Heat Mass Transfer (2016) 52:897–911 DOI 10.1007/s00231-015-1606-3
ORIGINAL
Thermal radiation and Hall effects on boundary layer flow past a non‑isothermal stretching surface embedded in porous medium with non‑uniform heat source/sink and fluid‑particle suspension B. J. Gireesha1,2 · B. Mahanthesh2 · Rama Subba Reddy Gorla1 · P. T. Manjunatha3
Received: 26 September 2014 / Accepted: 3 June 2015 / Published online: 14 June 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract Theoretical study on hydromagnetic heat transfer in dusty viscous fluid on continuously stretching nonisothermal surface, with linear variation of surface temperature or heat flux has been carried out. Effects of Hall current, Darcy porous medium, thermal radiation and nonuniform heat source/sink are taken into the account. The sheet is considered to be permeable to allow fluid suction or blowing, and stretching with a surface velocity varied according to a linear. Two cases of the temperature boundary conditions were considered at the surface namely, PST and PHF cases. The governing partial differential equations are transferred to a system of non-linear ordinary differential equations by employing suitable similarity transformations and then they are solved numerically. Effects of various pertinent parameters on flow and heat transfer for both phases is analyzed and discussed through graphs in detail. The values of skin friction and Nusselt number for different governing parameters are also tabulated. Comparison of the
* Rama Subba Reddy Gorla
[email protected] B. J. Gireesha
[email protected] B. Mahanthesh
[email protected] P. T. Manjunatha
[email protected] 1
Department of Mechanical Engineering, Cleveland State University, Cleveland, OH 44114, USA
2
Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, Shimoga 577 451, Karnataka, India
3
Department of Mathematics, Govt. Science College, Chitradurga 577 501, Karnataka, India
present results with known numerical results is presented and an excellent agreement is found. List of symbols A, b, D Constants A∗ , B∗ Parameters of space and temperature dependent heat generation Magnetic induction vector B B0 Magnetic field Cfx , Cfz Local skin friction coefficient along x and z directions cp Specific heat coefficient of fluid (J kg−1 K−1) cm Specific heat coefficient of dust particles (J kg−1 K−1) e Charge of electron E Electric field Ec Eckert number J Current density vector Jx, Jy, Jz Current density components along x, y and z directions f Dimensionless axial velocity of fluid F Dimensionless axial velocity of dust h Dimensionless transverse velocity of fluid phase H Dimensionless transverse velocity of fluid phase K Stokes drag coefficient k ∗ Permeability of porous medium k0 Porous parameter k Thermal conductivity (W m−1 k−1) K+ Mean absorption coefficient (m−1) l Dust particle mass concentration parameter l∗ Characteristic length m Hall parameter M2 Magnetic parameter mp Mass of dust particle per unit volume N Number density of dust particles
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Nu Local Nusselt number Pr Prandtl number pe Electronic pressure Space and temperature dependent heat q′′′ generation/absorption qw Heat flux qr Radiative heat flux (Wm−2) R Thermal radiation parameter r Radius of dust particles Rex Local Reynolds number S Suction/blowing parameter Sh Sherwood number T Fluid phase temperature (K) u, v, w Fluid phase velocity components along x, y and z directions (m s−1) Uw Stretching sheet velocity Vw Suction/injection velocity x, y, z Coordinates (m) Greek symbols βv Fluid-particle interaction parameter for velocity βT Fluid-particle interaction parameter for temperature ν Kinematic viscosity (m2 s−1) μ Dynamic viscosity (kg m−1 s−1) σ Electrical conductivity of the fluid σ ∗ Stefan–Boltzmann constant (Wm−2 K−4) θ Dimensionless fluid phase temperature (K) η Similarity variable τe Electron collision time τT Thermal equilibrium time τw Surface shear stress τv Relaxation time of the dust particles τwx Surface shear stress in x-direction τwz Surface shear stress in z-direction γ Specific heat ratio ρ Density of the fluid (kg m−3) θ Non-dimensional temperature Superscript ′ Derivative with respect to η Subscript p Dust phase w Fluid properties at the wall ∞ Fluid properties at ambient condition
1 Introduction Boundary layer flow and heat transfer of an electrically conducting viscous fluid induced by a continuously moving or stretching surface is relevant to many manufacturing and
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industrial processes such as polymers involving the cooling of continuous strips or filaments by drawing them through a quiescent fluid. Further, glass blowing, continuous casting of metals and spinning of fibers involve the flow due to a stretching surface in an ambient fluid. The quality of the final sheeting material, as well as the cost of production is affected by the speed of collection and the heat transfer rate. Also, in several engineering processes, materials manufactured by extrusion processes and heat treated materials traveling between a feed roll and a wind up roll on convey belts possess the characteristic of a moving continuous surface. Due to these facts, Sakiadis [1] first who discussed the boundary layer flow over a continues solid surface moving with a constant speed. Later, Crane [2] generalized the problem of [1] for a stretching sheet and obtained the closed form solution. Various direction of boundary layer flow over a stretching sheet