Afr. Mat. (2012) 23:229–241 DOI 10.1007/s13370-011-0031-0
Unsteady hydromagnetics boundary layer flow and heat transfer of dusty fluid over a stretching sheet B. J. Gireesha · S. Manjunatha · C. S. Bagewadi
Received: 11 December 2010 / Accepted: 24 May 2011 / Published online: 22 June 2011 © African Mathematical Union and Springer-Verlag 2011
Abstract The aim of this paper is to investigate the effect of magnetic field on an unsteady boundary layer flow and heat transfer of a dusty fluid over a stretching surface. The fluid is assumed to be viscous and incompressible. The governing partial differential equations are reduced to coupled non-linear ordinary differential equations by similarity transformation. Numerical solutions of these coupled non-linear equations are obtained by using RKF-45 method. The solution is found to be dependent on governing parameters including Hartman number (M), unsteadiness parameter (A), Prandtl number (P r), dust interaction parameter (β), suction parameter (R) and Eckert number (Ec). Comparison of numerical results is made with previously published results under the special cases, and found to be in good agreement. Keywords Unsteady flow and heat transfer · Boundary layer flow · Stretching porous surface · Dusty fluid · Numerical solution
Mathematics Subject Classification (2000)
76T15 · 80A20
1 Introduction From the technological point of view, the study of boundary layer flow and heat transfer of dusty fluid on continuous, moving solid surfaces is always important. The analysis of such flow finds numerous and wide-range applications in industrial manufacturing processes, such as polymer extrusion, drawing of copper wires, continuous stretching of plastic films and artificial fibers, hot rolling, wire drawing, glass-fiber, metal extrusion, and metal
B. J. Gireesha (B) · S. Manjunatha · C. S. Bagewadi Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, Shimoga 577 451, Karnataka, India e-mail:
[email protected]
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spinning. The investigations were made by many researchers such as, Sakiadis [1] initiated the study of the boundary layer flow over a stretched surface moving with a constant velocity and formulated a boundary-layer equation for two-dimensional and axisymmetric flows. Tsou et al. [2] analyzed the effect of heat transfer in the boundary layer on a continuous moving surface with a constant velocity and experimentally confirmed the numerical results of Sakiadis [1]. Carragher and Crane [3] investigated the heat transfer in the flow over a stretching surface in the case when the temperature difference between the surface and the ambient fluid is proportional to a power of distance from the fixed point. Vajravelu and Nayfeh [4] has discussed hydromagnetic flow of a dusty fluid over a stretching sheet. Sharidan et al. [5] has studied similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet. Abel et al. [6] have studied the flow and heat transfer in a viscoelastic boundary layer flow over a stretching sheet with prescribed surface temperature (PST) case and prescribed heat flux (PHF) case. Elbashbeshy and Aldawody [7] has studied the effect of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over a porous stretching surface. Grubka and Bobba [8] has studied the temperature field in the flow over a stretching surface subjected to a uniform heat flux. Ogulu and Makinde [9] has studied unsteady hydromagnetic free convection flow of a dissipative and radiating fluid past a vertical plate with constant heat flux. Makinde [10] has discussed similarity solution of hydromagnetic heat and mass transfer over a vertical plate with a convective surface boundary condition. Andersson et al. [11] presented a new similarity solution for the temperature fields is devised, which transforms the time-dependent thermal energy equation to an ordinary differential equation. Aziz [12] obtained the numerical solution for laminar thermal boundary over a flat plate with a convective surface boundary condition using the symbolic algebra software Maple. The governing similarity equations contains Prandtl number, Eckret number, number density and unsteadiness parameter. Although a similarity solution is accomplished by these authors, some physically unrealistic phenomena are encountered for specific values of the unsteadiness parameter. The aim of the analysis is to study the unsteady flow and heat transfer of dusty fluid over a stretching sheet in a viscous and incompressible fluid which is at rest under the similarity conditions considered by Elbashbeshy and Bazid [13]. In addition, both the variable wall temperature (VWT) and variable heat flux (VHF) conditions have been considered. The governing equations are solved numerically using RKF-45 method with the help of Maple.
