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Accepted Manuscript Effects on Heat Transfer of Multiphase Magnetic Fluid due to Circular Magnetic Field over a Stretching Surface with Heat Source/Sink and Thermal Radiation A. Zeeshan, A. Majeed, C. Fetecau, S. Muhammad PII: DOI: Reference:
S2211-3797(17)30886-0 http://dx.doi.org/10.1016/j.rinp.2017.08.047 RINP 897
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
22 May 2017 19 August 2017 28 August 2017
Please cite this article as: Zeeshan, A., Majeed, A., Fetecau, C., Muhammad, S., Effects on Heat Transfer of Multiphase Magnetic Fluid due to Circular Magnetic Field over a Stretching Surface with Heat Source/Sink and Thermal Radiation, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.08.047
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Effects on Heat Transfer of Multiphase Magnetic Fluid due to Circular Magnetic Field over a Stretching Surface with Heat Source/Sink and Thermal Radiation a
A. Zeeshan a , A.Majeed a , C. Fetecau b , S. Muhammad c Department of Mathematics and Statistics, FBAS, International Islamic University, Islamabad 44000, Pakistan b Academy of Romanian Scientists, Bucharest, 050094, Romania c Department of Mathematics, Abdual wali khan university Mardan, Pakistan
Abstract The purpose of the current article is to explore the boundary layer heat transport flow of multiphase magnetic fluid with solid impurities suspended homogeneously past a stretching sheet under the impact of circular magnetic field. Thermal radiation effects are also taken in account. The equations describing the flow of dust particles in fluid along with point dipole are modelled by employing conservation laws of mass, momentum and energy, which are then converted into non-linear coupled differential equations by mean of similarity approach. The transformed ODE’s are tackled numerically with the help of efficient Runga-Kutta method. The influence of ferromagnetic interaction parameter, viscous dissipation, fluid-particle interaction parameter, Eckert number, Prandtl number, thermal radiation parameter and number of dust particles, heat production or absorption parameter with the two thermal process namely, prescribed heat flux (PHF) or prescribed surface temperature (PST) are observed on temperature and velocity profiles. The value of skin-friction coefficient and Nusselt number are calculated for numerous physical parameters. Present results are correlated with available for a limited case and an excellent agreement is found.
Keywords: Ferromagnetic interaction parameter; Dusty magnetic fluid, Stretching sheet; Magnetic dipole; Heat source/sink; Thermal radiation
Nomenclature Ψ Pr
Stream function Prandtl number Heat generation/absorption coefficient Magnetic field Thermal conductivity Pyromagnetic constant Magnetization Local Nusselt number Skin friction coefficient
Q0
H k K
∗
M Nux
Cf (A,D) uw
(u, v)
(u ,v ) p
p
Positive constant Wall velocity Velocity component of ferro phase
Velocity component of dust phase
(T, Tp ) Temperature of ferro and dust phase Tc
α1
β
γ
ε η, ξ
Curie temperature Dimensionless distance form surface Ferromagnetic parameter Magnetic field strength Dimensionless curie temperature Dimensionless coordinates
r w
T K1
H1 Tr Ec
Relative density Shear stress Magnetic potential Thermal equilibrium time Stoke resistance coefficient Stefan--Boltzmann constant Heat source/sink parameter Thermal radiation Eckert number
Stretching rate N Number density (c1 , c3 ) Local fluid particle interaction parameters for heat transfer (cp , cm ) Specific heat of ferro and dust particles c2 Local fluid particle interaction parameter for velocity 0 Magnetic permeability c
m c
l
Mass concentration Stretching constant Dimensionless temperature Viscous dissipation parameter Dynamic viscosity Characteristic length
1. Introduction In the most recent couple of decades, boundary layer flow and energy transport phenomena due to stretched surface have conquered significant importance because of wider range of applications in engineering and modern industrial procedures, particularly, in continuous stretching of plastic films, polymer extrusion, glass blowing, tinning and annealing of copper wires, chemical industries such as metallurgy process like metal extrusion and metal spinning, heat removal from nuclear fuel debris, artificial fibres, hot rolling and so on. Crane [1] was the first one, who gave analytical solution for a steady incompressible quiescent fluid flow over a stretching surface. Later for more general case Grubka and Bobba [2] have continued this study to power law temperature distribution. They found a several closed form temperature solutions for specified conditions in
terms of Kummer’s functions. Further on, expansions to non-Newtonian fluids like micropolar, power-law fluids or viscoelastic have been analyzed by (Siddappa and Khapate [3]; Chain [4]; Andersson and Dandapat [5]). Vajravelu and Hadjinicolaou [6] investigated the heat transfer analysis on viscous boundary layer flow past a stretching surface with the influence of internal heating and viscous dissipation. Hayat et al. [7] designates the Brownian motion and thermophoresis aspects in nonlinear flow of micropolar nanoliquid over a stretching sheet under the consideration of thermal radiation. Saffman [8] seems to be the first, who modelled the governing flow equations for dusty liquid and analyzed the stability of dusty gas flow carrying small dust particles distributed uniformly. More recently, Gireesha et al. [9] have reported heat transfer characteristics of the dust particle over a stretching surface with non-uniform heat source/sink effect. Prasannakumara et al. [10] considered melting flow and energy transport of dusty viscous fluid near stagnation-point in the attendance of applied magnetic field and thermal radiation. Vajravelu and Nayfeh [11] have carried out hydromagnetic flow of a dust particle liquid flow in view of adequate light and impact of suction and particle loading over a stretching surface. In addition, ferrofluids are a unique class of magnetic nanoparticles suspended in single domain and carrier liquid [12]. Ferromagnetic fluid and the flow of energy transport can be control owing to external magnetic field. This fluid has attracted many researchers and scientists because it has uncountable applications in biomedicine, chemical engineering, lithographic patterning, and micro electro mechanical system (MEMS) [13-17]. Neuringer [18] examined the dipole magnetic field impact on stagnation point flow against a two-dimensional parallel flow of a heated ferrofluid toward a wall with surface temperature linearly decreasing. Convective flow of ferromagnetic fluid with linear temperature in a cavity was accomplished by Kefayati [19]. He verified that by increasing the concentration of ferro particles reduction occur in the transfer of heat by applying LBM technique. Khan et al. [20] reported MHD nonlinear thermally radiative three-dimensional Jeffrey nanoliquid flow through Newtonian heat and mass conditions in bidirectional surface by using homotopy procedure. Sheikholeslami and Rashidi [21] have explored the heat transfer characteristics and non-uniform magnetic effects on the magnetite (Fe3O4) nanofluid suspended with water in semi annulus enclosure. Recently, articles related to ferro and nano liquid flow along with external magnetic field have been presented by [22-29]. In spite of all the above mentioned applications, the persistence of current note is to investigate
dusty ferrofluid flow and heat transport past a stretchable surface under the impact of radiation and magnetic dipole field. The governing PDE’s are transformed into ODE’s by considering suitable similarity procedure. These highly nonlinear ODE’s are solved numerically by applying Runge-Kutta shooting technique using MATLAB package. Additional, the influence of various pertinent parameter like Eckert number, viscous dissipation, heat source/sink parameter, number density of dust particles, Prandtl number, and radiation parameter on velocity distribution and temperature distribution is studied for two set of boundary conditions, such as prescribed heat flux (PHF) or prescribed surface temperature (PST). The results entitles that Nusselt number enhances against two thermal heating situations PST and PHF for higher values of H1 . Also temperature of ferrofluid phase and dust phase increases with the variation of magneto-hydrodynamic interaction parameter due to line dipole field.
2. Mathematical Modelling Let us consider an incompressible, two-dimensional viscous ferromagnetic dusty liquid flow over a stretchable surface. The sheet is taken toward x-axis with velocity u w normal to y-axis (see Fig.1). A magnetic dipole is situated at a distance “ a ” from the sheet with its centre on y-axis. The direction of the magnetic dipole in the positive x-direction gives rise to magnetic field to drench the ferromagnetic liquid. It is also supposed that uniform temperature at wall is Tw and Curie temperature Tc , while temperature of ambient ferromagnetic fluid away from the wall is Tw = Tc .
Fig.1 Geometry of the problem.