problem was studied by several researchers [3–6]. Flow and transport of heat through a porous medium occurs in numerous practical applications. Such investigations finds their applications over a broad spectrum of science and engineering disciplines, especially in chemical catalytic reactors, grain storage, migration of moisture through the air contained fibrous insulations, heat exchange between soil and atmosphere, salt leaching in soils, solar power collectors, electrochemical processes, insulation of nuclear reactors, regenerative heat exchangers and geothermal systems and many others. In fact, boundary layer flow and transport of heat through a porous medium is abundant [7]. Representative studies dealing with boundary layer flow through porous medium have been reported by many authors like Cortell [8], Anghel et al. [9] and Alsaedi et al. [10]. On the other hand, in all above said studies the electrical conductivity of the fluid was assumed to be uniform and low magnetic field strength. Nevertheless, in an ionized fluid where the density is low and thereby magnetic field strength is very strong, the conductivity normal to the magnetic field is reduced due to the spiraling of electrons and ions about the magnetic lines of force before collisions take place and a current induced in a direction normal to both the electric and magnetic fields, this phenomena is known as Hall effect. The study of MHD flows with Hall current has important applications in the problem of Hall accelerators as well as flight magnetohydrodynamics. The current trend for the application of magnetohydrodynamics is towards a strong magnetic field and low density of the gas. Under this condition Hall effect becomes significant. Watanabe and Pop [11] investigated the MHD boundarylayer flow over a continuously moving semi-infinite flat plate with Hall current. Hydrodynamic free convection and mass transfer of an electrically conducting viscous fluid past an infinite vertical porous plate has been reported by Singh and Gorla [12]. Later on, many authors like Fakhar
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[13], Aziz [14] and Ali et al. [15] have shown their interest to investigate the influence of Hall current on different fluids with different aspects. A considerable interest has been shown in the study of thermal radiation on boundary layer flow and heat transfer in fluids due to its significant effects in the surface heat transfer. Further, thermal radiation effects on flow and heat transfer processes are of major importance in the space technology and high temperature processes. Bataller [16] studied the effects of thermal radiation on the Blasius flow. The heat and mass transfer in stagnation point flow towards a stretching surface in the presence of thermal radiation have been investigated by Pal [17]. Mukhopadhyay [18] reported the effects of thermal radiation and variable fluid viscosity on stagnation point flow past a porous stretching sheet. Later, Magyari and Pantokratoras [19] examined the effect of thermal radiation using in the linearized Rosseland approximation on the heat transfer characteristics in boundary layer flow. Also, the effect of heat source/sink play a significant role in controlling heat transfer in the production of quality product as it depends on the heat controlling factor. Cortell [20] investigated the effect of internal heat generation/absorption on flow and heat transfer of a fluid through a porous medium over a stretching surface. AboEldahab and Aziz [21] analyzed the effect of internal heat source/sink on mixed convection boundary layer flow and heat transfer over an inclined continuously stretching surface with transpiration cooling. Then, a comprehensive survey of heat transport uniform or non-uniform heat source/ sink was conducted by [22, 23]. The study of boundary layer flow with fluid-particle suspension has become an increasing importance in last few years. This is mainly due to their applications in atmospheric fallout, powder technology, rain erosion, petroleum transport, nuclear reactor cooling, dust collection, sedimentation, environmental pollution, guided missiles, paint spraying, food technologies, fluidization transport of solid particles by a liquid and liquid slurries in chemical and nuclear processing, soil pollution, control of the cooling rate of sheets and etc. Saffman [24] first who studied the stability of laminar flow of gas in which dust particles are uniformly distributed. Datta and Mishra [25] discussed boundary layer flow of an electrically conducting dusty viscous fluid over a semi-infinite flat plate. Effect of uniform suction on hydrodynamic boundary layer flow of a dusty fluid over a stretching sheet was numerically analyzed by Vajravelu and Nayfeh [26]. Gireesha et al. [27–29] studied the boundary layer two-phase flow and heat transfer of dusty fluid over stretching surface with different aspects. The current study is a theoretical investigation of the fully developed two-phase boundary layer flow and heat transfer of radiating dusty viscous fluid, using the Saffman [24] model, in the presence of non-uniform heat source/
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sink and Hall current. In this article, we employ an extensively validated, highly efficient numerical method called, fourth–fifth order Runge–Kutta–Fehlberg method to study this problem. The influence of various pertinent parameters on the different flow fields are presented and discussed though several plots and tables.
