1 − ξ ) η φ ′ = 0 (19)

Subject to the boundary conditions:

F ′ ( ξ , 0 ) = δ F ″ ( ξ , 0 ) , S ( ξ , 0 ) = 1 , F ( ξ , 0 ) = 0 , θ ′ ( ξ , 0 ) = B i ( θ ( ξ , 0 ) − 1 ) , N b φ ′ ( ξ , 0 ) + N t θ ′ ( ξ , 0 ) = 0

F ′ ( ξ , ∞ ) = 1 , S ( ξ , ∞ ) = 0 , θ ( ξ , ∞ ) = 0 , φ ( ξ , ∞ ) = 0 (20)

In the second step, all the neglected terms in previous system are restored and the following new variables are introduced:

F 1 = ∂ F ∂ ξ , S 1 = ∂ S ∂ ξ , θ 1 = ∂ θ ∂ ξ and φ 1 = ∂ φ ∂ ξ (21)

Thus, the governing equations are:

F ‴ + ξ F F ″ + 1 4 η ( 1 − ξ ) F ″ + 1 2 ξ ( 1 − F ′ 2 + λ S 2 ) + 1 2 M ξ ( 1 − F ′ ) + 1 2 ξ γ ( θ − N r r φ ) = 1 2 ξ ( 1 − ξ ) F ′ 1 (22)

S ″ + ξ ( S ′ F − F ′ S ) + 1 4 η ( 1 − ξ ) S ′ − 1 2 M ξ S = 1 2 ξ ( 1 − ξ ) S 1 (23)

1 P r ( 1 + 4 3 R ) θ ″ + 1 4 η ( 1 − ξ ) θ ′ + ξ F θ ′ + N b φ ′ θ ′ + N T θ ′ 2 = 1 2 ξ ( 1 − ξ ) θ 1 (24)

φ ″ + N T N b θ ″ + L e ξ F φ ′ + L e 4 ( 1 − ξ ) η φ ′ = L e 2 ξ ( 1 − ξ ) φ 1 (25)

Subject to the boundary conditions:

F ′ ( ξ , 0 ) = δ F ″ ( ξ , 0 ) , S ( ξ , 0 ) = 1 , F ( ξ , 0 ) = 0 , θ ′ ( ξ , 0 ) = B i ( θ ( ξ , 0 ) − 1 ) , N b φ ′ ( ξ , 0 ) + N t θ ′ ( ξ , 0 ) = 0

F ′ ( ξ , ∞ ) = 1 , S ( ξ , ∞ ) = 0 , θ ( ξ , ∞ ) = 0 , φ ( ξ , ∞ ) = 0 (26)

There are four additional equations are needed with appropriate boundary conditions to evaluate the four new variables F 1 , S 1 , θ 1 and φ 1 . These can be obtained by differentiating Equations (22)-(26) with respect to ξ and neglecting

the terms ∂ F 1 ∂ ξ , ∂ S 1 ∂ ξ , ∂ θ 1 ∂ ξ and ∂ φ 1 ∂ ξ , it can be obtained:

F ‴ 1 + F F ″ + ξ ( F 1 F ″ + F F ″ 1 ) + 1 4 η ( 1 − ξ ) F ″ 1 − 1 4 η F ″ + 1 2 ( 1 − F ′ 2 + λ S 2 + ξ ( − 2 F ′ F ′ 1 + 2 λ S S 1 ) ) + 1 2 M ( 1 − F ′ ) − 1 2 M ξ F ′ 1 + 1 2 γ ( θ − N r r φ ) + 1 2 ξ γ ( θ 1 − N r r φ 1 ) = 1 2 ( 1 − ξ ) F ′ 1 − 1 2 ξ F ′ 1 (27)

S ″ 1 + ( S ′ F − F ′ S ) + ξ ( S ′ 1 F + S ′ F 1 − F ′ 1 S − F ′ S 1 ) − 1 4 η S ′ + 1 4 η ( 1 − ξ ) S ′ 1 − 1 2 M ( ξ S 1 + S ) = 1 2 ( 1 − ξ ) S 1 − 1 2 ξ S 1 (28)

1 P r ( 1 + 4 3 R ) θ ″ 1 − 1 4 η θ ′ + 1 4 η ( 1 − ξ ) θ ′ 1 + F θ ′ + ξ ( F 1 θ ′ + F θ ′ 1 ) + N b ( φ ′ 1 θ ′ + φ ′ θ ′ 1 ) + 2 N T θ ′ θ ′ 1 = 1 2 ( 1 − ξ ) θ 1 − 1 2 ξ θ 1 (29)

φ ″ 1 + N T N b θ ″ 1 + L e F φ ′ + L e ξ ( F 1 φ ′ + F φ ′ 1 ) − L e 4 η φ ′ + L e 4 ( 1 − ξ ) η φ ′ 1 = L e 2 ( 1 − ξ ) φ 1 − L e 2 ξ φ 1 (30)

Subjected to the boundary conditions:

F ′ 1 ( ξ , 0 ) = δ F ″ 1 ( ξ , 0 ) , S 1 ( ξ , 0 ) = 0 , F 1 ( ξ , 0 ) = 0 , θ ′ 1 ( ξ , 0 ) = B i θ 1 ( ξ , 0 ) , N b φ ′ 1 ( ξ , 0 ) + N t θ ′ 1 ( ξ , 0 ) = 0