The boundary layer equations of momentum and energy transport in terms of ferro and dusty phase are [30] ∂u ∂v + = 0, ∂x ∂y
(1)
∂u ∂u µ0 ∂H µ ∂ 2 u ∂ 2u K1 N u + v = M + + (u p − u ), + ∂y ρ ∂x ρ ∂x 2 ∂y 2 ρ ∂x
(2)
∂v ∂v µ 0 ∂H µ ∂ 2 v ∂ 2 v K1 N M + + (v p − v ), + u + v = ∂y ρ ∂y ρ ∂x 2 ∂y 2 ρ ∂x
(3)
up
up up K
v p m1 u u p , x y
up
∂v p ∂x
+ vp
∂v p ∂y
=
K1 (v − v p ), m
(4)
(5)
∂( ρ p u p ) ∂x
∂( ρ p vp )
+
∂y
= 0,
(6)
∂T ∂T µ0 ∂M ∂H ∂H k ∂ 2T 1 ∂qr +v T +v − + u + u = 2 ρ c p ∂y ∂y ρ c p ∂T ∂x ∂y ρ c p ∂y ∂x
µ ρcp
2 2 ∂u 2 ∂v ∂v ∂u Nc p N Q (Tp − T ) + (u p − u ) + 0 (Tc − T ), 2 + 2 + + + ρ c pτ v ρcp ∂x ∂y ∂x ∂y ρ c pτ T
up
∂Tp ∂x
∂Tp
+ vp
∂y
=
cp cmτ T
(Tc − T ),
(7)
(8)
where ( u, v ) and ( u p ,v p ) are velocities of ferro and dust particles along the coordinate axis, (T ,Tp ) are the ferrofluid temperature and particle temperature, K1 is the Stoke resistance
coefficient, Q0 is the heat generation/absorption coefficient. Further ( ρ , ρ p ) are densities of ferro and dust phase, µ the dynamic viscosity, µ 0 is the magnetic permeability, H is the magnetic field and M is the magnetization. The terms µ0 M
∂H ∂x
and µ0 M ∂∂Hy in Eqs (2) and (3)
denote magnetic force components, which rely on the presence of the magnetic gradient. These forces will not survive without magnetic gradient. The 2nd term in Eq. (7) on the left side signifies heating owing to is caloric magnetization, k is thermal conductivity, τ v = m / K1 is the relaxation time of dust particle, τ T is the thermal equilibrium time, N and m are the number density and mass concentration of the particle phase, and (c p , cm ) are the specific heat of ferromagnetic liquid and dust particles. The corresponding boundary equations are for PST for PHF
at y = 0 ,
(9)
u p → 0, v p → 0 Tp → Tc , ρ p → E ρ
as y → ∞ ,
(10)
u = uw ( x) = cx,
v = 0,
T = Tw = Tc − A ( −k ∂∂Ty = qw = D ( u → 0, T → Tc ,
x 2 l
)
x 2 l
)
The magnetic scalar potential of magnetic dipole is
Φ=
γ 2π
x x 2 + ( y + a )2
,
(11)
The components of magnetic field (H) are Hx and Hy 2 2 ∂Φ γ x − ( y + a ) Hx = − = ∂x 2π x 2 + y + a 2 ( )
(
)
∂Φ γ 2 x ( y + a ) Hy = − = ∂y 2π x 2 + y + a 2 ( )
(
2
)
2
,
(12)
,
(13)
The resultant magnitude H is given by γ H = 2π
1
2 2 x − ( y + a ) 2 x 2 + ( y + a )
(
)
2
2
γ 2 x ( y + a ) + 2π 2 2 x + ( y + a )
(
)
2
2
2 .
(14)
Taking the partial derivative and open the series about x = 0, and neglecting terms with cube and higher power of x, we get, ∂H γ 2x , =− ∂x 2π ( y + a ) 4
(15)
∂H γ −2 4 x2 , = + ∂y 2π ( y + a )3 ( y + a )5
(16)
The variation of M in term of temperature is taken as [31] M = K ∗ (Tc − T ) ,
(17)
where Tc is the Curie temperature and K ∗ is pyromagnetic constant. However, the following point is essential for the occurrence of ferro-hydrodynamic interaction: (i) fluid at temperature T is diverse from Tc and (ii) external magnetic field is non-homogeneous. Once the ferromagnetic fluid approaches to Curie temperature, there is no further more magnetization. Characteristic for
physical significance is very important, as Tc is very high. Employing Rosseland approximation of radiation [32], we have −4σ ∗ ∂T 4 , 3k1∗ ∂y
(18)
T 4 ≅ 4Tc3T − 3Tc4 ,
(19)
qr =
After simplification, we get
3. Similarity Transformation The following similarity transformations are introduce [31] Ψ (ξ ,η ) = νξ f (η )
θ (ξ ,η ) ≡
(20)
Tc − T = θ1 (η ) + ξ 2θ2 (η ) , Tc −Tw
x where Tc − Tw = A l Tc − Tw =
θ p (η ) =
Tc − Tp Tc −Tw
(21)
2
D x kl
for PST case 2
ν c
for PHF case
The non-dimensional variables are defined as [31] ξ=
c x, ν
η=
c y ν
(22)
Ψ (ξ ,η ) , θ (ξ ,η ) and θ p (η ) are non-dimensional stream function and temperature of ferrofluid
phase and dust phase temperature. The components of velocity defined as u=
∂Ψ ∂Ψ = cxf ′ (η ) , v = − = − cν f (η ) ∂y ∂x
u p = cxF (η ), v p = cν G (η ) , ρ r =
ρp = W (η ) . ρ
(23)
(24)
Where r is the relative density. Substituting Eqs. (20) to (24) in Eqs. (2) to (8) and comparing coefficients of ξ up to ξ 2 power, we get:
f ′′′ + ff ′′ − f ′2 −
f ′′ + ff ′ +
2βθ1
(η + α1 )
2 βθ1 3
(η + α1 )
4
+ l1αW ( F − f ′) = 0,
− l1αW (G + f ) = 0,
(25)
(26)
Pr( Ec f ′′2 + c1 N (θ p − θ1 ) + c2 N ( F − f ′) 2 − 2 f ′θ1 + f θ1′ + H1θ1 ) + = 0, (1 + Tr )θ1′′+ 2 λβ (θ1 −ε ) f − 4λ ( f ′) 2 3 (η +α1 )
(27)
λβ θ − ε 2 f ′ + 4 f + ) (η +α1 )4 (η +α1 )5 ( 1 (1 + Tr )θ2′′ + Pr ( −c1 Nθ2 − 4 f ′θ 2 + f θ2′ + H1θ2 ) − = 0, 2 λβ f θ2 ′′ 2 3 − λ( f ) (η +α1 )
(28)
F 2 + F ′G − α ( f ′ − F ) = 0,
(29)
GG ′ + α ( f + G ) = 0,
(30)
2 Fθ p + Gθ p′ + c3 (θ p − θ1 ) = 0,
(31)
FW + W ′G + WG′ = 0.
(32)
The corresponding boundary equations (9) and (10) are renovated as f = 0, f ′ = 1, θ1 = 1, θ 2 = 0, ′
f = 0, f ′ = 1, θ1′ = −1, θ2 = 0,
for PST Case for PHF Case
f ′ → 0, F → 0, G → − f , W → E θ1 → 0, θ 2 → 0, θ p → 0
at η = 0,
as η → ∞.
(33)
(34)
The non-dimension quantities appearing in the equations (25) to (32) are 2 2 λ = ρk (cTµc −Tw ) , α1 = cµρ a, l1 = τ1c , Ec = c p (UTcw−T ) , . cp 16σ ∗Tc3 Tc 1 1 Tr = 3kk ∗ , ε = Tc −Tw , c1 = ρ cτT , c2 = ρ cτ v , c3 = cmcτT , 1
β=
γρ
2πµ 2
µ0 K ∗ (Tc − Tw ) , Pr =
µc p k
, H1 = c Qρ c0 p ,
(35)
The parameters in (35) are: the ferromagnetic interaction parameter β , the Prandtl number Pr , H1 the heat absorption
( H1 > 0)
and heat generation
( H1 < 0)
parameter, the viscous
dissipation parameter λ , the non-dimensional distance from surface to dipole α 1 , the mass concentration of dust particles l1 , the Eckert number Ec, the thermal radiation parameter Tr , the dimensionless curie temperature ratio ε , for heat transfer local fluid particle parameters are c1 and c3 , for velocity local fluid particle parameter is c2 , the fluid interaction parameter α
and W is the relative density. The imperative physical quantities are the skin friction coefficient and Nusselt number can be expressed as ∂u ; τw = µ , ρ ( cx ) ∂y y = 0
(36)
∂T xqw ; qw = − , − k (Tc − T ) ∂y y = 0
(37)
C fx =
−2τ w
2
and Nu x =
Applying equations (20) to (24), we have Nux / Re = − θ1′ ( 0 ) + ξ θ 2′ ( 0 ) (PST) , Nu x / Re1/x 2 = 1 / θ1 ( 0 ) + ξ 2θ 2 ( 0 ) (PHF) ′′ Re1/2 x C f = −2 f ( 0 ) 1/2 x
( (
2
) )
where f ′′ ( 0 ) is the dimensionless skin-friction coefficient,
Re x =
(38)
ρ cx 2 µ
indicates the local
Reynolds number, and −θ ′(ξ ,0) = − θ1′ ( 0 ) + ξ 2θ 2′ ( 0 ) is the non-dimensional heat transfer rate.
