JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org
ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(4): Pg.340-350
Three Dimensional Free Convective Flow with Heat and Mass Transfer of a Visco-elastic Fluid Along a Vertical Porous Plate Rita Choudhury1 and Hillol Kanti Bhattacharjee2 1,2
Department of Mathematics, Gauhati University, Guwahati, Assam, INDIA. (Received on: July 7, 2014) ABSTRACT The analytical study on the steady three dimensional flow of a visco-elastic fluid with heat and mass transfer along a vertical plate in presence of transverse sinusoidal suction velocity and uniform free stream velocity has been presented. The regular perturbation scheme has been used to solve the governing equations. The influence of various parameters entering into the problem on the velocity, temperature and concentration fields are discussed. The skin friction, the rate of heat and mass transfer coefficients at the walls are also obtained. The velocity and skin friction coefficients are illustrated graphically and the results are physically interpreted. It is observed that the effect of visco-elastic parameter is prominent in the flow field. Keywords: Free convection, heat transfer, mass transfer, viscoelastic, Walters liquid( Model ܤ′ ), Grashof number.
1. INTRODUCTION In the recent years, the phenomenon of three dimensional free convective flow through a porous medium with heat and mass transfer has attracted the interest of many researchers because of its possible applications in many branches of science and technology, particularly in natural
sciences, engineering sciences and industries. In nature and industrial applications, many transport processes exit where the heat and mass transfer takes place. Simultaneous diffusion of thermal energy and chemical species have many applications in geophysics and engineering. Such phenomena are observed in buoyancy induced motions in the atmosphere, in
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Rita Choudhury, et al., J. Comp. & Math. Sci. Vol.5 (4), 340-350 (2014)
bodies of water, quasi-solid bodies such as earth etc. The process of heat and mass transfer is encountered in aeronautical engineering in view of its application to reduce drag and hence the vehicle power requirement by a substantial amount. Initially this subject has been developed by Lachmann (1961). The theoretical and experimental research have shown that the transition from laminar to the turbulent flow which causes the drag co-efficient to increase, may be prevented by suction of the fluid and the heat and mass transfer from the boundary layer to the wall. Many works have been presented by the researchers in this areas viz. Raptis and Kafousias14, Acharya et al.2, Singh et al.15, Al-Qadat et al.3, Zueco et al.21 and Makinda10 etc. Again, to get any desired reduction in the drag by increasing suction alone is uneconomical as the energy consumption of the suction pump will more. In view of these investigations, Mishra et al.16 have studied the flow of viscous incompressible fluid along an infinite porous plate by applying the transverse sinusoidal suction velocity distribution fluctuating with time. Singh and Verma17 and Singh et al.18 have studied the effect of such a transverse permeability distribution of the porous medium bounded by horizontal flat plate.