2 Mathematical formulation and solution of the problem Consider the two-dimensional unsteady boundary layer flow of a dusty viscous and incompressible fluid (with electric conductivity σ0 ) past semi-infinite stretching sheet in the region y > 0. Two equal and opposite forces are introduced along the x-axis so that the wall is stretched with a speed proportional to the distance from the origin. A uniform magnetic field B0 is imposed along the y-axis (Fig. 1). The unsteady two-dimensional boundary layer equations of dusty fluid in usual notation are: ∂u ∂v + = 0, (2.1) ∂x ∂y σ B02 ∂u ∂u μ ∂ 2 u KN ∂u +u +v = (u − up ) − u, + 2 ∂t ∂x ∂y ρ ∂y ρ ρ
123
(2.2)
Unsteady hydromagnetics boundary layer flow
231
Fig. 1 Schematic diagram of the flow geometry
∂up ∂up ∂up K + up + vp = (u − up ), ∂t ∂x ∂y m
(2.3)
∂vp ∂vp ∂vp K + up + vp = (v − vp ), ∂t ∂x ∂y m
(2.4)
∂(ρp up ) ∂(ρp vp ) + = 0. ∂x ∂y
(2.5)
We have the following nomenclature: (u, v) and (up , vp ) denote the velocity components of the fluid and particle phase along the x- and y-axes, respectively. Furthermore μ, ρ, B0 , ρp and τ are the coefficients of viscosity of fluid, density of the fluid, induced magnetic field, density of particle phase, and the relaxation time of particle, respectively. In deriving these equations, the Stokesian drag force is considered for the interaction between the fluid and particle phase and the induced magnetic field is neglected. It is also assumed that the external electric field is zero and the electric field due to polarization of charges is negligible. The boundary conditions applicable to the above problem are: u = Uw (x, t), v = Vw (x, t) at y = 0, u −→ 0, up −→ 0, vp −→ v, ρp −→ kp as y −→ ∞,
(2.6)
v0 cx is velocity of sheet, Vw = − √1−αt is suction velocity and c is where Uw (x, t) = 1−αt stretching rate being a positive constant, α is positive constant which measures the unsteadiness. Equations (2.1)–(2.5) subjected to boundary condition (2.6), admit self-similar solution in terms of the similarity function f and the similarity variable η defined by
cx cν u= f (η), v = − f (η), 1 − αt 1 − αt cν cx up = F (η), vp = G(η), 1 − αt 1 − αt c y, ρr = H (η), η= ν(1 − αt)
(2.7)
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where a prime denotes the differentiation with respect to η. Substituting the equation (2.7) into equations (2.1)–(2.5) gives η 2 f (η) + f (η)f (η) − f (η) − A f (η) + f (η) 2 + lβH F (η) − f (η) − Mf (η) = 0, η A F (η) + F (η) + G(η)F (η) + F (η)2 + β F (η) − f (η) = 0, 2
(2.8)
A G(η) + ηG (η) + G(η)G (η) + β [f (η) + G(η)] = 0, 2
(2.10)
H (η)F (η) + G (η)H (η) + G(η)H (η) = 0,
(2.11)
where ρr =
ρp ρ
is relative density, A =
measures unsteadiness, l =
mN ρp
α c
(2.9)
is non-dimensional unsteady parameter which
is mass concentration, M =
σ B02 ρc (1
− αt) is magnetic
1 τ c (1 − αt) is fluid particle interaction parameter. The corresponding bound-
parameter, β = ary conditions are transformed to:
f (η) = 1, f (η) = R, at η = 0, f (η) = 0, F (η) = 0, G(η) = −f (η), H (η) = E, as η −→ ∞, (2.12) 0 is suction parameter. where R = √vνc If A = 0, the analytical and numerical solution of equations (2.1)–(2.5) was given by Vajravelu and Nayfeh [4].
