^{1}

^{2}

^{*}

To analyze the thermal convection of ferrofluid along a flat plate is the persistence of this study. The two-
dimensional laminar, steady, incompressible flow past a flat plate subject to convective surface boundary condition, slip velocity in the presence of radiation has been studied where the magnetic field is applied in the transverse direction to the plate. Two different kinds of magnetic nanoparticles, magnetite Fe_{3}O_{4} and cobalt ferrite CoFe_{2}O_{4} are amalgamated within the base fluids water and kerosene. The effects of various physical aspects such as magnetic field, volume fraction, radiation and slip conditions on the flow and heat transfer characteristics are presented graphically and discussed. The effect of various dimensionless parameters on the skin friction coefficient and heat transfer rate are also tabulated.
To investigate this particular problem, numerical computations are done using the implicit finite difference method based Keller-Box Method.

Owing to low thermal conductivity of the customary heat transfer fluids such as water, oil and ethylene glycol, the act of engineering apparatus such as heat exchangers and electronic devices face complications. To recover this problem, fluids with higher thermal conductivity like nanofluids are used as additional to these fluids [

Among various kinds of explores on nanofluids, some of the studies have been absorbed on the nanofluids prepared by diffusing magnetic nanoparticles in a transporter liquid. These are called “ferrofluids”. This is a liquid that becomes highly magnetized in the presence of a magnetic field. Ferrofluids are colloidal liquids made of nanoscale particles (diameter usually 10 nanometers or less) of magnetite, hematite or some other ferromagnetic (metallic) such as iron (Fe), cobalt (Co), and nickel (Ni) as well as their oxides such as magnetite (Fe_{3}O_{4}) and ferrites (MnZn, Co ferrites) particles suspended in a carrier fluid (usually an organic solvent or water) [

The technology for production for nanoparticles and suspensions, synthesis of nanofluids, thermal transport in stationary fluids, thermal conductivity in nanofluids, convective heat transfer under both natural and forced flow and future developments in nanotechnology have been thoroughly studied by many researchers [

Consider the convective heat transfer of selected ferrofluids along a stationary flat plate in a constant magnetic field. The plate is embedded in a medium saturated with water- or kerosene-based ferrofluids. The flow is assumed to be laminar, steady, incompressible and two-dimensional. The base fluids and the selected nanoparticles are assumed to be in thermal equilibrium. In the presence of magnetic field, the ferroparticles moments almost instantly orient along the magnetic field lines and when the magnetic field is removed, the particles moments are quickly randomized. This orientation along the magnetic-field lines shows a certain precise positioning of the ferroparticles depending upon the position of the magnetic field (

The hydrodynamic slip is assumed at the fluid-solid interface with convective surface boundary condition. Here, we consider a situation where the work done by a fluid on adjacent layers due to action of shear forces that is transformed into heat is negligible compared to radiation. The viscous dissipation is insignificant for low viscous flow such as laminar flow. So, we have neglected the viscous dissipation in this study. The constant temperature T w is assumed to be greater than the ambient temperature T ∞ . Using an order-of-magnitude analysis, the standard boundary layer equations for this problem can be written as follows:

∂ u ∂ x + ∂ v ∂ y = 0 (1)

u ∂ u ∂ x + v ∂ u ∂ y = ν n f ∂ 2 u ∂ y 2 − σ B 2 ( x ) ρ n f ( u − U ∞ ) (2)

u ∂ T ∂ x + v ∂ T ∂ y = α n f ∂ 2 T ∂ y 2 − ( 1 ρ C p ) n f ∂ q r ∂ y (3)

The transverse magnetic field is assumed to be a function of the distance from

the origin and is defined as B ( x ) = B o x − 1 2 , B o ≠ 0 , where x is the coordinate along the plate and B o is the magnetic field strength [

Using the Rosseland approximation [

q r = − 4 σ * 3 k * ∂ T 4 ∂ y (4)

We assume that the temperature differences within the flow region are sufficiently small, so that the term T 4 can be expressed as a linear function of temperature. The best linear approximation of T 4 is obtained by expanding it in a Taylor series about T ∞ and neglecting higher order terms, i.e. T 4 ≅ 4 T ∞ 3 T − 3 T ∞ 4 .