4. Numerical Results and Discussion Equations (25) to (32) along with their corresponding boundary conditions Eqs. (33) and (34) are highly nonlinear. It is difficult to obtain closed form solutions, so we must solve them numerically. Resulting system of higher order ODE’s are then solved as an initial value problem (IVP) by applying robust Runge-Kutta based shooting technique. Choose suitable initial guesses for f ′′(0), θ1′(0) , θ 2′ (0), F (0) and θ p (0) in such a way that the system of first order ODE’s fulfil the
boundary conditions and found the desired solution. In order to assure asymptotic convergence with an error 10−5 we choose η max = 15 . The influence of several pertinent parameters on ferro-dusty liquid flow and heat transport are stimulated pictorially and elaborated in tabular form. Default values for the present study are Pr = 0.72, λ = 0.01, ε = 2.0, α 1 = 1.0, l1 = 1 , c1 = c2 = c3 = 1, E = 1, N = 0.5, α = 0.2.
To confirm the validity and accuracy of the current study, values of θ1′(0) for some values of Prandtl number (Pr) is specified in Table.1. The present results show an excellent arrangement with those available of Pal and Mondal [33] and Roopa et al. [34]. Furthermore, Table. 2 demonstrate skin friction and Nusselt number and wall temperature for two cases against different physical parameters. Temperature profiles for ferrofluid phase θ1 (η ) and dust phase θ p (η ) for several values of ferromagnetic interaction parameter β in case PST and PHF are presented graphically in Fig.2. Result shows that temperature increases as β increases for ferrofluid and dust phase in both cases. In fact, ferromagnetic liquid essentially has a carrier liquid with tinny size suspended particles of ferrite which enhance the viscosity of fluid and consequently the flow velocity diminishes for large β . Heat transfer is also increased via decay motion. There is an inaction between the fluid motion and the applied dipole field, which tends to reduce the velocity and rising the frictional heating within the fluid layers. This justify the improvement in the thermal transport as seen in Fig.2. Physical point of view an impact of magnetic field develops the opposite force to the flow, is called the Lorentz force. This force has ability to flattening the momentum boundary layer and enhance the thermal boundary layer thickness Also it is observed that temperature for ferro phase is higher than dust phase.
Fig.3 is plotted to perceive the behavior of temperature field θ1 (η ) and θ p (η ) for several values of heat source/sink parameter H1 . We conclude from this figure that the thermal boundary produces the energy and consequently rise in temperature for higher values of heat source ( H1 > 0) parameter for PST and PHF, whereas temperature decreases with respect to heat sink
parameter ( H1 < 0) . From practical interest heat source/sink parameters recitals as heat generator and heat absorption coefficients, heat generator discharge the thermal energy of flow, because of this act an enhancement is seen in temperature field.
Fig.4 shows the temperature distributions under the impact of Eckert number ( Ec) for both PST and PHF. It is obviously seen form graph that the temperature distribution is an increasing function of Ec for PST as well as PHF. It is because of the fact that due to frictional heating, heat energy is preserved in liquid form. Enhancement is seen at any point in temperature by increasing Ec , which is true for both cases. Fig.5 displays the effects of number density ( N ) on the temperature distributions of ferro and dust phase for two cases PST and PHF. From this figure, it is noted that temperature decreases for increasing values of N and it indicates that reduction in the thickness of thermal boundary layer for rising the density of dust particles. It is also observed that temperature for ferro phase is greater than dust phase. Fig. 6 depicts the temperature profiles with the variation of Prandtl number ( Pr ) for PST and PHF. In the problem of heat transport Pr supervises the momentum boundary layer thickness. When Pr is low, it infers that heat diffuses very rapidly as compared with velocity. This implies that for liquid metals boundary layer thickness for thermal is much higher as compared to velocity. Henceforth Prandtl number is used to enhance the rate of cooling. An Increase of Pr values implies a diminution of the temperature and hence a reduction occur in the thickness of thermal boundary layer. Consequence of temperature profiles for different values of viscous dissipation parameter ( λ ) versus η is graphically displayed in Fig.7. It is vibrant from figure that large values of viscous dissipation parameter diminishes the thickness of thermal boundary layer for two heat process. Additionally, the values of temperature are lower in case of PST as compared to PHF. Extraordinary behavior of ferrofluid leads to difference in flow pattern then that of hydrodynamic case, where increasing the value of λ , temperature profile is enhanced. Fig.8 is sketched to see the impact of radiation parameter ( Tr ) on temperature fields for PST and PHF. The result illuminates that temperature is an increasing function of Tr , because due to reduction in Rosseland radiative absorptivity parameter k1∗ , produces increment in divergence of the radiative heat flux ∂qr / ∂y, which boost up heat transfer rate of fluid, this is the bases for fluid temperature to rise.
Pal and Mondal [33] Roopa et al. [34] Present work
Pr
0.72
--------
1.08855
1.08863
1 2
1.33333 1.99999
1.33333 1.99999
1.33334 1.99999
3
2.50971
2.50971
2.50967
4 5
2.93878 3.31647
2.93878 3.31648
2.93873 3.31645
6
3.65776
3.65777
3.65773
7 8
3.97150 4.26345
3.97151 4.26345
3.97145 4.26337
9
4.53760
4.53760
4.53749
10
4.79687
4.79687
4.79671
Table.1. Comparison of Nusselt number α = N = H1 = β = Ec = Tr = λ = 0 .
−θ1′ ( 0 )
1
2 PST-Case
0.9 0.8
α = 0.2 λ = 0.01 N = 0.5 H1= 0.2 Ec = 2.0 Tr = 0.1 Pr = 0.72
β = 0.2, 0.8, 1.5
0.4 0.3 0.2
Ferrof luid phase velocity Dust phase velocity
1.4
θ1( η ), θ p( η)
θ1( η ), θ p( η)
1.6
0.6 0.5
PHF-Case
1.8
Ferrof luid phase velocity Dust phase velocity
0.7
1.2
α = 0.2 λ = 0.01 N = 0.5 H1= 0.2 Ec = 2.0 Tr = 0.1 Pr = 0.72
1 β = 0.2, 0.8, 1.5 0.8 0.6 0.4
0.1 0 0
for different values of Pr when
0.2 1
2
3 η
4
5
6
0 0
1
2
Fig.2. Impact of β on temperature profiles versus η for PST and PHF.
3 η
4
5
6
1.5
1 0.9 0.8
Ferrof luid phase velocity Dust phase velocity
0.7
Ferrof luid phase velocity Dust phase velocity 1
0.6
θ1( η ), θ p( η)
θ1(η), θ p(η)
PHF-Case
PST-Case
α = 0.2 β = 0.2 λ = 0.01 N = 0.5 Ec = 2.0 Tr = 0.1 Pr = 0.72
0.5 H1 = -0.2, 0, 0.2 0.4 0.3 0.2
α = 0.2 β = 0.2 λ = 0.01 N = 0.5 Ec = 2.0 Tr = 0.1 Pr = 0.72
H1 = -0.2, 0, 0.2 0.5
0.1 0 0
1
2
3 η
4
5
0 0
6
1
2
3 η
4
5
6
Fig.3. Impact of H1 on temperature profiles versus η for PST and PHF.
1
1.5 PHF-Case
PST-Case
0.9 0.8
Ferrof luid phase velocity Dust phase velocity
Ferrof luid phase velocity Dust phase velocity 1
θ1( η ), θ p( η)
θ1( η ), θ p( η)
0.7 0.6 α = 0.2 β = 0.2 λ = 0.01 N = 0.5 H1 = 0.2 Tr = 0.1 Pr = 0.72
Ec = 0.5, 1.0, 2.0
0.5 0.4 0.3 0.2
α = 0.2 β = 0.2 λ = 0.01 N = 0.5 H1 = 0.2 Tr = 0.1 Pr = 0.72
Ec = 0.5, 1.0, 2.0 0.5
0.1 0 0
1
2
3 η
4
5
6
0 0
1
2
Fig.4. Impact of Ec on temperature profiles versus η for PST and PHF.