2 Flow analysis of the problem Consider a steady two dimensional boundary layer twophase flows and heat transfer of an incompressible, electrically conducting, dusty viscous fluid past a non-isothermal stretching surface embedded in saturated porous medium in the presence of applied strong magnetic field. Fluid is assumed to be Newtonian embedded with dust particles. The number density of suspended dust particles is assumed to be constant throughout the flow and volume fraction of dust particles is neglected. We assume a Cartesian coordinate system taking x-axis along the stretching sheet and the y-axis is taken normal to it. The leading edge of stretching sheet is taken as coincident with z-axis is as shown in Fig. 1. The flow is caused by stretching of the sheet which moves in its own plane with the surface velocity Uw = bx, where b > 0 is an initial stretching rate. It is considered that the surface temperature Tw of the sheet is suddenly raised from T∞ to Tw (Tw > T∞), or there is a suddenly imposed heat flux qw at the wall. An external strong magnetic field is applied in the positive y-direction. In general, for an electrically conducting fluid, Hall current affects the flow in the presence of a strong magnetic field. The effect of Hall current gives rise to a force in z-direction, which induces a cross flow in that direction and hence the flow becomes three-dimensional.
Fig. 1 A schematic representation of the physical model and coordinates system
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To simplify the problem, we assume that there is no variation of flow quantities in z-direction. This assumption is considered to be valid if the surface be of infinite extent in z-direction. The generalized Ohm’s law including Hall currents is given in the form [14]; � =σ E � + V� × B � + 1 Pe �J + ωe τe × J� × B (2.1) B0 ene where J� = Jx , Jy , Jz is the current density vector, is the intensity vector V� = (u, v, w) is the velocity vector, E � = (0, B0 , 0) is the magnetic induction of the electric field, B vector, σ , τe , e, ne and pe are respectively, electrical conductivity, electron collision time, charge of electron, number density of electrons and electronic pressure. Since, no applied or polarization voltage is imposed on the flow field, � = 0. For a weakly ionized gases, the electric field vector E generalized Ohm’s law under aforementioned conditions gives Jy = 0 everywhere in the flow. Hence, under these assumptions, equating the x and z components in (2.1) and solving for the current density components Jx and Jz are read as,
Jx = Jy =
σ B0 (mu − w), 1 + m2
σ B0 (u + mw), 1 + m2
(2.2)
(2.3)
here, u, v and w are x, y and z components of the velocity vector V respectively and m = ωe τe is Hall parameter. Under these assumptions, two-phase boundary layer flow equations are governed by the following system;
∂u ∂v + = 0, ∂x ∂y ∂u ∂ 2u ∂u +v = µ 2 + KN up − u ρ u ∂x ∂y ∂y
σ B02 µ (u + mw) − ∗ u, − 2 k 1+m
(2.4)
∂vp ∂up + = 0, ∂x ∂y
σ B02 µ (mu − w) − ∗ w, 2 k 1+m
∂up ∂up + vp = KN u − up , ρp up ∂x ∂y ∂wp ∂wp + vp = KN w − wp , ρp up ∂x ∂y
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u = Uw (x), v = Vx (x), w = 0 at y = 0, u → 0, up → 0, vp → v, w → 0, wp → 0 as y → ∞,
(2.10) where Uw-stretching sheet velocity and Vw-sucion/injection velocity. It should be mention that Vw > 0 corresponds to suction, Vw < 0 corresponds to blowing and Vw = 0 corresponds to stretching surface is impermeable. To solve the set of partial differential Eqs. (2.4)–(2.9), we adopt the similarity technique by introducing the following similarity variables;
√
Uw y, u = bxf (η), v = − νbf (η), w = bxh(η), η = νx √ up = bxF ′ (η), vp = − νbF(η), wp = bxH(η), (2.11) ′
In view of the above Eqs. (2.4) and (2.7) are identically satisfied and one can see that Eqs. (2.5), (2.6), (2.8) and (2.9) are reduce to following set of non-linear ordinary differential equations, f ′′′ (η) + f ′′ (η)f (η) − f ′ (η)2 + lβv F ′ (η) − f ′ (η) − k0 f ′ (η) −
M2 ′ f (η) + mh(η) = 0, 2 1+m
(2.12)
h′′ (η) + h′ (η)f (η) − f ′ (η)h(η) + lβv (H(η) − h(η)) − k0 h(η)
(2.5)
∂w ∂ 2w ∂w +v = µ 2 + KN wp − w ρ u ∂x ∂y ∂y +
where (u, v, w) and (up, vp, wp) are respectively, fluid and dust phase velocity components along x, y and z-directions. ν-kinematic viscosity of the fluid, μ dynamic viscosity of the fluid, ρ density of the fluid, ρp density of dust particles, N number density of dust particles, mp mass of the dust particles, K = 6πμr the Stoke’s drag constant, r radius of dust particles, σ electric conductivity and k ∗ permeability of porous medium. For above boundary layer equations, the relevant boundary conditions are;
(2.6)
+
M2 ′ mf (η) − h(η) = 0, 1 + m2
(2.13)
F ′′ (η)F(η) − F ′ (η)2 + βv f ′ (η) − F ′ (η) = 0,
(2.14)
H ′ (η)F(η) − H(η)F ′ (η) + βv (h(η) − H(η)) = 0.