So, the energy Equation (3) becomes

u ∂ T ∂ x + v ∂ T ∂ y = α n f ∂ 2 T ∂ y 2 + ( 1 ρ C p ) n f 16 σ * 3 k * T ∞ 3 ∂ 2 T ∂ y 2 (5)

The effective properties of ferrofluids may be expressed in terms of the properties of base fluid, ferroparticles and the volume fraction of solid nanoparticles as follows [

{ ν n f = μ n f ρ n f , μ n f = μ f ( 1 − φ ) 2.5 , α n f = k n f ρ n f ( C p ) n f , ρ n f = ( 1 − φ ) ρ f + φ ρ s , ( ρ C p ) n f = ( 1 − φ ) ( ρ C p ) f + φ ( ρ C p ) s , k n f k f = k s + 2 k f − 2 φ ( k f − k s ) k s + 2 k f + φ ( k f − k s ) . (6)

The boundary conditions for the problem are given by

u = γ ∂ u ∂ y , v = 0 , − k f ∂ T ∂ y = h f ( T w − T ) at y = 0 , (7a)

u → U ∞ , v → 0 , T → T ∞ as y → ∞ , (7b)

where γ is the slip parameter. The bottom of the surface is heated due to the convective heat transfer from a hot fluid at a temperature T w , yielding a heat transfer coefficient h f as a function of x, with its strength given as

h f ( x ) = h f 0 x − 1 2 , h f 0 ≠ 0 [

We look for a similarity solution of Equation (1), Equation (2) and Equation (5) of the following form:

ψ = ν f R e x f ( η ) , η = y x R e x , θ ( η ) = T − T ∞ T w − T ∞ , (8)

where η is the similarity variable, R e x = U ∞ x ν f is the local Reynolds number based on the free stream velocity and ν f is the kinematic viscosity of the base fluid. The stream function ψ is defined as

u = ∂ ψ ∂ y , v = − ∂ ψ ∂ x .

Employing the similarity variables (8), Equation (1), Equation (2) and Equation (5) reduce to a nonlinear system of ordinary differential equations:

f ‴ + ( 1 − φ ) 2.5 [ { ( 1 − φ ) + φ ρ s ρ f } 1 2 f f ″ + M ( 1 − f ′ ) ] = 0 , (9)

k n f k f [ ( 1 − φ ) + φ { ( ρ C p ) s / ( ρ C p ) f } ] 1 P r ( 1 + 4 3 R ) θ ″ + 1 2 f θ ′ = 0. (10)

The associated boundary conditions are

f ( 0 ) = 0 , f ′ ( 0 ) = β f ″ ( 0 ) , θ ′ ( 0 ) = − a { 1 − θ ( 0 ) } , at η = 0 , (11a)

f ′ ( η ) → 1 , θ ( η ) → 0 as η → ∞ . (11b)

Here, primes denote differentiation with respect to η , M = σ B 0 2 / ρ U ∞ is the magnetic parameter, P r = μ C p k is the Prandtl number and R = 4 σ * α k * ( 1 ρ C p ) T ∞ 3 is the radiation parameter, a = h f o k f γ f U ∞ is the dimensionless thermal slip parameter and β is the dimensionless slip velocity. We have to take γ = c x 1 2 , where c is a constant of dimension L 1 2 . Thus we take β as the dimensionless slip parameter ranging from zero (total adhesion) to infinity (full slip) defined by β = c U ∞ γ f [

C f = τ w x ρ f U ∞ 2 , N u x = x q w k f ( T w − T ∞ ) .

Here, τ w x is the surface shear stress along the x direction and q w is the heat flux given by

τ w x = μ n f ( ∂ u ∂ y ) y = 0 , q w = − k n f ( ∂ T ∂ y ) y = 0 .

Reducing dimensionless form we get,

C f R e x 1 2 = f ″ ( 0 ) ( 1 − φ ) 2.5 , N u x R e x − 1 2 = k n f k f a { 1 − θ ( 0 ) } .

The coupled non-linear two-point boundary value problem (9) and (10) combined with the boundary conditions (11a) and (11b) is solved numerically using the implicit finite difference scheme, the Keller-Box method.