3 η
4
5
6
1
1.5
0.9
PHF-Case
PST-Case
0.8
Ferrof luid phase velocity Dust phase velocity
Ferrof luid phase velocity Dust phase velocity
0.7
0.5
θ1( η ), θ p( η)
θ1( η), θ p( η)
1 0.6
α = 0.2 β = 0.2 λ = 0.01 H1 = 0.2 Ec = 2.0 Tr = 0.1 Pr = 0.72
N = 0.5, 1.0, 1.5
0.4 0.3 0.2
α = 0.2 β = 0.2 λ = 0.01 H1 = 0.2 Ec = 2.0 Tr = 0.1 Pr = 0.72
N = 0.5, 1.0, 1.5 0.5
0.1 0 0
1
2
3 η
4
5
0 0
6
1
2
3 η
4
5
6
Fig.5. Impact of N on temperature profiles versus η for PST and PHF.
1
1.5
0.9 0.8
Ferrof luid phase velocity Dust phase velocity
0.7
Ferrof luid phase velocity Dust phase velocity 1
θ1( η ), θ p( η)
θ1( η ), θ p( η)
PHF-Case
PST-Case
0.6 α = 0.2 β = 0.2 λ = 0.01 N = 0.5 Ec = 2.0 H1 = 0.2 Tr = 0.1
0.5 Pr =0.72, 1.0, 2.0
0.4 0.3 0.2
α = 0.2 β = 0.2 λ = 0.01 N = 0.5 Ec =2.0 H1= 0.2 Tr = 0.1
Pr = 0.72, 1.0, 2.0 0.5
0.1 0 0
1
2
3 η
4
5
6
0 0
1
2
Fig.6. Impact of Pr on temperature profiles versus η for PST and PHF.
3 η
4
5
6
1
1.5 PST-Case
0.9 0.8
PHF-Case
Ferrof luid phase velocity Dust phase velocity
Ferrof luid phase velocity Dust phase velocity 1
θ1( η ), θ p( η)
θ1( η ), θ p( η )
0.7 0.6 α = 0.2 β = 0.2 N = 0.5 Ec = 2.0 H1= 0.2 Tr = 0.1 Pr = 0.72
0.5 λ = 0.01, 0.5, 1.5
0.4 0.3 0.2
α = 0.2 β = 0.2 N = 0.5 Ec = 2.0 H1 = 0.2 Tr = 0.1 Pr = 0.72
λ =0.01, 0.5, 1.5 0.5
0.1 0 0
1
2
3 η
4
5
0 0
6
1
2
3 η
4
5
6
Fig.7. Impact of λ on temperature profiles versus η for PST and PHF.
1
1.6 PST-Case
0.9 0.8
PHF-Case
1.4
Ferrof luid phase velocity Dust phase velocity
Ferrof luid phase velocity Dust phase velocity
1.2
α = 0.2 β = 0.2 λ = 0.01 Ec = 2.0 N = 0.5 H1 = 0.2 Pr = 0.72
0.6 0.5
Tr = 0.1, 0.3, 0.5
0.4 0.3
θ1( η ), θ p( η)
θ1( η ), θ p( η)
0.7 α = 0.2 β = 0.2 λ = 0.01 Ec = 2.0 N = 0.5 H1 = 0.2 Pr = 0.72
1 0.8 Tr = 0.1, 0.3, 0.5 0.6 0.4
0.2 0.2
0.1 0 0
1
2
3
4
5
6
7
0 0
1
2
η
Fig.8. Impact of Tr on temperature profiles versus η for PST and PHF.
3
4
η
5
6
7
☺
H1
Ec
N
Pr
T r f 0 1 0PST 1/1 0PHF
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
0.4
1.2111
-0.5044
0.6957
0.6
1.2532
-0.4986
0.6917
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
0.8
1.3840
-0.2213
0.6339
1.5
1.6576
-0.2863
0.5046
0.2 0.2 -0.2 2.0 0.5 0.72 0.1 1.1596
-0.2798
0.5717
0
1.1573
-0.4074
0.6424
0.2
1.1557
-0.5065
0.6989
0.2 0.2 0.2 0.5 0.5 0.72 0.1 1.1503
-1.0074
1.0064
1.5
1.1539
-0.6739
0.7792
2.0
1.1557
-0.5065
0.6989
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
1.0
1.1558
-0.4640
0.6976
2.0
1.1559
-0.3807
0.6945
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
1.0
1.1548
-0.3773,
0.7608
1.5
1.1531
-0.2822
0.8223
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
0.3 1.1565
-0.4776
0.6640
0.5 1.1572
-0.4525
0.6327
Table.2. Values of skin friction and Nusselt number for various values of physical parameters for (PST) and wall temperature (PHF).
5. Concluding Remarks In this article, a Mathematical analysis is presented to investigate heat transfer analysis of ferromagnetic and dusty fluid flow past a stretching surface with the effect of thermal radiation along with viscous dissipation and heat source/sink. The influence of Eckert number ( Ec) , Prandtl number ( Pr ) , viscous dissipation ( λ ) , number density of dust particles ( N ) ferromagnetic interaction parameter ( β ) and thermal radiation parameter (Tr ) on velocity and temperature profiles is also underlined and discussed. Present numerical results agreed very well with those
available in literature (Pal and Mondal [33], Roopa et al. [34]) when α = N = H1 = β = Ec = Tr = 0 . Some of the most important observations from the current study are as follow: •
Dimensionless temperatures θ1 (η ) and θ p (η ) , corresponding to ferro and dust phases, are increasing functions with respect to the parameters β , Tr , H1 , Ec and decrease for higher values of λ and N for PST and PHF .
•
The rates of heat transfer −θ1′(0) and 1 / θ1 (0) corresponding to both PST and PHF increases for large values of H1 .
•
As estimated, thermal boundary layer thickness reduces for increasing values of Prandtl number Pr .
References [1] Crane LJ. Flow past a stretching plate. Z Angew Math Phy 1970; 21: 645-7. [2] Grubka LJ, Bobba KM. Heat transfer characteristics of a continuous, stretching surface with variable temperature. J Heat Transfer 1985; 107: 248--250. [3] Siddappa BB, Khapate S. Rivlin-Ericksen fluid flow past a stretching plate. Rev Roum Sci Techn Mrc Appl 1976; 21: 497-505. [4] Chain LC. Micropolar fluid flow over a stretching sheet. ZAMM. 1982; 62: 565-568. [5] Andersson HI, Dandapat BS. Flow of a power-law fluid over a stretching sheet. Stability Appl Anal Cont Media 1991; 1: 339-347. [6] Vajravelu K, Hadjinicolaou A. Heat Transfer in a Viscous Fluid over a Stretching Sheet with Viscous Dissipation and Internal Heat Generation. Int Comm Heat Mass Transfer 1993; 20: 417-430. [7] Hayat T, Khan MI, Waqas M, Alsaedi A, Khan MI. Radiative flow of micropolar nanofluid accounting thermophoresis and Brownian moment. International Journal of Hydrogen Energy 2017: 42:16821-16833. [8] Saffman PG. On the stability of laminar flow of a dusty gas. J Fluid Mech 1962; 13. [9] Gireesha BJ, Ramesh GK, Abel MS, Bagewadi CS. Boundary layer flow and heat transfer of a dusty fluid flow over a stretching sheet with non-uniform heat source/sink. Int J Multiphase Flow 2011; 37: 977-982. [10] Prasannakumara BC, Gireesha BJ, Manjunatha PT. Melting phenomenon in MHD stagnation
point flow of dusty fluid over a stretching sheet in the presence of thermal radiation and non-uniform heat source/sink. Int J Comput Methods in Eng Sci and Mech 16(5), 2015; 5: 265-274. [11] Vajravelu K, Nayfeh J. Hydromagnetic flow of a dusty fluid over a stretching sheet. Int J Nonlin Mech 1992; 27: 937-945. [12] Xuan Y, Ye M, Li Q. Mesoscale simulation of ferrofluid structure. Int J Heat and Mass Transfer 2005:12: 2443-2451. [13] Rosensweig RE. Heating magnetic fluid with alternating magnetic field. J magn and magnetic materials 252, 2002: 370-374. [14] Ashouri M, Ebrahimi B, Shafii M, Saidi M. Correlation for Nusselt number in pure magnetic convection ferrofluid flow in a square cavity by a numerical investigation. J Magn Magn Mater 2010;322: 3607-3613. [15] Aminfar H, Mohammadpourfard M, Mohseni F. Two-phase mixture model simulation of the hydro-thermal behavior of an electrical conductive ferrofluids in the presence of magnetic fields. J Magn Magn Mater 2012; 324: 830-842. [16] Jue TC. Analysis of combined thermal and magnetic convection ferrofluid flow in a cavity. Int Commun Heat Mass Transfr 2006;33: 846-852. [17] Bissell P, Bates P, Chantrell R, Raj K, Wyman J. Cavity magnetic field measurements in ferrofluids J Magn Magn Mater 1983; 39: 27-29. [18] Neuringer JL. Some viscous flows of a saturated ferrofluid under the combined influence of thermal and magnetic field gradients. Int J Non-linear Mech 1966;1: 123-137. [19] Kefayati G. Natural convection of ferrofluid in a linearly heated cavity utilizing LBM. J Mol Liquids 2014; 191: 1-9. [20] Khan MI, Alsaedi A, Shehzad SA, Hayat T. Hydromagnetic nonlinear thermally radiative nanoliquid flow with Newtonian heat and mass conditions. Results in Physics 2017 ;(in press). [21] Sheikholeslami M, Rashidi MM. Effect of space dependent magnetic field on free convection of Fe3O4-water nanofluid. J Taiwan Inst Chem Engs 2015; 56: 6-15. [22] Tzirtzilakis EE, Kafoussias NG, Raptis A. Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 2010; 5: 929-947.