3 Heat transfer analysis The unsteady thermal boundary layer equations in space for two dimensional flow is given by [14] ρcp
N cp ∂ 2T ∂T ∂T ∂T N +u +v = k∗ 2 + (Tp − T ) + (up − u)2 , ∂t ∂x ∂y ∂y τT τv ∂Tp ∂Tp ∂Tp N cp (Tp − T ), N cm + up + vp =− ∂t ∂x ∂y τT
(3.1) (3.2)
where T and Tp is the temperature of the fluid and dust particle, cp and cm are the specific heat of fluid and dust particles, τT is the Thermal equilibrium time and is time required by the dust cloud to adjust its temperature to the fluid, τv is the relaxation time of the of dust particle i.e., the time required by the a dust particle to adjust its velocity relative to the gas, k ∗ is the thermal conductivity. The solution of equations (3.1)–(3.2) depends on the nature
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Unsteady hydromagnetics boundary layer flow
233
of the prescribed boundary condition. The two types of heating processes are discussed below. Case 1: Variable wall temperature (VWT) For this heating process, the variable wall temperature is assumed to be a quadratic function of x and it is given by T = Tw = T∞ + T0 T −→ T∞ ,
cx 2 ν
(1 − αt)−2 at y = 0,
Tp −→ T∞ as y −→ ∞,
(3.3)
where Tw is the temperature of the wall and T∞ is the constant temperature far away from the sheet. In order to obtain similarity solution for the temperatures θ (η) and θp (η), we define dimensionless temperature variables as follows: θ (η) =
Tp − T∞ T − T∞ , θp (η) = , Tw − T∞ Tw − T∞
(3.4)
where T − T∞ = T0 ( cxν )(1 − αt)−2 θ (η). Using equations (3.3) and (3.4) in the equations (3.1) and (3.2), we get 2
A θ (η) + P r f (η)θ (η) − 2f (η)θ (η) − P r 4θ (η) + ηθ (η) 2 + N P ra1 θp (η) − θ (η) + N P rEca2 [F (η) − f (η)]2 = 0, G(η)θp (η) + 2F (η)θp (η) + where P r = c
μcp k∗
(3.5)
A 4θp (η) + ηθ (η) + b1 θp (η) − θ (η) = 0, (3.6) 2 ν2 cp T 0
is Prandtl number, Ec =
is the Eckret number, a1 = τT1ρc (1 − αt),
b1 = τT cpm c (1 − αt) are local fluid particle interaction parameter for heat transfer and a2 = 1 τv ρν (1 − αt) is local fluid particle interaction parameter of velocity. Using the equations (3.3) and (3.4) the corresponding boundary conditions for θ (η) and θp (η) reduces to following form θ (η) = 1 at η = 0, θ (η) −→ 0, θp (η) −→ 0 as η −→ ∞.
(3.7)
Case 2: Variable heat flux (VHF) In this heating process we employ the following variable heat flux boundary conditions. ∂T qw(x,t) at y = 0, =− ∗ ∂y k T −→ T∞ , Tp −→ T∞ as y −→ ∞ where qw (x, t) = qw0 x 2 ( νc ) 2 (1 − αt) 3
−5 2
(3.8)
.
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In order to obtain similarity solution for temperature we define the dimensionless q cx 2 temperature variables in VHF case as in equation (3.4) where as Tw − T∞ = kw∗0 ν(1−αt) 2 θ (η). With this dimensionless variable the temperature equations (3.1) and (3.2) takes the form A θ (η) + P r f (η)θ (η) − 2f (η)θ (η) − P r 4θ (η) − ηθ (η) 2 + N P ra1 θp (η) − θ (η) + N P rEc a2 [F (η) − f (η)]2 = 0, G(η)θp (η) + 2F (η)θp (η) +
A 4θp (η) + ηθ (η) + b1 θp (η) − θ (η) = 0. 2
(3.9) (3.10)
The corresponding boundary conditions becomes θ (η) = −1 at η = 0,
(3.11)
θ (η) = 0, θp (η) = 0 as η −→ ∞.