As described in Cebeci and Bradshaw [

f ′ = u (12a)

u ′ = v (12b)

s ′ = t (12c)

v ′ + ( 1 − φ ) 2.5 [ ( 1 − φ + φ ρ s ρ f ) 1 2 f v + M ( 1 − u ) ] = 0 (12d)

k n f k f [ 1 − φ + φ { ( ρ C p ) s / ( ρ C p ) f } ] 1 P r ( 1 + 4 3 R ) t ′ + 1 2 f t = 0 (12e)

In terms of the new dependent variables, the boundary conditions become

f ( x , 0 ) = 0 , u ( x , 0 ) = β v ( x , 0 ) , t ( x , 0 ) = − a { 1 − s ( x , 0 ) } (13a)

u ( x , ∞ ) = 1 , s ( x , ∞ ) = 0 (13b)

We approximate the quantities ( f , u , v , s , t ) at points ( x n , η j ) of the net by ( f j n , u j n , v j n , s j n , t j n ) which we shall call net functions. Then the system of Equation (12) is written in the finite difference form as (

f j − f j − 1 − h j 2 ( u j + u j − 1 ) = 0 , (14a)

u j − u j − 1 − h j 2 ( v j + v j − 1 ) = 0 , (14b)

s j − s j − 1 − h j 2 ( t j + t j − 1 ) = 0 , (14c)

v j − v j − 1 + h j ( 1 − φ ) 2.5 [ ( 1 − φ + φ ρ s ρ f ) 1 8 ( f j + f j − 1 ) ( v j + v j − 1 ) + M { 1 − 1 2 ( u j + u j − 1 ) } ] = ( R 1 ) j − 1 2 n − 1 , (14d)

k n f k f [ 1 − φ + φ { ( ρ C p ) s / ( ρ C p ) f } ] 1 P r ( 1 + 4 3 R ) ( t j − t j − 1 ) + h j 8 [ ( f j + f j − 1 ) ( t j + t j − 1 ) ] = ( R 2 ) j − 1 2 n − 1 . (14e)

Equation (14) are imposed for j = 1 , 2 , ⋯ , J and the transformed boundary layer thickness, η j , is sufficiently large so that it is beyond the edge of the boundary layer [

f 0 n = 0 , u 0 n = β v 0 n , t 0 n = − a ( 1 − s 0 n ) , u J n = 1 , s J n = 0. (15)

The nonlinear system of Equation (14) is linearized using Newton’s method and the following iterates are introduced

f j i + 1 = f j i + δ f j i , u j i + 1 = u j i + δ u j i , v j i + 1 = v j i + δ v j i , s j i + 1 = s j i + δ s j i , t j i + 1 = t j i + δ t j i (16)

Then we obtain the following tridiagonal system,

δ f j − δ f j − 1 − h j 2 ( δ u j + δ u j − 1 ) = ( r 1 ) j − 1 2 (17a)

δ u j − δ u j − 1 − h j 2 ( δ v j + δ v j − 1 ) = ( r 2 ) j − 1 2 (17b)

δ s j − δ s j − 1 − h j 2 ( δ t j + δ t j − 1 ) = ( r 3 ) j − 1 2 (17c)

( a 1 ) j δ v j − ( a 2 ) j δ v j − 1 + ( a 3 ) j δ f j + ( a 4 ) j δ f j − 1 − ( a 5 ) j δ u j − ( a 6 ) j δ u j − 1 = ( r 4 ) j − 1 2 (17d)

( b 1 ) j δ t j − ( b 2 ) j δ t j − 1 + ( b 3 ) j δ f j + ( b 4 ) j δ f j − 1 = ( r 5 ) j − 1 2 (17e)

where,

( a 1 ) j = 1 + 1 4 h j ( 1 − φ ) 2.5 ( 1 − φ + φ ρ s ρ f ) f j − 1 2 , ( a 2 ) j = 2 − ( a 1 ) j ,

( a 3 ) j = 1 4 h j ( 1 − φ ) 2.5 ( 1 − φ + φ ρ s ρ f ) v j − 1 2 , ( a 4 ) j = ( a 3 ) j , (18a)

( a 5 ) j = 1 2 M h j ( 1 − φ ) 2.5 , ( a 6 ) j = ( a 5 ) j

( b 1 ) j = k n f k f [ 1 − φ + φ { ( ρ C p ) s / ( ρ C p ) f } ] 1 P r ( 1 + 4 3 R ) + h j 4 f j − 1 2 ,

( b 2 ) j = 2 k n f k f [ 1 − φ + φ { ( ρ C p ) s / ( ρ C p ) f } ] 1 P r ( 1 + 4 3 R ) − ( b 1 ) j , (18b)

( b 3 ) j = 1 4 h j t j − 1 2 , ( b 4 ) j = ( b 3 ) j

and

( r 1 ) j − 1 2 = f j − 1 − f j + h j u j − 1 2 ,

( r 2 ) j − 1 2 = u j − 1 − u j + h j v j − 1 2 ,

( r 3 ) j − 1 2 = s j − 1 − s j + h j t j − 1 2 , (18c)