[23] Strek T, Jopek H. Computer simulation of heat transfer through a ferrofluid. physica status solidi (b) 2007; 244: 1027-1037. [24] Zeeshan A, Majeed A. Non Darcy mixed convection flow of magnetic fluid over a permeable stretching sheet with Ohmic dissipation. J Magnetics 2016; 21: 153-158. [25] Majeed A, Zeeshan A, Ellahi R. Chemical reaction and heat transfer on boundary layer Maxwell Ferro-fluid flow under magnetic dipole with Soret and suction effects. Eng Sci Techn Int J 2016. [26] Zeeshan A, Majeed A. Effect of magnetic dipole on radiative non-Darcian mixed convective flow over a stretching sheet in porous medium. J Nanofluids 2016; 4: 617-626. [27] Hayat T, Khan MI, Farooq M, Yasmeen T, Alsaedi A. Stagnation point flow with Cattaneo-Christov heat flux and homogeneous-heterogeneous reactions. Journal of Molecular Liquids 2016; 220; 49-55. [28] Farooq M, Khan MI, Waqas M, Hayat T, Alsaedi A, Khan MI. MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects. Journal of Molecular Liquids 2016; 221; 1097-1103. [29] Hayat T, Khan MI, Farooq M, Alsaedi A, Waqas M, Yasmeen T. Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface. International Journal of Heat and Mass Transfer 2016; 99; 702-710.
[30] Turkyilmazoglu M. Magnetohydrodynamic two-phase dusty fluid flow and heat model over deforming isothermal surfaces. Physics of Fluids 2017; 29; 013302. [31] Andersson HI, Valnes OA. Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole. Acta Mechanica 1998;128: 39-47. [32] Brewster MQ. Thermal radiative transfer properties. John Wiley and Sons 1972. [33] Pal D, Mondal H. Hydromagnetic non-Darcy flow and heat transfer over a stretching sheet in the presence of thermal radiation and Ohmic dissipation. Commu Nonlinear Sci Numer Simu 2010;15: 1197-1209. [34] Roopa GS, Gireesha BJ, Bagewadi CS. Effect of viscous dissipation on MHD flow and heat transfer of a dusty fluid over an unsteady stretching sheet. Int J Math Archive 2011; 2: 2229-5046.
S2211-3797(17)30886-0 http://dx.doi.org/10.1016/j.rinp.2017.08.047 RINP 897
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
22 May 2017 19 August 2017 28 August 2017
Please cite this article as: Zeeshan, A., Majeed, A., Fetecau, C., Muhammad, S., Effects on Heat Transfer of Multiphase Magnetic Fluid due to Circular Magnetic Field over a Stretching Surface with Heat Source/Sink and Thermal Radiation, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.08.047
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effects on Heat Transfer of Multiphase Magnetic Fluid due to Circular Magnetic Field over a Stretching Surface with Heat Source/Sink and Thermal Radiation a
A. Zeeshan a , A.Majeed a , C. Fetecau b , S. Muhammad c Department of Mathematics and Statistics, FBAS, International Islamic University, Islamabad 44000, Pakistan b Academy of Romanian Scientists, Bucharest, 050094, Romania c Department of Mathematics, Abdual wali khan university Mardan, Pakistan
Abstract The purpose of the current article is to explore the boundary layer heat transport flow of multiphase magnetic fluid with solid impurities suspended homogeneously past a stretching sheet under the impact of circular magnetic field. Thermal radiation effects are also taken in account. The equations describing the flow of dust particles in fluid along with point dipole are modelled by employing conservation laws of mass, momentum and energy, which are then converted into non-linear coupled differential equations by mean of similarity approach. The transformed ODE’s are tackled numerically with the help of efficient Runga-Kutta method. The influence of ferromagnetic interaction parameter, viscous dissipation, fluid-particle interaction parameter, Eckert number, Prandtl number, thermal radiation parameter and number of dust particles, heat production or absorption parameter with the two thermal process namely, prescribed heat flux (PHF) or prescribed surface temperature (PST) are observed on temperature and velocity profiles. The value of skin-friction coefficient and Nusselt number are calculated for numerous physical parameters. Present results are correlated with available for a limited case and an excellent agreement is found.
Keywords: Ferromagnetic interaction parameter; Dusty magnetic fluid, Stretching sheet; Magnetic dipole; Heat source/sink; Thermal radiation
Nomenclature Ψ Pr
Stream function Prandtl number Heat generation/absorption coefficient Magnetic field Thermal conductivity Pyromagnetic constant Magnetization Local Nusselt number Skin friction coefficient
Q0
H k K
∗
M Nux
Cf (A,D) uw
(u, v)
(u ,v ) p
p
Positive constant Wall velocity Velocity component of ferro phase
Velocity component of dust phase
(T, Tp ) Temperature of ferro and dust phase Tc
α1
β
γ
ε η, ξ
Curie temperature Dimensionless distance form surface Ferromagnetic parameter Magnetic field strength Dimensionless curie temperature Dimensionless coordinates
r w
T K1
H1 Tr Ec
Relative density Shear stress Magnetic potential Thermal equilibrium time Stoke resistance coefficient Stefan--Boltzmann constant Heat source/sink parameter Thermal radiation Eckert number
Stretching rate N Number density (c1 , c3 ) Local fluid particle interaction parameters for heat transfer (cp , cm ) Specific heat of ferro and dust particles c2 Local fluid particle interaction parameter for velocity 0 Magnetic permeability c
m c
l
Mass concentration Stretching constant Dimensionless temperature Viscous dissipation parameter Dynamic viscosity Characteristic length
1. Introduction In the most recent couple of decades, boundary layer flow and energy transport phenomena due to stretched surface have conquered significant importance because of wider range of applications in engineering and modern industrial procedures, particularly, in continuous stretching of plastic films, polymer extrusion, glass blowing, tinning and annealing of copper wires, chemical industries such as metallurgy process like metal extrusion and metal spinning, heat removal from nuclear fuel debris, artificial fibres, hot rolling and so on. Crane [1] was the first one, who gave analytical solution for a steady incompressible quiescent fluid flow over a stretching surface. Later for more general case Grubka and Bobba [2] have continued this study to power law temperature distribution. They found a several closed form temperature solutions for specified conditions in
terms of Kummer’s functions. Further on, expansions to non-Newtonian fluids like micropolar, power-law fluids or viscoelastic have been analyzed by (Siddappa and Khapate [3]; Chain [4]; Andersson and Dandapat [5]). Vajravelu and Hadjinicolaou [6] investigated the heat transfer analysis on viscous boundary layer flow past a stretching surface with the influence of internal heating and viscous dissipation. Hayat et al. [7] designates the Brownian motion and thermophoresis aspects in nonlinear flow of micropolar nanoliquid over a stretching sheet under the consideration of thermal radiation. Saffman [8] seems to be the first, who modelled the governing flow equations for dusty liquid and analyzed the stability of dusty gas flow carrying small dust particles distributed uniformly. More recently, Gireesha et al. [9] have reported heat transfer characteristics of the dust particle over a stretching surface with non-uniform heat source/sink effect. Prasannakumara et al. [10] considered melting flow and energy transport of dusty viscous fluid near stagnation-point in the attendance of applied magnetic field and thermal radiation. Vajravelu and Nayfeh [11] have carried out hydromagnetic flow of a dust particle liquid flow in view of adequate light and impact of suction and particle loading over a stretching surface. In addition, ferrofluids are a unique class of magnetic nanoparticles suspended in single domain and carrier liquid [12]. Ferromagnetic fluid and the flow of energy transport can be control owing to external magnetic field. This fluid has attracted many researchers and scientists because it has uncountable applications in biomedicine, chemical engineering, lithographic patterning, and micro electro mechanical system (MEMS) [13-17]. Neuringer [18] examined the dipole magnetic field impact on stagnation point flow against a two-dimensional parallel flow of a heated ferrofluid toward a wall with surface temperature linearly decreasing. Convective flow of ferromagnetic fluid with linear temperature in a cavity was accomplished by Kefayati [19]. He verified that by increasing the concentration of ferro particles reduction occur in the transfer of heat by applying LBM technique. Khan et al. [20] reported MHD nonlinear thermally radiative three-dimensional Jeffrey nanoliquid flow through Newtonian heat and mass conditions in bidirectional surface by using homotopy procedure. Sheikholeslami and Rashidi [21] have explored the heat transfer characteristics and non-uniform magnetic effects on the magnetite (Fe3O4) nanofluid suspended with water in semi annulus enclosure. Recently, articles related to ferro and nano liquid flow along with external magnetic field have been presented by [22-29]. In spite of all the above mentioned applications, the persistence of current note is to investigate
dusty ferrofluid flow and heat transport past a stretchable surface under the impact of radiation and magnetic dipole field. The governing PDE’s are transformed into ODE’s by considering suitable similarity procedure. These highly nonlinear ODE’s are solved numerically by applying Runge-Kutta shooting technique using MATLAB package. Additional, the influence of various pertinent parameter like Eckert number, viscous dissipation, heat source/sink parameter, number density of dust particles, Prandtl number, and radiation parameter on velocity distribution and temperature distribution is studied for two set of boundary conditions, such as prescribed heat flux (PHF) or prescribed surface temperature (PST). The results entitles that Nusselt number enhances against two thermal heating situations PST and PHF for higher values of H1 . Also temperature of ferrofluid phase and dust phase increases with the variation of magneto-hydrodynamic interaction parameter due to line dipole field.