(3.12)
4 Numerical solution The equations (2.8) and (2.11) together with the boundary condition (2.12) forms highly nonlinear ordinary differential equations. In order solve these non-linear equations numerically we adopted symbolic software Maple which was given by Aziz [12] and it is very efficient in using the well known Runge Kutta Fehlberg fourth-fifth order method (RKF45 Method). In accordance with the boundary layer analysis, the boundary condition (2.12) at η = ∞ were replaced by η = 5. The coupled boundary value problem equations (2.8)–(2.11) and either equations (3.5) and (3.6) or (3.9) and (3.10) were solved by RKF45 method. The accuracy of this numerical method was validated by direct comparison with the numerical results reported by Vajravelu and Nayfeh [4] for the steady-state flow case (A = 0, M = 3, R = 2, β = 0). Table 1 represent results of this comparison for f (η), f (η) , f (η). It can be seen from this table that a very good agreement between the results.
5 Results and discussion An unsteady dusty boundary layer problem for momentum and heat transfer with space for an incompressible fluid flow over a stretching sheet is examined in this paper. The boundary layer equations of momentum and heat transfer are solved numerically. The temperature profile θ (η) in VWT case and in θ (η) VHF case depicted graphically. The computation through employed numerical scheme has been carried out for various values of the parameters such as unsteadiness parameter A, magnetic parameter M, suction parameter R, fluid particle interaction parameter β, Prandtl number P r, number density N and Eckert number Ec. From Table 1 we note that there is close agreement with the results of previously published work by Vajravelu and Nayfeh [4]. And thus verify the accuracy of the method used.
123
Unsteady hydromagnetics boundary layer flow
235
Table 1 Comparison of numerical solutions of f (η), f (η), f (η) with A = 0, M = 3, R = 2, β = 0. η
Vajravelu and Nayfeh f (η)
f (η)
Present result f (η)
f (η)
f (η)
f (η)
0.00000
2.000000
1.000000
−3.236068
2.000000
1.00000
−3.236068
0.408163
2.226537
0.266910
−0.863739
2.226537
0.266910
−0.863739
0.816327
2.287002
0.071241
−0.230540
2.287002
0.071241
−0.230540
1.224490
2.303141
0.019015
−0.061534
2.303141
0.019015
−0.061534
1.632653
2.307448
0.005075
−0.016424
2.307449
0.005075
−0.016424
2.040816
2.308598
0.001355
−0.04384
2.308598
0.001355
−0.004384
2.448980
2.308905
0.000362
−0.001170
2.308905
0.000362
−0.001170
2.857143
2.308987
0.000096
−0.000312
2.308987
0.000097
−0.000312
3.265306
2.309009
0.000026
−0.000083
2.309009
0.000026
−0.000083
3.673469
2.309015
0.000007
−0.000022
2.309015
0.000007
−0.000022
4.081633
2.309016
0.000002
−0.000006
2.309016
0.000002
−0.000006
4.489796
2.309017
0.000000
−0.000002
2.309017
0.000000
−0.000002
4.897959
2.309017
0.000000
−0.000001
2.309017
0.000000
0.000000
5.000000
2.309017
0.000000
0.000000
2.3090167
0.000000
0.000000
Fig. 2 Effect of unsteady parameter A for fluid and dust velocity
Figure 2 represents horizontal velocity profile of both fluid and dust particles for various value of A when P r = 0.72, R = 2, M = 3, N = 3 and β = 0.5. From this one can observed that the velocity decreases with the increase of the unsteady parameter A. It is interesting to note that the thickness of boundary deceases with increasing values of A. This is due the fluid flow is caused solely by the stretching sheet. From the Fig. 3 it is observed that the velocity decreasing with increase of magnetic parameter M. This is because of fact that the introduction of transverse magnetic field to an electrically conducting fluid gives rise to a resistive type of force, known as Lorentz force.