(2.15)
f ′ (η) = 1, f (η) = S, h(η) = 0 at η = 0, f ′ (η) → 0, F ′ (η) → 0, F(η) → f (η), h(η) → 0, H(η) → 0 as η → ∞,
(2.16)
The boundary conditions (2.10) will becomes;
(2.7)
(2.8) (2.9)
where l = mN/ρ mass concentration of dust particles, τv = mp/K relaxation time of the dust particle, βv = 1/τvb fluid-particle interaction parameter for
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velocity, M2 = σB20/ρb magnetic parameter, k0 = ν/k ∗ b √ porous parameter and S = −Vw (x)/ bν suction/blowing parameter. Physical quantities of interest in engineering point of view is the local skin friction coefficient Cfx in x-direction and in Cfz in z-direction, are defined as,
Cfx =
τwx , ρUw2
Cfz =
τwz , ρUw2
(2.17)
where τwx and τwz are surface shear stress in x and z-directions respectively, which are given by
τwx
∂u , =µ ∂y y=0
τwz
∂w =µ , ∂y y=0
(2.18)
Using Eqs. (2.11) and (2.18) in (2.17), we obtain;
Rex Cfx = f ′′ (0),
Rex Cfz = h′ (0)
(2.19)
where Rex = Uw x/ν is the local Reynolds number. It is important to note that, tangential velocity Eq. (2.12) in the absence of M2, k0 and l reduced to boundary-layer flow past a stretching sheet whose analytical solution has been reported by Crane [2] as follows:
f (η) = 1 − e−η .
(2.20)
f ′′ (0) = −1.
(2.21)
In terms of above equation, the skin friction co-efficient in x-direction is given by
3 Heat transfer analysis The fluid and dust particle phase boundary layer equations for energy in the presence of thermal radiation and nonuniform heat source sink are given by;
∂T ∂ 2T ρp cm ∂T +v =k 2 + ρcp u Tp − T ∂x ∂y ∂y τT 2 ∂qr ρp + up − u − + q′′′ , (3.1) τv ∂y
ρp cm up
∂Tp ∂Tp + vp ∂x ∂y
=−
ρp cm Tp − T , τT
(3.2)
where T and Tp are respectively the temperature of clean fluid and dust particles, cp and cm respectively are the specific heat of clean fluid and dust particles, τT thermal equilibrium time i.e., the time required by the dust cloud to adjust its temperature to the clean fluid, k thermal
conductivity and q′′′ space and temperature dependent heat generation/absorption, which can be expressed as [23];
q′′′ =
kUw ∗ A (Tw − T∞ )f ′ (η) + B∗ (T − T∞ ) xν
(3.3)
4σ ∗ ∂T 4 , 3k + ∂y
(3.4)
where A∗ and B∗ are parameters of space and temperature dependent heat generation or absorption. Here, if A∗ > 0 and B∗ > 0 correspond to internal heat generation, whereas A∗ < 0 and B∗ < 0 correspond to internal heat absorption. According to the Rosseland diffusion approximation, the radiative heat flux qr is of the form,
qr =
where σ ∗ is the Stefan–Boltzmann constant and k+ is the mean absorption coefficient. It is noted that the optically thick radiation limit is considered in this model. Assuming that the temperature differences within the flow are sufficiently small such that T4 may be expressed as a linear function of temperature and then by neglecting the higher order terms beyond the first degree in (T −T∞ ), one 3 T − 3T 3 . Using this in (3.4), one can we get T 4 ∼ = 4T∞ ∞ can get 4 σ ∗ ∂ 2T 16T∞ ∂qr = . ∂y 3k + ∂y2
