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The focus of the study is to examine thermal radiation and viscous dissipative heat transfers of magnetohydrodynamics (MHD) stagnation point flow past a permeable confined stretching cylinder with non-uniform heat source or sink. The formulated equation governing the flow is non-dimensional. The dimensionless momentum and energy equation are solved using shooting technique coupled with fourth-order Runge-kutta integrated scheme which satisfied smoothness conditions at the edge of the boundary layer. The result for the velocity and temperature distributions are presented graphically and discussed to portray the effects of some important embodiment parameters on the flow. The Nusselt number and skin friction were obtained and compared with the previous scholars’ results in others to validate the present research work.

The science dealing with the motion of electrically conducting fluids through magnetic fields is known as magnetohydrodynamics. Fluid mechanics and magnetic theory are harmonized as magnetohydrodynamics (MHD), the field which was initiated in [

The study of chemical reaction and viscous dissipation for free convective hydromagnetic flow past an inclined porous stretching surface was reported in the works of (Alam, M.S) [

Consider a laminar, two dimensional, hydromagnetic, incompressible viscous fluid flow in an electrically conducting, heated and progressing vertical thin resistant semi-infinite solid cylinder of the radius R. The cylindrical coordinates ( z , r ) are presumed to be the axis of the cylinder which is parallel to the uniform composite free stream flow. In addition, at the center of leading edge of the cylinder is the origin of the coordinate system. A uniform weak magnetic field density B 0 is applied normal to the flow direction. The cylinder moves linearly with steady velocity about its axis in the same direction as that of the free stream velocity with the temperature T of the free stream fluid. The cylinder surface is maintained at stable temperature T w while T ∞ is the ambient temperature (

as well as the equations governing the momentum and energy balance, are as follows:

∂ ( r u ) ∂ z + ∂ ( r v ) ∂ r = 0 (1)

u ∂ u ∂ z + v ∂ u ∂ r = − 1 ρ d P e d z + ν r ∂ ∂ r ( r ∂ u ∂ r ) − σ B 0 2 u ρ + g β ( T − T ∞ ) − m r u 2 (2)

u ∂ T ∂ z + v ∂ T ∂ r = k ρ c p 1 r ∂ ∂ r ( r ∂ T ∂ r ) − 1 ρ c p 1 r ∂ ∂ r ( r q r ) + ν c p ( ∂ u ∂ r ) 2 + q ‴ ρ c p (3)

Subject to slip boundary conditions

u ( z , r ) = U w ( z ) = u 0 z a , v ( z , r ) = − v w , T ( z , r ) = T w ( z ) = T ∞ + ( z a ) n Δ T at r = 0 u ( z , r ) = u e ( z ) = u ∞ z a , T ( z , r ) = T ∞ at r → ∞ (4)

From the above, u and v are the velocity elements in the z and r directions respectively also the following parameters P e , ν , ρ , σ , B 0 , g, β and m represent the elongating pressure gradient, kinematic viscosity, fluid density, electrical conductivity, uniform magnetic field, the acceleration due to gravity, the coefficient of expansivity and fluid mass. k, c p , q r , u 0 and u ∞ stand for the thermal conductivity, specific heat at constant pressure, thermal radiation, stretching velocity and free stream velocity respectively.

The term d P e d z in Equation (2) can be defined as

d p e d z = − ρ u e d u e d z − σ B 0 2 u e (5)

According to Rosseland diffusion approximation for radiation, q r can be defined as (Salawu and Amoo (2016));

q r = − 4 σ 0 3 δ ∂ T 4 ∂ r (6)

where σ 0 and δ are the Stefan-Boltzmann and the mean absorption coefficient respectively. Let heat difference in the flow be satisfactory low, T 4 may be expressed linearly as a function of temperature, applying Taylor series to expand T 4 is expanded about T ∞

T 4 ≅ T ∞ 4 + 4 T ∞ 3 ( T − T ∞ ) + 6 T ∞ 2 ( T − T ∞ ) 2 + ⋯ (7)

Neglecting higher-order terms of T − T ∞ in Equation (8) to get

T 4 ≈ 4 T ∞ 3 T − 3 T ∞ 4 (8)

Using Equation (8) on Equation (6) to obtain

q r = − 16 T ∞ 3 σ 0 3 δ ∂ T ∂ r (9)

The non-uniform heat source is defined as (Salawu and Fatunmbi (2017))

q ‴ = k u w ( z ) z ν [ λ ( T − T ∞ ) + λ * ( T w − T ∞ ) e − η ] (10)

where λ * and λ denote the heat source and space coefficients temperature dependent respectively while T ∞ is the free stream temperature. Here, λ > 0 and λ * > 0 represent heat source but λ < 0 and λ * < 0 depict heat sink.

