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Computational treatment of free convection effect on flow of elastico-viscous fluid past an accelerated plate with constant heat flux
Applied Mathematics and Computation 217 (2010) 685–688
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Computational treatment of free convection effect on flow of elastico-viscous fluid past an accelerated plate with constant heat flux J. Singh a, S.K. Gupta a,*, Srinivasan Chandrasekaran b a b
Hydraulics & Water Resources Engineering, Department of Civil Engineering, Institute of Technology, BHU, Varanasi 221005, UP, India Department of Ocean Engineering, Indian Institute of Technology Madras, Chennai, India
a r t i c l e
i n f o
Keywords: Free convection Visco-elastic fluid Laplace transform
a b s t r a c t Effect of free convection on the visco-elastic fluid (Walter – B’ type) flow past an infinite vertical plate accelerating in its own plane with constant heat flux is examined analytically. It is found that for given values of Grashof number, Prandtl number and Newtonian parameter; flow velocity at any point increases with the increase in time and non-Newtonian parameter, however, it decreases with both, the heating and cooling of the plate. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Several industrial applications involve the flow of non-Newtonian fluids, and thus the flow behaviour of such fluids finds a great relevance. Molten metals, plastic, pulps, emulsions, slurries and raw materials in fluid state are some examples to mention. Non-Newtonian flow also finds practical applications in bio-engineering, wherein blood circulation in human/animal artery is explained by an appropriate visco-elastic fluid model of small elasticity. The study of a visco-elastic pulsatile flow helps in understanding the mechanism of dialysis of blood through an artificial kidney. The constitutive equations of certain class of non-Newtonian fluids with short memories have been proposed by Walters [1] and Beard and Walters [2] for elastic-viscous fluid, referred to as Walters Liquid B’. The flow of viscous incompressible fluid past an impulsively started infinite horizontal plate in its own plane was first studied by Stokes [3]. Raptis et al. [4], using these equation, studied the influence of free convection and mass transfer on flow through a porous medium. Raptis and Predikis [5] studied the influence of free convection and mass transfer on oscillatory flow through a porous medium. The flow of visco-elastic incompressible and electrically conducting fluid past an infinite plate in presence of a transverse magnetic field, when the plate executes simple harmonic motion parallel to itself has been discussed by Sheriet and Ezzat [6]. The effects of suction, free oscillations and free convection currents on flow have been studied by Soundalgekar and Patil [7]. Singh [8] studied the mass transfer effect on the flow past an accelerated vertical plate with constant heat flux. Singh and Singh [9] studied the transient hydromagnetic free convection flow past an impulsively started vertical plate. Singh [10] studied the flow of elasto-viscous fluid past an accelerated porous plate. Recently, researchers [11–15] studied the several problems related to the flow of Walter’s Liquid B’. The present study investigates the effect of free convection on the flow of Walters Liquid B’ past an accelerating (in its own plane) infinite vertical plate subjected to the constant heat flux. 2. Problem formulation and solution We consider unsteady free convection flow of an incompressible elastic-viscous fluid past an infinite vertical plate. The x0 axis is taken along the plate in the upward direction and y0 -axis is taken normal to it. Initially the plate is at rest but at time * Corresponding author. E-mail address: [email protected] (S.K. Gupta). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.06.005