( r 4 ) j − 1 2 = ( R 1 ) j − 1 2 n − 1 − ( v j − v j − 1 ) − 1 2 h j ( 1 − φ ) 2.5 ( 1 − φ + φ ρ s ρ f ) f j − 1 2 v j − 1 2 − M h j ( 1 − φ ) 2.5 + M h j ( 1 − φ ) 2.5 u j − 1 2 ,

( r 5 ) j − 1 2 = ( R 2 ) j − 1 2 n − 1 − k n f k f [ 1 − φ + φ { ( ρ C p ) s ( ρ C p ) f } ] 1 P r ( 1 + 4 3 R ) ( t j − t j − 1 ) − 1 2 h j f j − 1 2 t j − 1 2 ,

with

( R 1 ) j − 1 2 n − 1 = − h j [ v j − v j − 1 h j + ( 1 − φ ) 2.5 [ ( 1 − φ + φ ρ s ρ f ) 1 2 ( f v ) j − 1 2 + M ( 1 − u j − 1 2 ) ] ] n − 1 ,

( R 2 ) j − 1 2 n − 1 = − h j [ k n f k f [ 1 − φ + φ { ( ρ C p ) s / ( ρ C p ) f } ] 1 P r ( 1 + 4 3 R ) t j − t j − 1 h j + 1 2 ( f t ) j − 1 2 ] n − 1 .

To complete the system (17) we recall the boundary conditions (15) which can be satisfied exactly with no iteration [

δ f 0 = 0 , δ u 0 = 0 , δ t 0 = 0 , δ u J = 0 , δ s J = 0. (19)

The linear system (17) can now be solved numerically by the block-elimination method [

A δ = r (20)

The coefficient matrix A in (20) is known as a tridiagonal matrix due to the facts that all elements of A are zero except those on the three main diagonals. To apply the block elimination method, we assume that A is nonsingular and we seek a factorization of the form

[ A ] = [ L ] [ U ] (21)

The solution of equation (21) by the block-elimination method consists of two sweeps: forward sweep, backward sweep. Once the elements of δ are found, Equation (17) can be used to find the (i + 1)th iteration in Equation (16). Calculations are stopped when | δ v 0 ( i ) | < ε 1 , where ε 1 is a small prescribed value.

The proposed numerical method is programmed in MATLAB with a step size η = 0.005 and used to solve the coupled system in the interval 0 ≤ η ≤ η max , where η max is the finite value of the similarity variable η for the far-field boundary conditions. The effects of volume fraction of solid ferroparticles, magnetic field, radiation, velocity slip parameters and Biot number on the dimensionless velocity, temperature, skin friction and Nusselt numbers are investigated for the selected ferroparticles, magnetite (Fe_{3}O_{4}) and cobalt ferrite (CoFe_{2}O_{4}) with two different base fluids water and kerosene. The values of Prandtl number and viscosity for the base fluids water and kerosene are taken to be 6.2, 21.0 and 8.9 × 10 − 4 , 0.00164 respectively. The effect of volume fraction of solid ferroparticles φ is studied in the range 0 ≤ φ ≤ 0.2 , where φ = 0 represents the pure fluid water or kerosene.

The thermophysical properties of the base fluids (water and kerosene) and the ferroparticles magnetite (Fe_{3}O_{4}) and cobalt ferrite (CoFe_{2}O_{4}) are listed in

In order to validate the accuracy of our numerical procedure, the skin friction coefficient is computed for pure fluid, magnetite and cobalt ferrite for specific values of velocity slip, magnetic parameters (

Materials | ρ (kg/m^{3}) | C_{p} (J/(kg∙K)) | k (W/(m∙K)) |
---|---|---|---|

Water | 997 | 4179 | 0.613 |

Kerosene | 783 | 2090 | 0.15 |

Fe_{3}O_{4} | 5180 | 670 | 9.7 |

CoFe_{2}O_{4} | 4907 | 700 | 3.7 |

Materials | φ | β | M | Blasius [ | Cortell [ | Yazdi [ | Khan [ | Present Work |
---|---|---|---|---|---|---|---|---|