123
236 Fig. 3 Effect of Hartman number M for fluid and dust velocity
Fig. 4 Effect of unsteady parameter A on temperature distribution for VWT and VHF case with P r = 0.72, Ec = 2, R = 2, N = 3, β = 0.5
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B. J. Gireesha et al.
Unsteady hydromagnetics boundary layer flow
237
Fig. 5 Effect of prandtl number P r on temperature distribution for VWT and VHF case with A = 0.6, Ec = 2, R = 2, N = 3, β = 0.5
This force has tendency to slowdown the motion of fluid in the boundary layer and hence it leads to enhanced deceleration of the flow. Figure 4a, b represents the temperature distribution for VWT and VHF case, for different values of unsteady parameter A versus η. It is evident from these graphs that temperature of fluid and dust particle is found to be decrease with increase of unsteady parameter A. This shows an important fact that the rate of cooling is much faster for higher values of unsteady parameter whereas it may take longer time for cooling during steady flow. This is due to the reason that sheet surface temperature is higher than free stream temperature. Throughout the thermal analysis we have considered a1 = a2 = b1 = 2. Figure 5a, b depicts the temperature profiles θ (η) and θp (η) versus η, for different values of P r. We infer from these figures that temperature of fluid and dust particles decreases with the increase in P r which implies viscous boundary layer is thicker than the thermal boundary layer. This is because there would be a decrease of thermal boundary layer thickness with the increase of value of Prandtl number P r. The increase of Prandtl number means slow rate of thermal diffusion. However, the effect of increasing value of Prandtl number P r is to increase temperature distribution near the boundary and decrease everywhere
123
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B. J. Gireesha et al.
Fig. 6 Effect of Eckert number Ec on temperature distribution for VHF case with A = 0.6, P r = 0.72, R = 2, N = 3, β = 0.5
away from the boundary in VWT case. However, it may be other than the unity in VHF case due to adiabatic temperature boundary condition. The results of VWT cases are qualitatively similar to that of VHF case but quantitatively they are different. The temperature in both VWT and VHF cases asymptotically approaches to zero in the free stream region. Figure 6a, b indicates the temperature profile θ (η) and θp (η) versus η, for VWT and VHF cases respectively. Here the effect of increasing values of Ec is to enhance the temperature of fluid and dust particles at any point which is true for both the cases VWT and VHF. This is due to fact that the heat energy is stored in the considered liquid due to frictional heating. Figure 7a and b are graphs of temperature profiles of θ (η) and θp (η) versus η, for different values of Number density N for VWT and VHF cases respectively. From the Fig. 7a, b observes that the temperature of fluid and dust particles decreases with the increase of N .
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Unsteady hydromagnetics boundary layer flow
239
Fig. 7 Effect of number density N on temperature distribution for VWT and VHF case with A = 0.6, P r = 0.72, R = 2, Ec = 2, β = 0.5
6 Conclusions Mathematically analysis has been carried out on momentum and heat transfer characteristics in an incompressible viscous unsteady boundary layer flow of a dusty fluid over a stretching sheet. The highly non-linear momentum equations (2.1)–(2.5) and heat transfer boundary layer equations (3.1)–(3.2) are converted into coupled ordinary differential equations by using similarity transformations. Resultant coupled ordinary differential equations (2.8)–(2.11) and (3.5)–(3.6) for VWT case and (3.9)–(3.10) for VHF case has been solved numerically by method employed by Aziz [12] i.e., RKF45 method. The effect of various physical parameter like unsteady parameter A, prandtl number P r, Eckret number Ec, Hartmann number M and number density N on various momentum and heat transfer characteristics are obtained. The results of thermal characteristics at the wall values of the temperature gradient function θ (0) in VWT case and temperature function θ (0) in VHF case are documented in Table 2. It reveals that the effects of increasing the values of β, A, P r and N is to decreases the wall temperature gradient function θ (0) and temperature function θ (0) and the effect of