686
J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688
t0 = 0 the plate is accelerated with a velocity u0 = U t0 in its own plane and heat is also supplied to the plate at the constant rate. The equations which govern the free convection flow of an elastic-viscous fluid under the usual Boussinesq’s approximations in dimensionless form are given below:
@u @ 2 u @3u ¼ 2 þ Gr h 2 @t @y @y @t
ð1Þ
@h 1 @ 2 h ¼ @t Pr @y2
ð2Þ
where, h is the non-dimensional temperature; Pr, is Prandtl number; Gr is the Grashof number. The initial and boundary conditions for velocity field and temperature field are
t 6 0; u ¼ 0; h ¼ 0 for all y ( u ¼ t; yh ¼ 1 at y ¼ 0 t > 0; u ¼ 0; h ¼ 0 as y ! 1
ð3Þ
The non-dimensional equations introduced in the above equations are defined as,
y ¼ y0
1=3 U
m2
t ¼ t0
;
1=3 U
0 2=3 1=3 ðT T 1 Þ kU ; q 2=3
m
0
h¼
Pr ¼
m
lC p k
;
K¼
K0 h
Gr ¼
2=3 U
;
u¼
u0 ðmUÞ1=3
m2=3 qgb kU
ð4Þ
4=3
m2
where, q is the constant heat flux per unit area, m is kinematic viscosity, K is elastic parameter, u is flow velocity. The solution of Eq. (2) under the boundary condition (3) has been obtained by Soundalgekar and Patil [7] and Georgantopoulos et al. [16]. We will now solve Eq. (1) under the boundary condition (3). Since Eq. (1) is third order differential equation when K – 0; and for K = 0, it reduces to an equation governing the Newtonian fluid flow. Mathematically, we need three boundary conditions to solve the third order differential equation for a unique solution. But there are only two boundary conditions. Therefore, to overcome this difficulty, following Beard and Walters [2], u may be taken as:
u ¼ u0 þ Ku1
ð5Þ 2
Substituting Eq. (5) in Eq. (1) and equating the coefficients of different powers of K and neglecting the powers of K , we get
@u0 @ 2 u0 þ Gr h ¼ @t @y2 2
ð6Þ
3
@u1 @ u1 @ u0 2 ¼ @t @y2 @y @t
ð7Þ
The corresponding boundary conditions are:
t 6 0; u0 ¼ 0; u1 ¼ 0 for all y u0 ¼ t; u1 ¼ 0 at y ¼ 0 t > 0; u0 ¼ 0; u1 ¼ 0 as y ! 1
ð8Þ
Applying the Laplace transform technique, the solution of (6) and (7) under the boundary conditions (8) are:
u0 ¼ tð1 þ 2g2 ÞerfcðgÞ 2pgffiffiptffi expðg2 Þ hn 2 o 3=2 pgffiffiffi expðg2 Þ g ð3 þ 2g2 ÞerfcðgÞ þ Gr ð4tÞpffiffiffiffi 1þ 12 6 p ðPr 1Þ
Pr
n 2 pffiffiffiffiffioi pffiffiffiffiffi g ð3 þ 2g2 Pr Þ Pr erfc g Pr 1þ6pg ffiffipffiPr expðg2 Pr Þ 12 pffiffiffiffiffi hn o 2G Pt p1ffiffiffi expðg2 Þ g erfcðgÞ u1 ¼ ðPr 1Þ2r p r n pffiffiffiffiffioi pffiffiffiffiffi p1ffiffipffi expðg2 Pr Þ g Pr erfc g Pr pffi pgffiffipffi expðg2 Þ Gr g pt ffiffiffiffi erfcðgÞ ðPr 1Þ
Pr
ð9Þ
ð10Þ
J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688
687
Fig. 1. Velocity profiles for uniformly accelerated plate.
Fig. 2. Velocity profiles for uniformly accelerated plate under cooling and heating conditions.
Now using Eq. (5), we get
2gt u ¼ tð1 þ 2g2 ÞerfcðgÞ pffiffiffiffi expðg2 Þ
p
Gr ð4tÞ3=2 1 þ g2 g pffiffiffiffiffi pffiffiffiffi expðg2 Þ þ ð3 þ 2g2 ÞerfcðgÞþ 12 6 p ðPr 1Þ Pr
pffiffiffiffiffi pffiffiffiffiffi 1 þ g2 Pr g pffiffiffiffi expðg2 Pr Þ ð3 þ 2g2 Pr Þ Pr erfc g Pr 12 6 p " pffiffiffiffiffiffiffi 2Gr Pr t 1 2 p ffiffiffi ffi expð g Þ g erfcð g Þ þK p ðPr 1Þ2
pffiffiffiffiffi pffiffiffiffiffi 1 pffiffiffiffi expðg2 Pr Þ g Pr erfc g Pr p pffiffi g Gr g t pffiffiffiffiffi erfcðgÞ pffiffiffiffi expðg2 Þ p ðPr 1Þ Pr where, g ¼ 2y
pffiffi t.