(3.5)
To solve the Eqs. (3.1) and (3.2), the prescribed surface temperature and prescribed surface heat flux boundary conditions are given by x 2 ∂T = qw (PHF), T = Tw (x) = T∞ + A ∗ (PST), −k l ∂y at y = 0, T → T∞ , Tp → T∞ as y = ∞, (3.6)
2 where T = A( lx∗ )2 θ(η) + T∞ (PST), Tw − T∞ = Dk lx∗ ν b (PHF), Tw and T∞ are denote the temperature at the wall and at large distance from the wall respectively. We now define the non-dimensional fluid phase temperature θ(η) and dust phase temperature θp(η) as θ(η) =
T − T∞ , Tw − T ∞
θp (η) =
Tp − T∞ , Tw − T ∞
(3.7)
where T − T∞ = A( lx∗ )2 θ(η), Tw and T∞ denote the temperature at the wall and at large distance from the wall respectively. In view of the Eq. (3.5) and using Eq. (3.7) into (3.1) and (3.2) and to the boundary conditions (3.6), one can arrive at the following dimensionless system of equations and boundary conditions;
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Table 1 Comparison results for surface temperature gradient for −θ ′ (0) ordinary viscous fluid with M 2 = l = k0 = A∗ = B∗ = R = 0
Pr
Grubka et al. [3]
Chen [4]
Abel et al. [5]
Ali et al. [6]
El-Aziz et al. [14]
Present study
0.72 1.0 3.0
1.0885 1.3333 2.5097
1.0885 1.3333 2.5097
1.0885 1.3333
– 1.3333 2.5097
1.0885 1.3333 2.5097
1.0886 1.3333 2.5097
10.0
4.7969
4.7968
4.7968
4.7969
4.7968
4.7969
4 1 + R θ ′′ (η) + Pr f (η)θ ′ (η) − 2f ′ (η)θ (η) 3 2 + lPrβT γ θp (η) − θ (η) + EcPr f ′′ (η) + A∗ f ′ (η) 2 (3.8) + B∗ θ (η) + lEcPrβv F ′ (η) − f ′ (η) = 0,
2F ′ (η)θp (η) − θp′ (η)F(η) + βT θp (η) − θ (η) = 0, θ(η) = 1(PST ), θ ′ (η) = −1(PHF) at θ (η) → 0, θp (η) → 0 as η → ∞.
η = 0,
(3.9) (3.10)
where Pr = μcp/k Prandtl number, βT = 1/bτT fluid-particle 3 /K + k interaction parameter for temperature, R = 4σ ∗ T∞ thermal radiation parameter γ = cm/cp specific heat ratio,
end point singularities. In accordance with the standard boundary layer analysis, the asymptotic boundary conditions at η → ∞ were replaced by those at η = 5, in which the obtained solutions meet the far field condition asymptotically. We have repeatedly confirmed in our previous publications to judge the accuracy and robustness of the used numerical method. We compare our results of heat transfer coefficient −θ ′ (0) with available results of Grubka and Bobba [3], Chen [4], Abel et al. [5], Ali et al. [6] and ElAziz et al. [14] for different values of Prandtl number as a further check and validation on the accuracy of our numerical computations with M 2 = l = k0 = A∗ = B∗ = R = 0. These comparisons are presented in Table 1 and found to be excellent agreement.
3
Ec = l∗2 b/Acp (PST) and Ec = l∗2 b 2 k/Dcp ν 1/2 (PHF) Eckert number, The important physical parameter for heat transfer coefficient (Nusselt number) is defined as qw Nu = , k(Tw − T∞ )
(3.11)
where qw is the surface heat flux, which is given by
qw = −k
∂T ∂y
.