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Table 2 Wall temperature gradient θ (0) and temperature function θ (0) for different values of the parameters β, A, Pr, Ec, and N. β
A
Pr
Ec
N
θ (η)(V W T case)
θ (η)(V H F case)
0.3
0.6
0.72
2
0.2
−2.10614
0.63177
0.6
−2.23401
0.59554
1.0
−2.33726
0.56449
−2.11653
0.61847
0.3
−2.22701
0.59067
0.6
−2.33725
0.56449
−2.33725
0.56449
1.0
−2.88930
0.51358
2.0
−4.59047
0.44574
−3.07060
0.32566
0.5
−2.88726
0.38537
2.0
−2.33725
0.56449
0.2
−2.52285
0.40584
0.5
−2.75481
0.38396
1.0
−3.0058
0.36922
1
1
1
1
0
0.6
0.6
0.6
0.72
0.72
0.72
0.72
2
2
0
2
0.2
0.2
0.2
increasing the values of Ec is to increases the temperature gradient function θ (0) and temperature function θ (0) for both VWT and VHF cases. Acknowledgments We wishes to express our thanks to UGC (University Grant commission), New Delhi for financial support to pursue this work under a Major Research Project (F.No.36-147/2008/(SR)/dated:2603-2009).
References 1. Sakiadis, B.C.: Boundary layer behaviour on continuous solid surface; I Boundary-layer equations for two-dimensional and axisymmetric flow. A. I. Ch. E. J. 7, 26–28 (1961) 2. Tsou, F.K., Sparrow, E.M., Glodstein, R.J.: Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transf. 10, 219–235 (1967) 3. Carragher, P., Crane, L.J.: Heat transfer on aconinuous stretching surface. Int. J. Appl. Math. Mech. (ZAMM) 62, 564–565 (1982) 4. Vajravelu, K., Nayfeh, J.: Hydromagnetic flow of a dusty fluid over a stretching sheet. Int. J. Nonlinear Mech. 27(6), 937–945 (1992) 5. Sharidan, S., Mahmood, T., Pop, I.: Similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet. Int. J. Appl. Mech. Eng. 11(3), 647–654 (2006) 6. Subhas Abel, M., Siddeshwar, P.G., Nandeppanavar, M.M.: Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source. Int. J. Heat Mass Transf. 50, 960–966 (2007) 7. Elbashbeshy, E.M.A., Aldawody, D.A.: Effect of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over a porous stretching surface. Int. J. Nonlinear Sciences 9(4), 448–454 (2010) 8. Grubka, L.J., Bobba, K.M.: Heat transfer characteristics of a contineous stretching surface with variable temperature. Int. J. Heat Mass Transf. 107, 248–250 (1985) 9. Ogulu, A., Makinde, O.D.: Unsteady hydromagnetic free convection flow of a dissipative and radiating fluid past a vertical plate with constant heat flux. Chem. Eng. Commun. 196(4), 454–462 (2009)
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10. Makinde, O.D.: Similarity solution of hydromagnetic heat and mass transfer over a vertical plate with a convective surface boundary condition. Int. J. Phys. Sci. 5(6), 700–710 (2010) 11. Andersson, H.T., Aareseth, J.B., Dandapat, B.S.: Heat transfer in a liquid film on an unsteady stretching surface. Int. J. Heat Mass Transf. 43, 69–74 (2000) 12. Aziz, A.: A similarity solution for laminar thermal boundary layer over aflat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simulat. 14, 1064–1068 (2009) 13. Elbashbeshy, E.M.A., Bazid, M.A.A.: Heat transfer over an unsteady stretching surface. Int. J. Heat Mass Transf. 41, 1–4 (2004) 14. Shercliff, J.A.: A Text Book of Magneto-Hydromagnetics. Pergamon press, London (1965) 15. Saffman, P.G.: On the stability of laminar flow of a dusty gas. J Fluid Mech. 13, 120–128 (1962)
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