Using Equation (5), Equation (9) and Equation (10) along with the similarity variables to Equation (11) as defined in Hayat et al. (2017) on Equations (1) to (4) gives;

u = u 0 z a f ' ( η ) , v = − R r ( u 0 ν a ) 1 2 f ( η ) , θ ( η ) = T − T ∞ T w − T ∞ , ψ = ( ν z 2 u 0 a ) 1 2 R f ( η ) , (11)

Hence, the governing equations and the boundary conditions reduce to the nonlinear coupled ordinary differential equation as follows,

( 1 + 2 K η ) f ‴ ( η ) + 2 K f ″ ( η ) + f ( η ) f ″ ( η ) − ( 1 + Z ) ( f ′ ( η ) ) 2 − H 2 ( f ′ ( η ) − γ ) + γ 2 + G r θ ( η ) = 0 , (12)

( 1 + 4 3 N ) ( ( 1 + 2 K η ) θ ″ ( η ) + 2 K θ ′ ( η ) ) + P r f ( η ) θ ′ ( η ) − n P r f ′ ( η ) θ ( η ) + λ * e − η + λ θ + E c P r ( 1 + 2 K η ) ( f ″ ( η ) ) 2 = 0 (13)

Subject to the boundary conditions,

f ′ ( η ) = 1 , f = f w , θ = 1 , at η = 0 , f ′ = γ , θ = 0 , as η → ∞ (14)

where K = 1 R a ν u 0 is the curvature parameter, Z = z m r is the load potential difference, H 2 = σ B 0 2 ρ u 0 is the Hartman number, G r = a 2 g β ( T w − T ∞ ) z u 0 2 is the thermal Grashof number, γ = u e U w is the stagnation rate ratio N = 4 σ 0 T ∞ 3 δ k is the radiation parameter, P r = ρ C p k is the Prandtl number, E c = u 0 2 C p ( T w − T ∞ ) is the Echert number, f w = − v w a ν u 0 is the suction f w > 0 or injection f w < 0 parameter.

The engineering interest of the physical quantities are the skin friction and Nusselt number that is C f and N u defined as follow,

C f z = 2 τ w ρ u 0 2 , (15)

where τ w

N u z = z q w k ( T w − T ∞ ) (16)

and q w are defined as following

τ w = μ ( ∂ u ∂ r ) r = R = ( ∂ u ∂ r + ( v ∂ 2 u ∂ r 2 + u ∂ 2 u ∂ z ∂ r ) ) r = R (17)

q w = − k ( ∂ T ∂ r ) r = R = − ( k + 16 σ 0 T ∞ 3 3 δ ) ( ∂ T ∂ r ) r = R (18)

using Equation (15) on Equation (17) result to,

R e z 1 2 C f z = f ″ ( 0 ) (19)

Also, using Equation (16) on Equation (18) gives,

R e z − 1 2 N u z = − ( 1 + 4 3 N ) θ ′ ( 0 ) (20)

where R e z = z u 0 ν shows the local Reynolds number (

The set of non-linear differential equations alongside the boundary conditions are unraveled numerically by applying fourth-order Runge-Kutta integration technique couple with shooting method. The smoothness conditions are confirmed through checking at the edge of the boundary layer whether they are satisfied. Calculations are conducted for different values of the following default parameters based on existing research works, the following diverse computations values as set as the default values for the embedded parameters; f ″ ( 0 ) for K = 0 , H = 0 ,

γ | Pop et al. (2004) [ | Sharma (2009) [ | Hayat et al. (2014) [ | Present results |
---|---|---|---|---|

0.1 | −0.9694 | −0.969386 | −0.9679 | −0.9694 |

0.2 | −0.9181 | −0.9181069 | −0.9172 | −0.9181 |

0.5 | −0.6673 | −0.667263 | −0.6670 | −0.6673 |

2.0 | 2.0174 | 2.01749079 | 2.0174 | 2.0175 |

3.0 | 4.7293 | 4.72922695 | 4.7294 | 4.7293 |

Pr | n | Mukhopadhyay (2012) [ | Pal (2012) [ | Hayat et al. (2014) [ | Present results |
---|---|---|---|---|---|