(3.12)
y=0
In view of similarity variables and using Eq. (3.12) into (3.11), one can get
Re−0.5 Nu = −θ ′ (0). x
(3.13)
4 Numerical method and validation The system of non-linear differential Eqs. (2.12)–(2.15) and (3.8), (3.9) under the boundary conditions (2.16) and (3.10) have been solved numerically using fourth–fifth order Runge–Kutta–Fehlberg method implemented on Maple. The algorithm is proved to be precise and accurate in solving a wide range of mathematical and engineering problems especially fluid flow and heat transfer cases. In this package, there are two sub methods are available namely midpoint method and Trapezoidal method. Midpoint method is chosen as a sub method in our computation, since it is capable of handle harmless
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5 Results and discussion The hydromagnetic two-phase boundary layer flow and heat transfer on a non-isothermal permeable stretching sheet in the presence of magnetic field, Hall current, suction/blowing, thermal radiation and non-uniform heat source/sink effects have been investigated numerically. Further, two different heating processes namely PST and PHF cases are considered in heat transfer analysis. A parametric study on different flow fields is also made to analyze the physical insight of the problem. It is worth mentioning that the present study reduces to the classical viscous fluid flow problem if l = 0 i.e., in the absence dust particles mass concentration. Figures 2, 4, 6, 8, 10, 12 and 3, 5, 7, 9, 11, 13 are respectively present the effects of m, M 2 , βv , l, S and k0 on axial and transverse velocity profiles of both fluid and dust particle phase. It is apparent from Figs. 2 and 3 respectively that, the axial and transverse velocity profiles for both fluid and dust phase are increases with increasing values of Hall parameter. Since, as effective conducρ tivity 1+m 2 decreases with increase in m this reduces the magnetic damping force on axial and transverse velocity profiles. Moreover, f ′ (η) and F ′ (η) profiles approach their classical hydrodynamic values when the Hall parameter tends to infinity. Since the magnetic force terms approach zero value for very large values of Hall parameter. This result is consistent with El-Aziz [14]. It is observed from
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Fig. 2 The effect of m on axial velocity profile
Fig. 4 The effect of M2 on axial velocity profile
Fig. 3 The effect of m on transverse velocity profile
Fig. 5 The effect of M2 on transverse velocity profile
Fig. 4 that, for increasing values of magnetic parameter results in flattening of f ′ (η) and F ′ (η) profiles. The transverse contraction of axial velocity boundary layer is due to the applied magnetic field which results in the Lorentz force producing considerable opposition to the motion. In the absence of the magnetic field (M2 = 0), there is no transverse velocity for both phases (h(η) = H(η) = 0) and as magnetic field increases, a cross flow in the transverse direction is greatly induced due to the Hall effect as depicted in the Fig. 3. In addition, plot 5 elucidate that, close to the sheet surface an increase in the values of M2 leads to an increase in lateral velocity profiles for both phases with shifting the maximum toward the sheet. Most part of the boundary layer for a fixed η, the lateral velocity profile along with boundary layer thickness decreases with M2 for both the phases.
The effect of fluid-particle interaction parameter on the main flow for both phases was the similar result observed by Gireesha et al. [29]. It is clear from Figs. 6 and 7 that, an increase in fluid-particle interaction parameter leads to enhance the axial and transverse velocity of dust phase whereas opposite phenomenon is observed for fluid phase. It is evident from plots 8 and 9 respectively that, increase in the dust particles mass concentration parameter is to decrease the axial and transverse velocity profiles for both the phases. This is because; the presence of dust particles produces friction force in the fluid, which retards the flow. Further, it is observed that the axial and transverse velocity profile is higher for ordinary viscous fluid (l = 0) than that of dusty viscous fluid (l ≠ 0). Figures 10 and 11 elucidate that, as expected the opposite results are found for suction
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Fig. 6 The effect of βv on axial velocity profile
Fig. 7 The effect of βv on transverse velocity profile
and blowing. Blowing causes increase in axial and transverse velocity profile for both phases, although by increasing suction parameter the axial and transverse velocity profiles notably decreases near the boundary. This is due to the fact that, while stronger blowing is provided, the heated fluid is pushed farther away from the sheet, where due to less effect of viscosity, the flow is accelerated. This effect acts to increase maximum axial and transverse velocity within the boundary layer. The same principle is acted but in reverse direction in the case of suction. It is noted from Figs. 12 and 13 respectively that, both axial and transverse velocity profiles are strictly decreasing for increasing values of porous parameter. This is due to fact that, the presence of porous medium is to increase the resistance to the flow, which causes the fluid velocity to
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Fig. 8 The effect of l on axial velocity profile
Fig. 9 The effect of l on transverse velocity profile
decrease, associated with a decrease in momentum boundary layer thickness. Figures 14, 15, 16, 17, 18, 19, 20, 21, 22 and 23 are respectively plotted to show the influence of A∗ , B∗ , m, l, Ec, Pr, R, S, M 2 and k0 on both fluid and dust particles temperature distribution in PST and PHF cases. It is important to note that, A∗ > 0, B∗ > 0 corresponds to internal heat generation and A∗ < 0, B∗ < 0 corresponds to internal heat absorption. It is the cumulative effect of the space-dependent and temperature-dependent heat source/ sink parameter that determines the extent to which the temperature falls or rises in the boundary layer region. The influence of A∗ and B∗ on thermal field is qualitatively consistent with Abel et al. [24]. From the plots 13 and 14, it is observed that, the energy is released for increasing values
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Fig. 10 The effect of S on axial velocity profile
Fig. 12 The effect of k0 on axial velocity profile
Fig. 11 The effect of S on transverse velocity profile
Fig. 13 The effect of k0 on transverse velocity profile
of A∗ > 0 and B∗ > 0 and this causes the magnitude of temperature to increase both in PST and PHF cases, where as energy is absorbed for increasing values of A∗ < 0 and B∗ < 0 resulting in temperature dropping significantly near the boundary layer. Further, it is possible to see that thermal boundary layer for heat source is thicker than heat sink case. The fluid and dust particles temperature profile notably decreases for increasing the hall parameter in both PST and PHF cases and it is depicted in the Fig. 16. Figure 17 represents that increase in the dust particles mass concentration parameter decreases the heat transfer in thermal boundary layer of both fluid and dust phase in PST and PHF cases. The central reason for this effect that, the presence of dust particles in clean viscous fluid tends to absorb the heat, when they come into contact, which
has the tendency to diminish the temperature profile. It is also note that, temperature of clean fluid i.e., l = 0 is higher than that of dusty fluid i.e., l ≠ 0, corresponding heat transfer rate is higher in dusty fluid than clean viscous fluid. Thus, dusty fluid is more preferable in the context of cooling processes. Figure 18 elucidates that increase in the Eckert number used to strictly increase the fluid and particle temperature profiles. This trend is quiet opposite for the effect of increasing values of Prandtl number on temperature distributions of both phases in PST and PHF cases, which is illustrated in the Fig. 19. Physically, the higher Prandtl number fluid has relatively low thermal conductivity, which reduce the conduction and thermal boundary layer thickness and as a result temperature profile trim downs.