ð11Þ
688
J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688
Fig. 3. Velocity profiles for uniformly accelerated plate under heating condition at different Prandtl numbers.
3. Results and discussion For the purpose of discussing the results the numerical calculations are carried out for different values of t, Gr, Pr and K, respectively. Fig. 1 shows that for fixed values of t, Gr, Pr, the velocity at any point decreases with the increase of non-Newtonian parameter K. It is also clear from Fig. 1 that for fixed values of Gr, Pr and K, flow velocity increases with the increase in time parameter t. Fig. 2 reveals that flow velocity decreases with the increase in non-Newtonian parameter K for both heating and cooling of the plate (i.e. Gr < 0 or Gr > 0). It is also clear from Fig. 2 that, in case of relatively greater heating of the plate, velocity decreases; whereas it increases with the greater cooling of the plate. Fig. 3 shows that for fixed values of Gr, t and K, the velocity at any point increases with the increase in Pr. 4. Conclusions
(1) The velocity of visco-elastic fluid (Walter’s fluid B’) is less in comparison to that of Newtonian fluid. (2) The flow velocity increases with the increase in time. (3) The flow velocity increases with the increase in Prandtl number.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
K. Walters, Second-order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, New York, 1964. p. 507. D. Beard, K. Walters, Elastico-viscous boundary-layer flows. Part I. Two dimensional flow near a stagnation point, Proc. Camb. Phil. Soc. 60 (1964) 667. G.G. Stokes, Flow of viscous incompressible fluid past an impulsively started infinite horizontal plate, Cambridge Phil. Trans. 9 (1851) 8. A. Raptis, G. Tzivanidis, N. Kafousias, Free convection and mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction, Lett. Heat Mass Transfer 8 (1981) 417. A. Raptis, C. Perdikis, Oscillatory flow through a porous medium by the presence of free convection flow, Int. J. Eng. Sci. 23 (1985) 51. H. Sherief, M. Ezzat, A problem of a viscoelastic magnetohydrodynamic fluctuating boundary-layer flow past an infinite porous plate, Can. J. Phys. 71 (1994) 97. V. Soundalgekar, V. Patil, Unsteady mass transfer flow past a porous plate, Ind. J. Pure Appl. Math. 139 (1982) 399. A.K. Singh, Mass transfer effects on the flow past an accelerated vertical plate with constant heat flux, Astrophys. Space Sci. 97 (1980) 57. J.N. Singh, A.K. Singh, Transient Hydromagnetic free convection flow past an impulsively started vertical plate, Astrophys. Space Sci. 176 (1991) 97– 104. J. Singh, Flow of elasto-viscous fluid past an accelerated porous plates, Astrophys. Space Sci. 106 (1984) 41–45. M. Ezzat, M. Abd-Elaal, Free convection effects on a viscoelastic boundary-layer flow with one relaxation time through a porous medium, J. Franklin Inst. 334B (1997) 685. K.A. Ames, L.E. Payre, Continuous dependent results for an ill posed problems in non-linear visco-elasticity, ZAMP 48 (1997) 1. M. Zakaria, Magnetohydrodynamic viscoelastic boundary layer flow past a stretching plate and heat transfer, Appl. Math. Comput. 155 (2004) 165– 177. A.A. El-Bary, Computational treatment of free convection effects on perfectly conducting viscoelastic fluid, Appl. Math. Comput. 170 (2005) 801–820. Y. Youbing, Ke-Qin Zhu, Oscillating flow of viscoelastic fluid in a pipe with the fractional Maxwell model, Appl. Math. Comput. 173 (2006) 231–242. G.A. Georgantopoulos, D.N. Nanousis, C.L. Goudas, The effect of the mass transfer on the free convection flow in Stokes problems, Astrophys. Space Sci. 66 (1979) 13.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Computational treatment of free convection effect on flow of elastico-viscous fluid past an accelerated plate with constant heat flux J. Singh a, S.K. Gupta a,*, Srinivasan Chandrasekaran b a b