1 | −2 | - | - | −1.0000 | −0.9889 |

−1 | - | - | −0.0 | −0.0025 | |

0 | 0.5821 | - | −0.5832 | −0.5826 | |

1 | 1.0000 | - | 1.0000 | 1.0002 | |

2 | 1.3332 | 1.333333 | 1.3332 | 1.3333 | |

10 | −2 | - | - | 10.0000 | 9.9964 |

−1 | - | - | 0.0 | 0.0 | |

0 | - | - | 2.3080 | 2.3080 | |

1 | - | - | 3.7207 | 3.7206 | |

2 | - | 4.796871 | 4.7969 | 4.7968 | |

0.7 | 2 | - | - | 1.0791 | 1.0702 |

0.6 | 2 | - | - | - | 0.9708 |

0.5 | 2 | - | - | - | 0.8637 |

G r = 0 , Z = 0 , λ = 0 , n = 0 , λ * = 0 , N = 0 , f w = 0 , P r = 1 , E c = 0.2 , η → ∞ .and θ ′ ( 0 ) for K = 0 , H = 0 , G r = 0 , Z = 0 , γ = 0 , λ = 0 , λ * = 0 , N = 0 , f w = 0 , P r = 1 , E c = 0.2 , η → ∞ .

All graphs satisfy the values except otherwise stated in the respective graph.

The impact of diverse values of the thermal Grashof number Gr on the momentum distributions is shown in

the relative thermal buoyancy force values with respect to that of viscous hydrodynamic force enhanced the flow in the system. This is because the fluid flow gets warmer as it moves along the boundary layer surface thereby reduces the flow resistance forces which then result in an increase in the fluid flow rate. For low buoyancy effects, the maximum flow velocity occurs at the surface.

The reaction of the ratio of rates on fluid velocity field is presented in

The effect of loss load Z on fluid velocity is shown in

temperature and momentum field. It is obtained that velocity and temperature distribution has diminishing behavior near the cylinder surface while it increases far away from it. However, it is observed that thickness of the boundary layer reduces. Physically it is proved that larger values of curvature parameter K reduce the cylinder radius. Consequently, the contact surface area of the cylinder with the flowing liquid and temperature decreases which provides large resistance to the motion of the fluid and increase the rate at which heat is diffuse out of the system thereby decreases the velocity and heat profiles.

The influence of diverse values of Prandtl number Pr on the temperature distribution is illustrated in

The influence of viscous dissipation parameter Ec on the temperature field is illustrated in

The effect of radiation parameter N on temperature distributions are displayed in

heat transfer. Hence, the flow temperature profiles decrease as N increases. Thermal radiation is the relative contribution of heat conduction transfer to thermal radiation. Hence, an increase in the magnitude of radiation parameter N reduces the thermal boundary layer. The reason for this is that the divergence of the radiative heat flux qr decreases as the Rosseland mean absorption coefficient δ increases. Thus, the rate of radiative heat transfer to the fluid decreases and then causing the temperature of the fluid to decline. Therefore, to have the cooling process at a faster rate, N should be enhancing. The effect of thermal radiation becomes more significant as N → ∞ .

The action of the ratio of momentum diffusivity to thermal diffusivity on the Nusselt number is displayed in

A model for characterization of magneto hydrodynamic (MHD) flow through a confined cylinder was achieved when validated with some existing scholars’ research work. However, formulated model was transformed to nonlinear coupled differential equation and solved using shooting technique of fourth-order

Runge-kutta method. In this study, Maple (2016) software is used for the computational analysis of the work with computer code developed for Equations (12)-(14). The effect of embedded fluid parameters on the flow momentum and energy balance was investigated and exhibited graphically. Likewise, the responses of the skin friction and Nusselt number to variation increase in some germane parameters are also illustrated graphically and quantitatively discussed. The detailed physical understanding is the scientific advantage of the analysis and discussion is furnished in Sections 3 and 4, which could not have been treasured easily from a global, overall solution of all the equations in one goal. The outcome of this analysis is very important because fluid flow past a permeable surface is useful in the areas of engineering, science and several transport processes in nature.

The authors appreciate the Tetfund Nigeria (Lagos state Polytechnic, Ikorodu, Lagos) for her financial contribution towards this publication.

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

Onanuga, O.K., Chendo, M.A.C. and Erusiafe, N.E. (2018) Thermal Radiation of Hydromagnetic Stagnation Gravity-Driven Flow through a Porous Confined Cylinder with Non-Uniform Heat Source. Open Journal of Fluid Dynamics, 8, 361-377. https://doi.org/10.4236/ojfd.2018.84023