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Fig. 14 The effect of A∗ on temperature profile in both PST and PHF case
Fig. 16 The effect of m on temperature profile in both PST and PHF case
Fig. 15 The effect of B∗ on temperature profile in both PST and PHF case
Fig. 17 The effect of l on temperature profile in both PST and PHF case
Figure 20 displays that increasing the radiation parameter increases both the fluid and dust particles temperature profiles in PST and PHF cases, consequently brings about decrease in rate of heat transfer. This is due to the fact that, as radiation parameter increases, mean Rosseland absorption co-efficient k ∗ decreases, this cause increase in temperature profiles. Thus radiation should
be at its minimum in order to facilitate the cooling process. Figure 21 shows that, both fluid and dust particles temperature profile decreases for blowing (S < 0) whereas enhances for influence the suction (S > 0). As expected that, the temperature profile of both fluid and particles phase increases for increasing values of magnetic parameter, which is illustrated in plot 21. Figure 23 elucidates
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Fig. 18 The effect of Ec on temperature profile in both PST and PHF case
Fig. 20 The effect of R on temperature profile in both PST and PHF case
Fig. 19 The effect of Pr on temperature profile in both PST and PHF case
Fig. 21 The effect of S on temperature profile in both PST and PHF case
that increase in porous parameter enhances the temperature profile of both phases, and corresponding thermal boundary layer thickness is increases. Physically speaking, the higher values of porous parameter produce higher restriction on the fluid motion, which is responsible for increasing the temperature profile. Most interestingly, it is observed from the all plots that, velocity and temperature
profiles of dust particles are parallel to that of fluid and fluid phase velocity and temperature profiles are higher than that of dust phase. Moreover, It is noted from the Figs. 14, 15, 16, 17, 18, 19, 20, 21, 22 and 23 that, the temperature profile is higher in PHF heating process as compared with PST, thus PHF thermal heating process is better suited for cooling processes.
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908
Heat Mass Transfer (2016) 52:897–911
Fig. 24 The effect of l on skin friction co-efficient versus S Fig. 22 The effect of M2 on temperature profile in both PST and PHF case
Fig. 25 The effect of m on skin friction co-efficient versus S
Fig. 23 The effect of k0 on temperature profile in both PST and PHF case
We now move over to a discussion on the skin friction at the stretching sheet along x-direction and z-direction. Figures 24 and 25 are plotted to depict the effect of dust particles mass concentration parameter and Hall parameter on f ′′ (0) and h′ (0) profiles versus suction/blowing parameter
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respectively. It is shown from plot 23 that, f ′′ (0) and h′ (0) profiles decreases as increase in l. Further, f ′′ (0) and h′ (0) profiles are higher for clean viscous fluid (l = 0) than that of dusty viscous fluid (l ≠ 0). On the other hand, as m increases f ′′ (0) and h′ (0) profiles also increases, which is presented in plot 24. It is also observed that, in the absence of Hall parameter, f ′′ (0) profile is smaller and h′ (0) profile vanishes. Furthermore, we can see from plots 23 and 24 that, higher values of suction/blowing parameter retards the f ′′ (0) and h′ (0) profiles. Influence of l and m on Nusselt number has been illustrated in Fig. 26. It is shown that, the
Heat Mass Transfer (2016) 52:897–911
909
rate of heat transfer is high for higher values of Hall parameter and rate of heat transfer is lower in ordinary viscous fluid (l = 0) than that of dusty viscous fluid (l ≠ 0). The rate of heat transfer −θ ′ (0) is decreases for increasing values of M2 and R is as shown in Fig. 27. Table 2 presents the numerical values of Nusselt number −θ ′ (0) and wall temperature θ(0) for various values of M2, R, A∗ and B∗ for clean fluid (l = 0) as well as dusty fluid (l ≠ 0). It reveals that, the rate of heat transfer is decreases for increasing values of M2, R, A∗ and B∗. This means that, in order to facilitate the cooling process M2, R, A∗ and B∗ are should be at its minimum. Further, it is noticed that the
Fig. 26 The effect of m and l on Nusselt number profile versus S Table 2 Numerical values −θ ′ (0) and θ(0) for different values of, M 2 , R, A∗ and B∗ for ordinary fluid and dusty fluid
M2
R
A∗
rate of heat transfer is higher for dusty fluid than that of clean fluid. Finally, Table 3 presents the numerical values of f ′′ (0) and h′ (0) for various values of k0, M2, l and S with Hall current (m ≠ 0) and without Hall current (m = 0). The f ′′ (0) profile higher in the presence of Hall current and h′ (0) profile vanishes in the absence of Hall current (Fig. 27).