2. Mathematical Modelling Let us consider an incompressible, two-dimensional viscous ferromagnetic dusty liquid flow over a stretchable surface. The sheet is taken toward x-axis with velocity u w normal to y-axis (see Fig.1). A magnetic dipole is situated at a distance “ a ” from the sheet with its centre on y-axis. The direction of the magnetic dipole in the positive x-direction gives rise to magnetic field to drench the ferromagnetic liquid. It is also supposed that uniform temperature at wall is Tw and Curie temperature Tc , while temperature of ambient ferromagnetic fluid away from the wall is Tw = Tc .
Fig.1 Geometry of the problem.
The boundary layer equations of momentum and energy transport in terms of ferro and dusty phase are [30] ∂u ∂v + = 0, ∂x ∂y
(1)
∂u ∂u µ0 ∂H µ ∂ 2 u ∂ 2u K1 N u + v = M + + (u p − u ), + ∂y ρ ∂x ρ ∂x 2 ∂y 2 ρ ∂x
(2)
∂v ∂v µ 0 ∂H µ ∂ 2 v ∂ 2 v K1 N M + + (v p − v ), + u + v = ∂y ρ ∂y ρ ∂x 2 ∂y 2 ρ ∂x
(3)
up
up up K
v p m1 u u p , x y
up
∂v p ∂x
+ vp
∂v p ∂y
=
K1 (v − v p ), m
(4)
(5)
∂( ρ p u p ) ∂x
∂( ρ p vp )
+
∂y
= 0,
(6)
∂T ∂T µ0 ∂M ∂H ∂H k ∂ 2T 1 ∂qr +v T +v − + u + u = 2 ρ c p ∂y ∂y ρ c p ∂T ∂x ∂y ρ c p ∂y ∂x
µ ρcp
2 2 ∂u 2 ∂v ∂v ∂u Nc p N Q (Tp − T ) + (u p − u ) + 0 (Tc − T ), 2 + 2 + + + ρ c pτ v ρcp ∂x ∂y ∂x ∂y ρ c pτ T
up
∂Tp ∂x
∂Tp
+ vp
∂y
=
cp cmτ T
(Tc − T ),
(7)
(8)
where ( u, v ) and ( u p ,v p ) are velocities of ferro and dust particles along the coordinate axis, (T ,Tp ) are the ferrofluid temperature and particle temperature, K1 is the Stoke resistance
coefficient, Q0 is the heat generation/absorption coefficient. Further ( ρ , ρ p ) are densities of ferro and dust phase, µ the dynamic viscosity, µ 0 is the magnetic permeability, H is the magnetic field and M is the magnetization. The terms µ0 M
∂H ∂x
and µ0 M ∂∂Hy in Eqs (2) and (3)
denote magnetic force components, which rely on the presence of the magnetic gradient. These forces will not survive without magnetic gradient. The 2nd term in Eq. (7) on the left side signifies heating owing to is caloric magnetization, k is thermal conductivity, τ v = m / K1 is the relaxation time of dust particle, τ T is the thermal equilibrium time, N and m are the number density and mass concentration of the particle phase, and (c p , cm ) are the specific heat of ferromagnetic liquid and dust particles. The corresponding boundary equations are for PST for PHF
at y = 0 ,
(9)
u p → 0, v p → 0 Tp → Tc , ρ p → E ρ
as y → ∞ ,
(10)
u = uw ( x) = cx,
v = 0,
T = Tw = Tc − A ( −k ∂∂Ty = qw = D ( u → 0, T → Tc ,
x 2 l
)
x 2 l
)
The magnetic scalar potential of magnetic dipole is
Φ=
γ 2π
x x 2 + ( y + a )2
,
(11)
The components of magnetic field (H) are Hx and Hy 2 2 ∂Φ γ x − ( y + a ) Hx = − = ∂x 2π x 2 + y + a 2 ( )
(
)
∂Φ γ 2 x ( y + a ) Hy = − = ∂y 2π x 2 + y + a 2 ( )
(
2
)
2
,
(12)
,
(13)
The resultant magnitude H is given by γ H = 2π
1
2 2 x − ( y + a ) 2 x 2 + ( y + a )
(
)
2
2
γ 2 x ( y + a ) + 2π 2 2 x + ( y + a )
(
)
2
2
2 .
(14)
Taking the partial derivative and open the series about x = 0, and neglecting terms with cube and higher power of x, we get, ∂H γ 2x , =− ∂x 2π ( y + a ) 4
(15)
∂H γ −2 4 x2 , = + ∂y 2π ( y + a )3 ( y + a )5
(16)
The variation of M in term of temperature is taken as [31] M = K ∗ (Tc − T ) ,
(17)
where Tc is the Curie temperature and K ∗ is pyromagnetic constant. However, the following point is essential for the occurrence of ferro-hydrodynamic interaction: (i) fluid at temperature T is diverse from Tc and (ii) external magnetic field is non-homogeneous. Once the ferromagnetic fluid approaches to Curie temperature, there is no further more magnetization. Characteristic for
physical significance is very important, as Tc is very high. Employing Rosseland approximation of radiation [32], we have −4σ ∗ ∂T 4 , 3k1∗ ∂y
(18)
T 4 ≅ 4Tc3T − 3Tc4 ,
(19)
qr =
After simplification, we get
3. Similarity Transformation The following similarity transformations are introduce [31] Ψ (ξ ,η ) = νξ f (η )
θ (ξ ,η ) ≡
(20)
Tc − T = θ1 (η ) + ξ 2θ2 (η ) , Tc −Tw
x where Tc − Tw = A l Tc − Tw =
θ p (η ) =
Tc − Tp Tc −Tw
(21)
2
D x kl
for PST case 2
ν c
for PHF case
The non-dimensional variables are defined as [31] ξ=
c x, ν
η=
c y ν
(22)
Ψ (ξ ,η ) , θ (ξ ,η ) and θ p (η ) are non-dimensional stream function and temperature of ferrofluid
phase and dust phase temperature. The components of velocity defined as u=
∂Ψ ∂Ψ = cxf ′ (η ) , v = − = − cν f (η ) ∂y ∂x
u p = cxF (η ), v p = cν G (η ) , ρ r =
ρp = W (η ) . ρ
(23)
(24)
Where r is the relative density. Substituting Eqs. (20) to (24) in Eqs. (2) to (8) and comparing coefficients of ξ up to ξ 2 power, we get:
f ′′′ + ff ′′ − f ′2 −
f ′′ + ff ′ +
2βθ1
(η + α1 )
2 βθ1 3
(η + α1 )
4
+ l1αW ( F − f ′) = 0,
− l1αW (G + f ) = 0,
(25)
(26)
Pr( Ec f ′′2 + c1 N (θ p − θ1 ) + c2 N ( F − f ′) 2 − 2 f ′θ1 + f θ1′ + H1θ1 ) + = 0, (1 + Tr )θ1′′+ 2 λβ (θ1 −ε ) f − 4λ ( f ′) 2 3 (η +α1 )
(27)
λβ θ − ε 2 f ′ + 4 f + ) (η +α1 )4 (η +α1 )5 ( 1 (1 + Tr )θ2′′ + Pr ( −c1 Nθ2 − 4 f ′θ 2 + f θ2′ + H1θ2 ) − = 0, 2 λβ f θ2 ′′ 2 3 − λ( f ) (η +α1 )
(28)
F 2 + F ′G − α ( f ′ − F ) = 0,
(29)
GG ′ + α ( f + G ) = 0,
(30)
2 Fθ p + Gθ p′ + c3 (θ p − θ1 ) = 0,
(31)
FW + W ′G + WG′ = 0.