F ′ 1 ( ξ , ∞ ) = 0 , S 1 ( ξ , ∞ ) = 0 , θ 1 ( ξ , ∞ ) = 0 , φ 1 ( ξ , ∞ ) = 0 (31)

Equations (16)-(19), with the boundary conditions (20) and Equations (27)-(30) with the boundary conditions (31) are solved numerically using MATLAB software. The value of η max is set to be equal 10 and the step size is Δ η = 0.02 . This method is found to be suitable and gives results those are very close to Ahmed and Mahdy [

and nanoparticle volume fraction profiles are illustrated. It is found that, the increase in Nt leads to an increase in the velocity profiles and temperature distributions but the nanoparticle volume fraction profiles decrease with the increase in δ near the sphere surface, whereas at η > 1 , the inverse behavior is observed. Also, the thermal boundary layer increases, slightly, along with the increase in Nt. In Figures 9-11, the effects of Biot number Bi on the velocity profiles in the x-direction, temperature distribution and the nanoparticle volume fraction profiles are presented. It can be noted that the increase in Bi enhances the velocity profiles and temperature distributions, while on contrary the profiles of nanoparticle volume fraction are reduced. Additionally, this mentioned behavior of ϕ is noted near the surface but far away from the sphere surface, the increase in Bi improves the distributions of ϕ. Here, it should be mentioned that when B i → ∞ , the constant wall temperature case (CWT) is obtained. From the

physical view, the increase in Bi enhances the temperature differences at the wall which in turn augments the mixed convection and consequently increases the velocity temperature distributions.

Figures 12-14 display effect of the radiation parameter R on the velocity in the x-direction, temperature and the nanoparticle volume fraction profiles. It is found that the increase in R leads to a reduction in the velocity and temperature profiles. Further, the nanoparticle volume fraction profiles decrease with increasing R near the surface up to a certain value of η but beyond this point, the opposite tendency is observed.

The effect of variations of Lewis Le on the velocity profiles in the x-direction for different values of slip parameter δ is examined with the help of

Also, the same effect is observed for the increase in δ, moreover the thickness of the thermal boundary layer decreases. As can be observed in

In

The effects of the rotation parameter λ and the slip parameter δ on the local skin friction coefficient F ″ ( ξ , 0 ) are presented in

The effects of slip parameter δ on the local Nusselt and Sherwood numbers are shown in Figures 22 and

In

This paper discussed the unsteady MHD mixed convection flow along rotating spheres with a stagnation point in the presence of thermal radiation, slip and convective boundary conditions. The problem was formulated using Navier stokes equations with boundary layer approximations and then solved numerically using MATLAB software. The obtained results were presented in terms of

profiles of velocity in the x-direction, temperature and nanoparticle volume fraction as well as skin friction coefficient, local Nusselt number and local Sherwood number. The key findings of the present investigation are summarized as follows:

1) The increase in slip parameter enhances the velocity profiles, skin friction coefficient and local Nusselt number, while the temperature distributions and local Sherwood number take the inverse behaviors.

2) An improvement in profiles of velocity and temperature can be obtained by increasing the thermophores parameter, while the profiles of nanoparticle volume fraction near the surface, local Nusselt and Sherwood numbers are reduced.

3) The increase in Biot number causes an increase in the hot nanofluid at the surface and consequently the mixed convection is improved.

4) The increase in radiation parameter resists the fluid flow and decreases the fluid temperature.

5) Regardless the values of slip parameter, the increase in Lewis number reduces the fluid velocity and the fluid temperature decreases, gradually, as the Prandtl number increases.

The authors extend their appreciation to the editors and reviewers for their comments to improve the quality of the paper.

Ahmed, S.E. and Rashed, Z.Z. (2018) Unsteady MHD Mixed Convection Flow with Slip of a Nanofluid in the Stagnation Region of an Impulsively Rotating Sphere with Effects of Thermal Radiation and Convective Boundary Conditions. World Journal of Mechanics, 8, 137-160. https://doi.org/10.4236/wjm.2018.85011

a Velocity gradient the edge

B Magnetic field

Bi Biot number

c_{p} Specific heat

D_{B} Brownian diffusion coefficient

D_{T} Thermophoretic diffusion coefficient

FD Imensionless stream function

g Gravity acceleration

Gr_{R} Grashof number

h_{f} Convective heat transfer coefficient

k Thermal conductivity of base fluid

k^{*} Absorption coefficient

Le Lewis number

M Magnetic field parameter

Nr Buoyancy ratio

Nb Brownian motion parameter

Nt Thermophoresis parameter

Nu Nusselt number

r Radius of the sphere

R Radiation parameter

Re_{R} Reynolds number

S Velocity component in-direction y

Sh Sherwood numbers

t Time

T Local fluid temperature

T_{f} Temperature of the hot fluid

(u, v, w) Velocity components in the x, y and z directions

U Ambient velocity

(x, y, z) Cartesian coordinates

β_{T} Coefficient of thermal expansion

μ Dynamic viscosity

ν Kinematic viscosity

λ Rotation parameter

σ Electrical conductivity

σ_{1} Stefan-Boltzman constant

ρ_{f} Density of base fluid

ρ_{p} Nanoparticle mass density

θ Dimensionless temperature

ϕ Dimensionless nanoparticle volume fraction

(ρc)_{f} Heat capacity of the base fluid

(ρc)_{p} Heat capacity of the nanoparticle material

γ Buoyancy parameter

δ Slip parameter

Ω Angular velocity

(ξ, η) Non-similarity variables

w Conditions at the surface

∞ Conditions in the free stream