effects of chemical reaction and thermo diffusion on MHD three dimensional free convection flows with heat absorption. Ling et al.9 discussed steady mixed convection boundary layer flow over a vertical flat plate in a porous medium filled with water at 40C. case of variable wall temperature. Again, the visco-elastic fluid has tremendous achievement in industrial applications. This type of fluid combines both viscous and elastic properties. There is a large body of literature on visco-elastic flows and some of the recent studies, for example, Hayat et al.8, Abel et al.1, Murthy et al.11, Mustafa et al.12, Choudhury et al.5,6, and Choudhury et al.7. The purpose of this paper is to investigate the three dimensional free convective flow with heat and mass transfer of a visco-elastic fluid along a vertical porous plate. The velocity and skin friction coefficient are illustrate graphically and the results are physically interpreted. It is observed that the effect of visco-elastic parameter is prominent in the flow field. The constitutive equation of the visco-elastic fluid characterized by Walters liquid (Model B′) is given by
Singh and Sharma19 have studied the effect of the porous medium on the three dimensional couette flow and heat transfer. They20 have also studied the effect of transverse periodic variation of the permeability of the heat transfer and the free convective flow of a viscous incompressible fluid through a highly porous medium bounded by a vertical porous plate. Balamurugan et al.4 have investigated the
where σik is the stress tensor, p is isotropic pressure, gik is the metric tensor of a fixed co-ordinate system xi, vi is the velocity vector, the contravarient form of e´ik is given by
σ୧୩ = −pg ୧୩ + 2η e୧୩ − 2k e′୧୩
e′୧୩ =
பୣౡ ப୲
+ v ୫ e୧୩ ,୫ − v ୧ ,୫ e୧୫ − v ୧ ,୫ e୫୩
(1)
(2)
It is the convected derivative of the deformation rate tensor eik defined by
Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)
342
Rita Choudhury, et al., J. Comp. & Math. Sci. Vol.5 (4), 340-350 (2014) ୧ 2e୧୩ = v,୩ + v,୧୩
(3)
Here η0 is the limiting viscosity at the small rate of shear which is given by η = Nሺτሻdτ and k = τNሺτሻdτ (4) ∞
∞
is the visco-elastic parameter. N(τ) being the relaxation spectrum as introduced by Walters (1960,1962) . This idealized model is a valid approximation of Walters liquid (Model B′) taking very short memories into account so that terms involving τ୬ Nሺτሻdτ, n ≥ 2 have been neglected. ∞
(5)
2. MATHEMATICAL FORMULATION Consider the three dimensional free convective flow with heat and mass transfer of a visco-elastic fluid through a porous medium which is bounded by a vertical porous plate. We introduce a co-ordinate system ൫ ݔ′ , ݕ′ , ݖ′ ൯ with ݔ′ -axis vertically upwards along the plate, ݕ′ -axis perpendicular to it directed into the fluid region and ݖ′ -axis along the width of the plate. Let ݑ, ݒ, ݓbe the velocity components in the direction of ݔ′ , ݕ′ , ݖ′ respectively. The suction velocity is taken as follows: ݒ௪′ ൫ ݖ′ ൯ =