Hydraulics & Water Resources Engineering, Department of Civil Engineering, Institute of Technology, BHU, Varanasi 221005, UP, India Department of Ocean Engineering, Indian Institute of Technology Madras, Chennai, India
a r t i c l e
i n f o
Keywords: Free convection Visco-elastic fluid Laplace transform
a b s t r a c t Effect of free convection on the visco-elastic fluid (Walter – B’ type) flow past an infinite vertical plate accelerating in its own plane with constant heat flux is examined analytically. It is found that for given values of Grashof number, Prandtl number and Newtonian parameter; flow velocity at any point increases with the increase in time and non-Newtonian parameter, however, it decreases with both, the heating and cooling of the plate. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Several industrial applications involve the flow of non-Newtonian fluids, and thus the flow behaviour of such fluids finds a great relevance. Molten metals, plastic, pulps, emulsions, slurries and raw materials in fluid state are some examples to mention. Non-Newtonian flow also finds practical applications in bio-engineering, wherein blood circulation in human/animal artery is explained by an appropriate visco-elastic fluid model of small elasticity. The study of a visco-elastic pulsatile flow helps in understanding the mechanism of dialysis of blood through an artificial kidney. The constitutive equations of certain class of non-Newtonian fluids with short memories have been proposed by Walters [1] and Beard and Walters [2] for elastic-viscous fluid, referred to as Walters Liquid B’. The flow of viscous incompressible fluid past an impulsively started infinite horizontal plate in its own plane was first studied by Stokes [3]. Raptis et al. [4], using these equation, studied the influence of free convection and mass transfer on flow through a porous medium. Raptis and Predikis [5] studied the influence of free convection and mass transfer on oscillatory flow through a porous medium. The flow of visco-elastic incompressible and electrically conducting fluid past an infinite plate in presence of a transverse magnetic field, when the plate executes simple harmonic motion parallel to itself has been discussed by Sheriet and Ezzat [6]. The effects of suction, free oscillations and free convection currents on flow have been studied by Soundalgekar and Patil [7]. Singh [8] studied the mass transfer effect on the flow past an accelerated vertical plate with constant heat flux. Singh and Singh [9] studied the transient hydromagnetic free convection flow past an impulsively started vertical plate. Singh [10] studied the flow of elasto-viscous fluid past an accelerated porous plate. Recently, researchers [11–15] studied the several problems related to the flow of Walter’s Liquid B’. The present study investigates the effect of free convection on the flow of Walters Liquid B’ past an accelerating (in its own plane) infinite vertical plate subjected to the constant heat flux. 2. Problem formulation and solution We consider unsteady free convection flow of an incompressible elastic-viscous fluid past an infinite vertical plate. The x0 axis is taken along the plate in the upward direction and y0 -axis is taken normal to it. Initially the plate is at rest but at time * Corresponding author. E-mail address: [email protected] (S.K. Gupta). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.06.005