Pure Fluid | 0.0 | 0.0 | 0.0 | 0.3321 | 0.33206 | - | 0.33206 | 0.33206 |

0.0 | 0.0 | 1.0 | - | - | 1.0440 | 1.04400 | 1.0440 | |

0.0 | 0.5 | 1.0 | - | - | 0.6987 | 0.69872 | 0.6989 | |

Fe_{3}O_{4} | 0.01 | 0.0 | 0.0 | - | - | - | 0.34324 | 0.34324 |

0.1 | 0.0 | 0.0 | - | - | - | 0.45131 | 0.45132 | |

CoFe_{2}O_{4} | 0.01 | 0.0 | 0.0 | - | - | - | 0.34278 | 0.34279 |

0.1 | 0.0 | 0.0 | - | - | - | 0.44694 | 0.44692 |

or Nusselt number is computed for pure fluids for specific values of Prandtl number, Biot number, magnetic parameter and velocity slip parameters (

From _{3}O_{4}) has higher skin friction and heat transfer rate or Nusselt number than cobalt ferrite (CoFe_{2}O_{4}). The same behavior was observed by Khan et al. [

The effects of parameters variation on the dimensionless velocity and temperature profiles of water-based and kerosene-based magnetite (Fe_{3}O_{4}) and cobalt ferrite (CoFe_{2}O_{4}) are plotted in Figures 3-7.

The variation of the dimensionless velocity and temperature profiles with magnetic parameter M is shown in

Pr | M | β | a | Aziz [ | Ishak [ | Yazdi [ | Present Work |
---|---|---|---|---|---|---|---|

1.0 | 1.0 | 0.0 | 0.01 | - | - | 0.0098 | 0.0098 |

1.0 | 0.5 | 0.5 | - | - | 0.2417 | 0.2417 | |

1.0 | 0.5 | 1.0 | - | - | 0.3188 | 0.3182 | |

0.72 | 0.0 | 0.0 | 0.1 | 0.0747 | 0.0747 | 0.0757 | 0.0748 |

0.0 | 0.0 | 1.0 | 0.2282 | 0.2282 | 0.2282 | 0.2287 |

Pr | M | φ | R | β | a | C_{f} (Fe_{3}O_{4}) | C_{f} (CoFe_{2}O_{4}) | Nu_{x} (Fe_{3}O_{4}) | Nu_{x} (CoFe_{2}O_{4}) |
---|---|---|---|---|---|---|---|---|---|