(32)
The corresponding boundary equations (9) and (10) are renovated as f = 0, f ′ = 1, θ1 = 1, θ 2 = 0, ′
f = 0, f ′ = 1, θ1′ = −1, θ2 = 0,
for PST Case for PHF Case
f ′ → 0, F → 0, G → − f , W → E θ1 → 0, θ 2 → 0, θ p → 0
at η = 0,
as η → ∞.
(33)
(34)
The non-dimension quantities appearing in the equations (25) to (32) are 2 2 λ = ρk (cTµc −Tw ) , α1 = cµρ a, l1 = τ1c , Ec = c p (UTcw−T ) , . cp 16σ ∗Tc3 Tc 1 1 Tr = 3kk ∗ , ε = Tc −Tw , c1 = ρ cτT , c2 = ρ cτ v , c3 = cmcτT , 1
β=
γρ
2πµ 2
µ0 K ∗ (Tc − Tw ) , Pr =
µc p k
, H1 = c Qρ c0 p ,
(35)
The parameters in (35) are: the ferromagnetic interaction parameter β , the Prandtl number Pr , H1 the heat absorption
( H1 > 0)
and heat generation
( H1 < 0)
parameter, the viscous
dissipation parameter λ , the non-dimensional distance from surface to dipole α 1 , the mass concentration of dust particles l1 , the Eckert number Ec, the thermal radiation parameter Tr , the dimensionless curie temperature ratio ε , for heat transfer local fluid particle parameters are c1 and c3 , for velocity local fluid particle parameter is c2 , the fluid interaction parameter α
and W is the relative density. The imperative physical quantities are the skin friction coefficient and Nusselt number can be expressed as ∂u ; τw = µ , ρ ( cx ) ∂y y = 0
(36)
∂T xqw ; qw = − , − k (Tc − T ) ∂y y = 0
(37)
C fx =
−2τ w
2
and Nu x =
Applying equations (20) to (24), we have Nux / Re = − θ1′ ( 0 ) + ξ θ 2′ ( 0 ) (PST) , Nu x / Re1/x 2 = 1 / θ1 ( 0 ) + ξ 2θ 2 ( 0 ) (PHF) ′′ Re1/2 x C f = −2 f ( 0 ) 1/2 x
( (
2
) )
where f ′′ ( 0 ) is the dimensionless skin-friction coefficient,
Re x =
(38)
ρ cx 2 µ
indicates the local
Reynolds number, and −θ ′(ξ ,0) = − θ1′ ( 0 ) + ξ 2θ 2′ ( 0 ) is the non-dimensional heat transfer rate.
4. Numerical Results and Discussion Equations (25) to (32) along with their corresponding boundary conditions Eqs. (33) and (34) are highly nonlinear. It is difficult to obtain closed form solutions, so we must solve them numerically. Resulting system of higher order ODE’s are then solved as an initial value problem (IVP) by applying robust Runge-Kutta based shooting technique. Choose suitable initial guesses for f ′′(0), θ1′(0) , θ 2′ (0), F (0) and θ p (0) in such a way that the system of first order ODE’s fulfil the
boundary conditions and found the desired solution. In order to assure asymptotic convergence with an error 10−5 we choose η max = 15 . The influence of several pertinent parameters on ferro-dusty liquid flow and heat transport are stimulated pictorially and elaborated in tabular form. Default values for the present study are Pr = 0.72, λ = 0.01, ε = 2.0, α 1 = 1.0, l1 = 1 , c1 = c2 = c3 = 1, E = 1, N = 0.5, α = 0.2.
To confirm the validity and accuracy of the current study, values of θ1′(0) for some values of Prandtl number (Pr) is specified in Table.1. The present results show an excellent arrangement with those available of Pal and Mondal [33] and Roopa et al. [34]. Furthermore, Table. 2 demonstrate skin friction and Nusselt number and wall temperature for two cases against different physical parameters. Temperature profiles for ferrofluid phase θ1 (η ) and dust phase θ p (η ) for several values of ferromagnetic interaction parameter β in case PST and PHF are presented graphically in Fig.2. Result shows that temperature increases as β increases for ferrofluid and dust phase in both cases. In fact, ferromagnetic liquid essentially has a carrier liquid with tinny size suspended particles of ferrite which enhance the viscosity of fluid and consequently the flow velocity diminishes for large β . Heat transfer is also increased via decay motion. There is an inaction between the fluid motion and the applied dipole field, which tends to reduce the velocity and rising the frictional heating within the fluid layers. This justify the improvement in the thermal transport as seen in Fig.2. Physical point of view an impact of magnetic field develops the opposite force to the flow, is called the Lorentz force. This force has ability to flattening the momentum boundary layer and enhance the thermal boundary layer thickness Also it is observed that temperature for ferro phase is higher than dust phase.
Fig.3 is plotted to perceive the behavior of temperature field θ1 (η ) and θ p (η ) for several values of heat source/sink parameter H1 . We conclude from this figure that the thermal boundary produces the energy and consequently rise in temperature for higher values of heat source ( H1 > 0) parameter for PST and PHF, whereas temperature decreases with respect to heat sink
parameter ( H1 < 0) . From practical interest heat source/sink parameters recitals as heat generator and heat absorption coefficients, heat generator discharge the thermal energy of flow, because of this act an enhancement is seen in temperature field.
Fig.4 shows the temperature distributions under the impact of Eckert number ( Ec) for both PST and PHF. It is obviously seen form graph that the temperature distribution is an increasing function of Ec for PST as well as PHF. It is because of the fact that due to frictional heating, heat energy is preserved in liquid form. Enhancement is seen at any point in temperature by increasing Ec , which is true for both cases. Fig.5 displays the effects of number density ( N ) on the temperature distributions of ferro and dust phase for two cases PST and PHF. From this figure, it is noted that temperature decreases for increasing values of N and it indicates that reduction in the thickness of thermal boundary layer for rising the density of dust particles. It is also observed that temperature for ferro phase is greater than dust phase. Fig. 6 depicts the temperature profiles with the variation of Prandtl number ( Pr ) for PST and PHF. In the problem of heat transport Pr supervises the momentum boundary layer thickness. When Pr is low, it infers that heat diffuses very rapidly as compared with velocity. This implies that for liquid metals boundary layer thickness for thermal is much higher as compared to velocity. Henceforth Prandtl number is used to enhance the rate of cooling. An Increase of Pr values implies a diminution of the temperature and hence a reduction occur in the thickness of thermal boundary layer. Consequence of temperature profiles for different values of viscous dissipation parameter ( λ ) versus η is graphically displayed in Fig.7. It is vibrant from figure that large values of viscous dissipation parameter diminishes the thickness of thermal boundary layer for two heat process. Additionally, the values of temperature are lower in case of PST as compared to PHF. Extraordinary behavior of ferrofluid leads to difference in flow pattern then that of hydrodynamic case, where increasing the value of λ , temperature profile is enhanced. Fig.8 is sketched to see the impact of radiation parameter ( Tr ) on temperature fields for PST and PHF. The result illuminates that temperature is an increasing function of Tr , because due to reduction in Rosseland radiative absorptivity parameter k1∗ , produces increment in divergence of the radiative heat flux ∂qr / ∂y, which boost up heat transfer rate of fluid, this is the bases for fluid temperature to rise.
Pal and Mondal [33] Roopa et al. [34] Present work
Pr
0.72
--------
1.08855
1.08863
1 2
1.33333 1.99999
1.33333 1.99999
1.33334 1.99999
3
2.50971
2.50971
2.50967
4 5
2.93878 3.31647
2.93878 3.31648
2.93873 3.31645
6
3.65776
3.65777
3.65773
7 8
3.97150 4.26345
3.97151 4.26345
3.97145 4.26337
9
4.53760
4.53760
4.53749
10
4.79687
4.79687
4.79671
Table.1. Comparison of Nusselt number α = N = H1 = β = Ec = Tr = λ = 0 .