686
J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688
t0 = 0 the plate is accelerated with a velocity u0 = U t0 in its own plane and heat is also supplied to the plate at the constant rate. The equations which govern the free convection flow of an elastic-viscous fluid under the usual Boussinesq’s approximations in dimensionless form are given below:
@u @ 2 u @3u ¼ 2 þ Gr h 2 @t @y @y @t
ð1Þ
@h 1 @ 2 h ¼ @t Pr @y2
ð2Þ
where, h is the non-dimensional temperature; Pr, is Prandtl number; Gr is the Grashof number. The initial and boundary conditions for velocity field and temperature field are
t 6 0; u ¼ 0; h ¼ 0 for all y ( u ¼ t; yh ¼ 1 at y ¼ 0 t > 0; u ¼ 0; h ¼ 0 as y ! 1
ð3Þ
The non-dimensional equations introduced in the above equations are defined as,
y ¼ y0
1=3 U
m2
t ¼ t0
;
1=3 U
0 2=3 1=3 ðT T 1 Þ kU ; q 2=3
m
0
h¼
Pr ¼
m
lC p k
;
K¼
K0 h
Gr ¼
2=3 U
;
u¼
u0 ðmUÞ1=3
m2=3 qgb kU
ð4Þ
4=3
m2
where, q is the constant heat flux per unit area, m is kinematic viscosity, K is elastic parameter, u is flow velocity. The solution of Eq. (2) under the boundary condition (3) has been obtained by Soundalgekar and Patil [7] and Georgantopoulos et al. [16]. We will now solve Eq. (1) under the boundary condition (3). Since Eq. (1) is third order differential equation when K – 0; and for K = 0, it reduces to an equation governing the Newtonian fluid flow. Mathematically, we need three boundary conditions to solve the third order differential equation for a unique solution. But there are only two boundary conditions. Therefore, to overcome this difficulty, following Beard and Walters [2], u may be taken as:
u ¼ u0 þ Ku1
ð5Þ 2
Substituting Eq. (5) in Eq. (1) and equating the coefficients of different powers of K and neglecting the powers of K , we get
@u0 @ 2 u0 þ Gr h ¼ @t @y2 2
ð6Þ
3
@u1 @ u1 @ u0 2 ¼ @t @y2 @y @t
ð7Þ
The corresponding boundary conditions are:
t 6 0; u0 ¼ 0; u1 ¼ 0 for all y u0 ¼ t; u1 ¼ 0 at y ¼ 0 t > 0; u0 ¼ 0; u1 ¼ 0 as y ! 1
ð8Þ
Applying the Laplace transform technique, the solution of (6) and (7) under the boundary conditions (8) are:
u0 ¼ tð1 þ 2g2 ÞerfcðgÞ 2pgffiffiptffi expðg2 Þ hn 2 o 3=2 pgffiffiffi expðg2 Þ g ð3 þ 2g2 ÞerfcðgÞ þ Gr ð4tÞpffiffiffiffi 1þ 12 6 p ðPr 1Þ
Pr
n 2 pffiffiffiffiffioi pffiffiffiffiffi g ð3 þ 2g2 Pr Þ Pr erfc g Pr 1þ6pg ffiffipffiPr expðg2 Pr Þ 12 pffiffiffiffiffi hn o 2G Pt p1ffiffiffi expðg2 Þ g erfcðgÞ u1 ¼ ðPr 1Þ2r p r n pffiffiffiffiffioi pffiffiffiffiffi p1ffiffipffi expðg2 Pr Þ g Pr erfc g Pr pffi pgffiffipffi expðg2 Þ Gr g pt ffiffiffiffi erfcðgÞ ðPr 1Þ
Pr
ð9Þ
ð10Þ
J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688
687
Fig. 1. Velocity profiles for uniformly accelerated plate.
Fig. 2. Velocity profiles for uniformly accelerated plate under cooling and heating conditions.
Now using Eq. (5), we get
2gt u ¼ tð1 þ 2g2 ÞerfcðgÞ pffiffiffiffi expðg2 Þ
p
Gr ð4tÞ3=2 1 þ g2 g pffiffiffiffiffi pffiffiffiffi expðg2 Þ þ ð3 þ 2g2 ÞerfcðgÞþ 12 6 p ðPr 1Þ Pr
pffiffiffiffiffi pffiffiffiffiffi 1 þ g2 Pr g pffiffiffiffi expðg2 Pr Þ ð3 þ 2g2 Pr Þ Pr erfc g Pr 12 6 p " pffiffiffiffiffiffiffi 2Gr Pr t 1 2 p ffiffiffi ffi expð g Þ g erfcð g Þ þK p ðPr 1Þ2
pffiffiffiffiffi pffiffiffiffiffi 1 pffiffiffiffi expðg2 Pr Þ g Pr erfc g Pr p pffiffi g Gr g t pffiffiffiffiffi erfcðgÞ pffiffiffiffi expðg2 Þ p ðPr 1Þ Pr where, g ¼ 2y
pffiffi t.