6 Concluding remarks Two-phase boundary layer flow of radiating dusty fluid past a non-isothermal stretching sheet in the presence of non-uniform internal heat source/sink and Hall current has been investigated. The resulting mathematical problems have been solved numerically and obtained results have an excellent agreement with existing one for some limiting case. There is decrease in momentum and thermal boundary layer thickness with an increase in l. It is observed that, the rate of heat transfer is higher for dusty viscous fluid than that of clean fluid, thus effect of suspended dust particles play a very important role. Fluid phase velocity suppresses for increase in fluid-particle interaction parameter but particle phase velocity accelerates. Moreover, both axial and transverse velocity increases and momentum boundary layer thickens with increase in Hall parameter, where as opposite trend has been observed for increase in M2, l, S and k0. The temperature profile of both phase and its corresponding thermal boundary layer thickness are increasing function of M 2 , k0 , A∗ , B∗ , R and Ec and decreasing function of l, m and Pr. The rate of transfer decreases for increase in M 2 , R, A∗ and B∗ thus in order to facilitate the cooling process M 2 , R, A∗ and B∗ are should be at its minimum. In the absence of Hall current h′ (0) profile get B∗
l = 0 −θ ′ (0)
0.0 1.0 2.0 0.5
1.0
0.1
0.1
0.0
0.1
0.1
0.5
1.0
0.1
0.5
1.0
−0.5 0.0 0.5 0.1
l = 1 θ(0)
−θ ′ (0)
θ(0)
−0.5 0.0
0.50046 0.47805 0.45154 0.84916 0.60605 0.37856 0.65494 0.51777 0.38061 0.75525 0.54821
1.94489 2.03446 2.15205 1.16651 1.61262 2.56677 1.66641 1.93132 2.19622 1.31447 1.78632
0.56042 0.54154 0.51795 0.95397 0.68466 0.42095 0.70029 0.57668 0.45306 0.79518 0.60325
1.72543 1.78402 1.86307 1.04403 1.42362 2.28572 1.50252 1.70978 1.91705 1.24539 1.61365
0.5
0.02220
17.16481
0.20106
4.13557
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910 Table 3 Numerical values of f ′′ (0) and h′ (0) for different values of k0, M2, l and S for with and without Hall current
Heat Mass Transfer (2016) 52:897–911 k0
0.0 1.0 2.0 0.5
M2
l
S
m = 0
0.5
0.5
−0.1
0.5
−0.1
0.5
0.0 0.5 1.5 0.5
−0.1
0.5
0.5
0.0 0.7 1.4 0.5
−0.1 0.0 0.1
Fig. 27 The effect of M2 and R on Nusselt number profile versus S
vanishes. Finally, the prescribed heat flux boundary condition is better suited for effective cooling of the stretching sheet. Acknowledgments One of the authors (B. J. Gireesha) is thankful to the University Grants Commission, India, for the financial support under the scheme of Raman Fellowship for Post-Doctoral Research for Indian Scholars in USA.
References 1. Sakiadis BC (1961) Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE 7:26–28
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m = 1
f ′′ (0)
h′ (0)
f ′′ (0)
h′ (0)
−1.23908 −1.58075 −1.86242 −1.23908 −1.41985 −1.72716 −1.36510 −1.44112 −1.51300 −1.41985 −1.47196
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
−1.14558 −1.50500 −1.79747 −1.23908 −1.33638 1.52501 −1.27838 −1.35886 −1.43460 −1.33638 −1.38840
0.12397 0.08799 0.07193 0.00000 0.10149 −0.25983 0.10537 0.10005 0.09547 0.10149 0.10108
−1.52542
0.0
−1.44188
0.10052
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