6.2 | 0.0 | 0.1 | 0.5 | 1.0 | 0.5 | 0.313843 | 0.310840 | 0.353972 | 0.340727 |

0.05 | 0.530258 | 0.528102 | 0.366343 | 0.352073 | |||||

0.08 | 0.654255 | 0.652579 | 0.372411 | 0.357636 | |||||

0.1 | 0.734906 | 0.733543 | 0.376039 | 0.360963 | |||||

0.08 | 0.0 | 0.572979 | 0.572979 | 0.317991 | 0.317991 | ||||

0.05 | 0.610823 | 0.610072 | 0.345406 | 0.338016 | |||||

0.1 | 0.654255 | 0.652579 | 0.372411 | 0.357636 | |||||

0.2 | 0.760744 | 0.756681 | 0.423586 | 0.394805 | |||||

0.1 | 0.0 | 0.654254 | 0.652579 | 0.426164 | 0.407135 | ||||

1.0 | 0.654254 | 0.652579 | 0.329848 | 0.318454 | |||||

2.0 | 0.654254 | 0.652579 | 0.261707 | 0.255749 | |||||

3.0 | 0.654254 | 0.652579 | 0.206436 | 0.204857 | |||||

0.05 | 0.5 | 0.0 | 0.703410 | 0.700109 | 0.247676 | 0.240386 | |||

0.5 | 0.608566 | 0.605774 | 0.338995 | 0.326481 | |||||

1.0 | 0.530258 | 0.528102 | 0.366343 | 0.352073 | |||||

1.5 | 0.474980 | 0.473301 | 0.379628 | 0.364471 | |||||

0.08 | 1.0 | 0.1 | 0.654255 | 0.652579 | 0.112832 | 0.106822 | |||

0.5 | 0.654255 | 0.652579 | 0.372411 | 0.357636 | |||||

1.0 | 0.654255 | 0.652579 | 0.481164 | 0.472623 | |||||

10.0 | 0.654255 | 0.652579 | 1.659979 | 1.229714 | |||||

21.0 | 0.0 | 0.5 | 0.329016 | 0.325351 | 0.499449 | 0.491256 | |||

0.05 | 0.541174 | 0.538537 | 0.504371 | 0.496067 | |||||

0.08 | 0.662775 | 0.660720 | 0.506870 | 0.498492 | |||||

0.1 | 0.741901 | 0.740194 | 0.508392 | 0.499990 | |||||

0.08 | 0.0 | 0.572979 | 0.572979 | 0.394507 | 0.394507 | ||||

0.05 | 0.614682 | 0.613739 | 0.448412 | 0.444551 | |||||

0.1 | 0.662775 | 0.660720 | 0.506870 | 0.498492 | |||||

0.2 | 0.781126 | 0.776217 | 0.640016 | 0.620159 | |||||

0.1 | 0.0 | 0.662775 | 0.660720 | 0.538307 | 0.529123 | ||||

1.0 | 0.662775 | 0.660720 | 0.481986 | 0.474245 | |||||

2.0 | 0.662775 | 0.660720 | 0.442288 | 0.435565 | |||||

3.0 | 0.662775 | 0.660720 | 0.410171 | 0.404272 | |||||

0.05 | 0.5 | 0.0 | 0.720163 | 0.716106 | 0.403942 | 0.397348 | |||

0.5 | 0.622702 | 0.619276 | 0.484083 | 0.476191 | |||||

1.0 | 0.541174 | 0.538537 | 0.504371 | 0.496067 | |||||

1.5 | 0.483512 | 0.481440 | 0.513724 | 0.505216 | |||||

0.08 | 1.0 | 0.1 | 0.662775 | 0.660720 | 0.123411 | 0.121167 | |||

0.5 | 0.662775 | 0.660720 | 0.506870 | 0.498492 | |||||

1.0 | 0.662775 | 0.660720 | 0.862235 | 0.849386 | |||||

10.0 | 0.662775 | 0.660720 | 4.903603 | 4.870995 |

increase of the magnetic parameter M results in the decrease of temperature profiles.

R increases θ ( η ) and hence the thermal boundary layer thickness. s variation of the dimensionless velocity profiles for magnetite and cobalt ferrite respectively, for various values of velocity slip parameter β . When there is no slip, i.e. β = 0 , the velocity is at the boundary of the wall. It is noted that the increase in velocity slip β corresponds to a rise in the fluid velocity adjacent to the wall. Temperature profiles for numerous values of velocity slip parameter β are shown in

number has no effect on the velocity profiles as radiation parameter regardless of the base fluid or ferroparticles. For each case, it is observed that due to the higher Prandtl number of kerosene, the thermal boundary layer thickness as well as the dimensionless surface temperature is smaller for kerosene-based ferrofluids.

The present study investigates the flow and heat transfer of ferrofluids over a flat plate with slip conditions and radiation using the Keller-Box method. For each ferrofluid, it is concluded that:

· The skin friction coefficient increases with the increase of magnetic field, volume fraction of solid ferroparticles and decreases with increasing velocity slip parameter, but remains the same for variation of radiation parameter and Biot number.

· The heat transfer rate or Nusselt number increases with increasing magnetic parameter, volume fraction, velocity slip parameter, Biot number and decreases with increasing radiation parameter.

· Kerosene-based ferrofluids have higher skin friction and heat transfer rate than the water-based ferrofluids.

· Kerosene-based magnetite (Fe_{3}O_{4}) provides the higher skin friction coefficient and heat transfer rate at the wall compared to the kerosene-based cobalt ferrite (CoFe_{2}O_{4}).

· In the slip flow regime, under low Biot number conditions, the permeability effects on heat transfer rate tend to be negligible.

· The dimensionless velocity profiles increase with the increase of magnetic parameter and velocity slip parameter, decrease with the increase of volume fraction of solid nanoparticles and there is no effect with the variation of radiation parameter and Biot number.

· The dimensionless temperature profile increases with the increase of volume fraction of solid nanoparticles, radiation parameter, Biot number and decreases with the increase of magnetic and velocity slip parameter.

For future work, investigation can be carried out to study the effects of magnetic field, radiation, slip conditions, viscous dissipation, internal heating, suction, injection of ferrofluids for unsteady case and turbulent flow.

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

Sejunti, M.I. and Khaleque, T.S. (2019) Effects of Velocity and Thermal Slip Conditions with Radiation on Heat Transfer Flow of Ferrofluids. Journal of Applied Mathematics and Physics, 7, 1369-1387. https://doi.org/10.4236/jamp.2019.76092