ð11Þ
688
J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688
Fig. 3. Velocity profiles for uniformly accelerated plate under heating condition at different Prandtl numbers.
3. Results and discussion For the purpose of discussing the results the numerical calculations are carried out for different values of t, Gr, Pr and K, respectively. Fig. 1 shows that for fixed values of t, Gr, Pr, the velocity at any point decreases with the increase of non-Newtonian parameter K. It is also clear from Fig. 1 that for fixed values of Gr, Pr and K, flow velocity increases with the increase in time parameter t. Fig. 2 reveals that flow velocity decreases with the increase in non-Newtonian parameter K for both heating and cooling of the plate (i.e. Gr < 0 or Gr > 0). It is also clear from Fig. 2 that, in case of relatively greater heating of the plate, velocity decreases; whereas it increases with the greater cooling of the plate. Fig. 3 shows that for fixed values of Gr, t and K, the velocity at any point increases with the increase in Pr. 4. Conclusions
(1) The velocity of visco-elastic fluid (Walter’s fluid B’) is less in comparison to that of Newtonian fluid. (2) The flow velocity increases with the increase in time. (3) The flow velocity increases with the increase in Prandtl number.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
K. Walters, Second-order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon, New York, 1964. p. 507. D. Beard, K. Walters, Elastico-viscous boundary-layer flows. Part I. Two dimensional flow near a stagnation point, Proc. Camb. Phil. Soc. 60 (1964) 667. G.G. Stokes, Flow of viscous incompressible fluid past an impulsively started infinite horizontal plate, Cambridge Phil. Trans. 9 (1851) 8. A. Raptis, G. Tzivanidis, N. Kafousias, Free convection and mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction, Lett. Heat Mass Transfer 8 (1981) 417. A. Raptis, C. Perdikis, Oscillatory flow through a porous medium by the presence of free convection flow, Int. J. Eng. Sci. 23 (1985) 51. H. Sherief, M. Ezzat, A problem of a viscoelastic magnetohydrodynamic fluctuating boundary-layer flow past an infinite porous plate, Can. J. Phys. 71 (1994) 97. V. Soundalgekar, V. Patil, Unsteady mass transfer flow past a porous plate, Ind. J. Pure Appl. Math. 139 (1982) 399. A.K. Singh, Mass transfer effects on the flow past an accelerated vertical plate with constant heat flux, Astrophys. Space Sci. 97 (1980) 57. J.N. Singh, A.K. Singh, Transient Hydromagnetic free convection flow past an impulsively started vertical plate, Astrophys. Space Sci. 176 (1991) 97– 104. J. Singh, Flow of elasto-viscous fluid past an accelerated porous plates, Astrophys. Space Sci. 106 (1984) 41–45. M. Ezzat, M. Abd-Elaal, Free convection effects on a viscoelastic boundary-layer flow with one relaxation time through a porous medium, J. Franklin Inst. 334B (1997) 685. K.A. Ames, L.E. Payre, Continuous dependent results for an ill posed problems in non-linear visco-elasticity, ZAMP 48 (1997) 1. M. Zakaria, Magnetohydrodynamic viscoelastic boundary layer flow past a stretching plate and heat transfer, Appl. Math. Comput. 155 (2004) 165– 177. A.A. El-Bary, Computational treatment of free convection effects on perfectly conducting viscoelastic fluid, Appl. Math. Comput. 170 (2005) 801–820. Y. Youbing, Ke-Qin Zhu, Oscillating flow of viscoelastic fluid in a pipe with the fractional Maxwell model, Appl. Math. Comput. 173 (2006) 231–242. G.A. Georgantopoulos, D.N. Nanousis, C.L. Goudas, The effect of the mass transfer on the free convection flow in Stokes problems, Astrophys. Space Sci. 66 (1979) 13.