−θ1′ ( 0 )
1
2 PST-Case
0.9 0.8
α = 0.2 λ = 0.01 N = 0.5 H1= 0.2 Ec = 2.0 Tr = 0.1 Pr = 0.72
β = 0.2, 0.8, 1.5
0.4 0.3 0.2
Ferrof luid phase velocity Dust phase velocity
1.4
θ1( η ), θ p( η)
θ1( η ), θ p( η)
1.6
0.6 0.5
PHF-Case
1.8
Ferrof luid phase velocity Dust phase velocity
0.7
1.2
α = 0.2 λ = 0.01 N = 0.5 H1= 0.2 Ec = 2.0 Tr = 0.1 Pr = 0.72
1 β = 0.2, 0.8, 1.5 0.8 0.6 0.4
0.1 0 0
for different values of Pr when
0.2 1
2
3 η
4
5
6
0 0
1
2
Fig.2. Impact of β on temperature profiles versus η for PST and PHF.
3 η
4
5
6
1.5
1 0.9 0.8
Ferrof luid phase velocity Dust phase velocity
0.7
Ferrof luid phase velocity Dust phase velocity 1
0.6
θ1( η ), θ p( η)
θ1(η), θ p(η)
PHF-Case
PST-Case
α = 0.2 β = 0.2 λ = 0.01 N = 0.5 Ec = 2.0 Tr = 0.1 Pr = 0.72
0.5 H1 = -0.2, 0, 0.2 0.4 0.3 0.2
α = 0.2 β = 0.2 λ = 0.01 N = 0.5 Ec = 2.0 Tr = 0.1 Pr = 0.72
H1 = -0.2, 0, 0.2 0.5
0.1 0 0
1
2
3 η
4
5
0 0
6
1
2
3 η
4
5
6
Fig.3. Impact of H1 on temperature profiles versus η for PST and PHF.
1
1.5 PHF-Case
PST-Case
0.9 0.8
Ferrof luid phase velocity Dust phase velocity
Ferrof luid phase velocity Dust phase velocity 1
θ1( η ), θ p( η)
θ1( η ), θ p( η)
0.7 0.6 α = 0.2 β = 0.2 λ = 0.01 N = 0.5 H1 = 0.2 Tr = 0.1 Pr = 0.72
Ec = 0.5, 1.0, 2.0
0.5 0.4 0.3 0.2
α = 0.2 β = 0.2 λ = 0.01 N = 0.5 H1 = 0.2 Tr = 0.1 Pr = 0.72
Ec = 0.5, 1.0, 2.0 0.5
0.1 0 0
1
2
3 η
4
5
6
0 0
1
2
Fig.4. Impact of Ec on temperature profiles versus η for PST and PHF.
3 η
4
5
6
1
1.5
0.9
PHF-Case
PST-Case
0.8
Ferrof luid phase velocity Dust phase velocity
Ferrof luid phase velocity Dust phase velocity
0.7
0.5
θ1( η ), θ p( η)
θ1( η), θ p( η)
1 0.6
α = 0.2 β = 0.2 λ = 0.01 H1 = 0.2 Ec = 2.0 Tr = 0.1 Pr = 0.72
N = 0.5, 1.0, 1.5
0.4 0.3 0.2
α = 0.2 β = 0.2 λ = 0.01 H1 = 0.2 Ec = 2.0 Tr = 0.1 Pr = 0.72
N = 0.5, 1.0, 1.5 0.5
0.1 0 0
1
2
3 η
4
5
0 0
6
1
2
3 η
4
5
6
Fig.5. Impact of N on temperature profiles versus η for PST and PHF.
1
1.5
0.9 0.8
Ferrof luid phase velocity Dust phase velocity
0.7
Ferrof luid phase velocity Dust phase velocity 1
θ1( η ), θ p( η)
θ1( η ), θ p( η)
PHF-Case
PST-Case
0.6 α = 0.2 β = 0.2 λ = 0.01 N = 0.5 Ec = 2.0 H1 = 0.2 Tr = 0.1
0.5 Pr =0.72, 1.0, 2.0
0.4 0.3 0.2
α = 0.2 β = 0.2 λ = 0.01 N = 0.5 Ec =2.0 H1= 0.2 Tr = 0.1
Pr = 0.72, 1.0, 2.0 0.5
0.1 0 0
1
2
3 η
4
5
6
0 0
1
2
Fig.6. Impact of Pr on temperature profiles versus η for PST and PHF.
3 η
4
5
6
1
1.5 PST-Case
0.9 0.8
PHF-Case
Ferrof luid phase velocity Dust phase velocity
Ferrof luid phase velocity Dust phase velocity 1
θ1( η ), θ p( η)
θ1( η ), θ p( η )
0.7 0.6 α = 0.2 β = 0.2 N = 0.5 Ec = 2.0 H1= 0.2 Tr = 0.1 Pr = 0.72
0.5 λ = 0.01, 0.5, 1.5
0.4 0.3 0.2
α = 0.2 β = 0.2 N = 0.5 Ec = 2.0 H1 = 0.2 Tr = 0.1 Pr = 0.72
λ =0.01, 0.5, 1.5 0.5
0.1 0 0
1
2
3 η
4
5
0 0
6
1
2
3 η
4
5
6
Fig.7. Impact of λ on temperature profiles versus η for PST and PHF.
1
1.6 PST-Case
0.9 0.8
PHF-Case
1.4
Ferrof luid phase velocity Dust phase velocity
Ferrof luid phase velocity Dust phase velocity
1.2
α = 0.2 β = 0.2 λ = 0.01 Ec = 2.0 N = 0.5 H1 = 0.2 Pr = 0.72
0.6 0.5
Tr = 0.1, 0.3, 0.5
0.4 0.3
θ1( η ), θ p( η)
θ1( η ), θ p( η)
0.7 α = 0.2 β = 0.2 λ = 0.01 Ec = 2.0 N = 0.5 H1 = 0.2 Pr = 0.72
1 0.8 Tr = 0.1, 0.3, 0.5 0.6 0.4
0.2 0.2
0.1 0 0
1
2
3
4
5
6
7
0 0
1
2
η
Fig.8. Impact of Tr on temperature profiles versus η for PST and PHF.
3
4
η
5
6
7
☺
H1
Ec
N
Pr
T r f 0 1 0PST 1/1 0PHF
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
0.4
1.2111
-0.5044
0.6957
0.6
1.2532
-0.4986
0.6917
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
0.8
1.3840
-0.2213
0.6339
1.5
1.6576
-0.2863
0.5046
0.2 0.2 -0.2 2.0 0.5 0.72 0.1 1.1596
-0.2798
0.5717
0
1.1573
-0.4074
0.6424
0.2
1.1557
-0.5065
0.6989
0.2 0.2 0.2 0.5 0.5 0.72 0.1 1.1503
-1.0074
1.0064
1.5
1.1539
-0.6739
0.7792
2.0
1.1557
-0.5065
0.6989
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
1.0
1.1558
-0.4640
0.6976
2.0
1.1559
-0.3807
0.6945
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
1.0
1.1548
-0.3773,
0.7608
1.5
1.1531
-0.2822
0.8223
0.2 0.2 0.2 2.0 0.5 0.72 0.1 1.1557
-0.5065
0.6989
0.3 1.1565
-0.4776
0.6640
0.5 1.1572
-0.4525
0.6327
Table.2. Values of skin friction and Nusselt number for various values of physical parameters for (PST) and wall temperature (PHF).
5. Concluding Remarks In this article, a Mathematical analysis is presented to investigate heat transfer analysis of ferromagnetic and dusty fluid flow past a stretching surface with the effect of thermal radiation along with viscous dissipation and heat source/sink. The influence of Eckert number ( Ec) , Prandtl number ( Pr ) , viscous dissipation ( λ ) , number density of dust particles ( N ) ferromagnetic interaction parameter ( β ) and thermal radiation parameter (Tr ) on velocity and temperature profiles is also underlined and discussed. Present numerical results agreed very well with those
available in literature (Pal and Mondal [33], Roopa et al. [34]) when α = N = H1 = β = Ec = Tr = 0 . Some of the most important observations from the current study are as follow: •
Dimensionless temperatures θ1 (η ) and θ p (η ) , corresponding to ferro and dust phases, are increasing functions with respect to the parameters β , Tr , H1 , Ec and decrease for higher values of λ and N for PST and PHF .
•
The rates of heat transfer −θ1′(0) and 1 / θ1 (0) corresponding to both PST and PHF increases for large values of H1 .
•
As estimated, thermal boundary layer thickness reduces for increasing values of Prandtl number Pr .
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