గ௭ ′ −ܸ ቂ1 + ߝܿ ݏ ቃ
where ߝ ≪ 1 and L is the wave length of the periodic suction. Since the plate is infinite in length in ݔ′ direction, therefore all the fluid properties are assumed to be independent of
ݔ′ , except possibly pressure. With the above assumptions and under the usual bounder layer, the equations governing the flow are given below. Continuity equation: డ௩ ′ డ௬ ′
+
డ௪ ′ డ௭ ′
=0
(6)
Momentum equations:
ݒ′ డ௬ ′ + ݓ′ డ௭ ′ = ݃ߚ൫ܶ ′ − ܶ∞′ ൯ + డ௨ ′
డ௨ ′
݃ߚ ′ (൫ܿ ′ − ܿ∞′ ൯ + ߥ ቀడ௬ ′మ + డ௭ ′మ ቁ − డమ ௨′
డమ ௨′
݇[ ݒ′ ቀడ௬ ′య + డ௬ ′ డ௭ ′మ ቁ + ݓ′ ቀ డ௭ ′య + డ௭ ′ డ௬ ′మ ቁ − డయ ௨′
డమ ௨′ డ௩ ′
డయ ௨′
డయ ௨′
డయ ௨′
డమ ௩ ′ డ௨ ′
డమ ௪ ′ డ௨ ′ − డ௭ ′ మ ′ ′ మ ′ ′ మ ′ ′ డ ௨ డ௪ డ ௪ డ௨ డ ௩ డ௨ 2 డ௭ ′మ డ௭ ′ − 3 డ௭ ′మ డ௭ ′ − డ௭ ′మ డ௬ ′ − డమ ௩ ′ డ௨ ′ డమ ௨′ డ௩ ′ డమ ௪ ′ డ௨ ′ 2 డ௬ ′ డ௭ ′ డ௭ ′ − 2 డ௬ ′ డ௭ ′ డ௭ ′ − 2 డ௬ ′ డ௭ ′ డ௬ ′ డ௪ ′ డమ ௨′ 2 ′ ′ ′] డ௬ డ௬ డ௭
2 డ௬ ′మ డ௬ ′ − 3 డ௬ ′మ డ௬ ′ − డ௬ ′మ
− (7)
ݒ′ డ௬ ′ + ݓ′ డ௭ ′ = − ఘ డ௬ ′ + ߥ ቀ2 డ௬ ′మ + డ௭ ′మ + డ௩ ′
డ௩ ′
ଵ డ′
డమ ௩ ′
డమ ௩ ′
డమ ௪ ′ డయ ௩ ′ డయ ௨′ ቁ − ݇ ቂ2 ݒ′ ′య + 2 ݓ′ ′ ′మ − డ௬ ′ డ௭ ′ డ௬ డ௭ డ௬ డ௩ ′ డమ ௩ ′ డమ ௩ ′ డ௩ ′ డయ ௩ ′ 6 డ௬ ′ డ௬ ′మ − 4 డ௬ ′ డ௭ ′ డ௭ ′ + ݒ′ డ௬ ′ డ௭ ′మ + డమ ௩ ′ డ௪ ′ డయ ௩ ′ డయ ௪ ′ డయ ௪ ′ ݓ′ డ௭ ′య − 2 డ௭ ′మ డ௭ ′ + ݒ′ డ௭ ′ డ௬ ′మ + ݓ′ డ௬ ′ డ௭ ′మ − డమ ௪ ′ డ௩ ′ డమ ௩ ′ డ௪ ′ డమ ௪ ′ డ௩ ′ 3 ′ ′ ′ − 3 ′ ′ ′ − 3 ′మ ′ − డ௬ డ௭ డ௬ డ௬ డ௭ డ௬ డ௭ డ௭ డమ ௪ ′ డ௪ ′ డమ ௪ ′ డ௩ ′ − డ௬ ′మ డ௭ ′ − డ௭ ′మ డ௬ ′ మ ′ ′ డ ௩ డ௩ ቃ (8) డ௭ ′మ డ௬ ′
Energy equation:
ߩܥ ቀ ݒ′ డ௬ ′ + ݓ′ డ௭ ′ ቁ = ݇ ቀడ௬ ′మ + డ௭ ′మ ቁ డ் ′
డ் ′
డమ ் ′
డమ ் ′
(9)
Mass diffusion equation:
ቀ ݒ′ డ௬ ′ + ݓ′ డ௭ ′ ቁ = ܦቀడ௬ ′మ + డ௭ ′మ ቁ డ் ′
డ் ′
డమ ் ′
డమ ் ′
The relevant boundary conditions are
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(10)
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Rita Choudhury, et al., J. Comp. & Math. Sci. Vol.5 (4), 340-350 (2014)
= ݕ ݐܣ0: ݑ′ = 0 , ݒ′ = ݒ௪′ , , ݓ′ = 0 , ܶ ᇱ = ܶ௪ᇱ , ܥᇱ = ܥ௪ᇱ ∞ → ݕ ݐܣ: ݑ′ = ܷ ′ , ݒ′ = −ܸ , ݓ′ = 0 (11) ᇱ ᇱ ܶ ᇱ = ܶஶᇱ , ܥᇱ = ܥஶ , ᇱ = ஶ We introduce the following non-dimensional quantities: =ݕ
=ݓ
௬′ ௭′ ௨′ ௩′ , = ݖ, = ݑ, = ݒ, బ బ ௪′ (் ′ ି்∞′) ( ′ି∞′ ) , ߠ = , ߶ = ′ ′ ି ′ ) ′ బ (்ೢ ି்∞) (ೢ ∞ ′ ି் ′ ) ఓ ఔ ఉ(்ೢ ∞
ܲ= ݎ
݉ܩ
=
, ܵܿ =
′ ି ′ ) ఉ ′ (ೢ ∞ = బమ ′
, = ݎܩ
, ܴ݁ =
బమ
బ ఔ
,
(12)
, ܷ=
௨′ బ
ഌ ಽ
ఘ( )మ
where Gr is the Grashof number for heat transfer, Gm is the Grashof number for mass transfer, Pr is the Prandtl number, Re is the Reynold’s number, ݇ is the dimensionless visco-elastic parameter, Sc is the Schmidt number, U is the non-dimensional free stream velocity, ߠ is the non-dimensional temperature,߶ is the non-dimensional concentration. The non-dimensional forms of the governing equations are డ௩ డ௬
+
డ௪ డ௭
డ௩ డ௩ ଵ డ ଵ డమ ௩ డమ ௩ + ݓడ௭ = − ோ మ డ௬ + ோ ቀ2 డ௬ మ + డ௭ మ + డ௬ డమ ௪ డయ ௩ డయ ௨ డ௩ డమ ௩ ቁ − ݇[2ݒ + 2ݓ − 6 − య మ డ௬డ௭ డ௬ డ௭డ௬ డ௬ డ௬ మ డమ ௩ డ௩ డయ ௩ డయ ௩ డమ ௩ డ௪ 4 డ௬డ௭ డ௭ + ݒడ௬డ௭ మ + ݓడ௭ య − 2 డ௭ మ డ௭ + డయ ௪ డయ ௪ డమ ௪ డ௩ ݒ′ +ݓ −3 − మ మ డ௭డ௬ డ௬డ௭ డ௬డ௭ డ௬ డమ ௩ డ௪ డమ ௪ డ௩ డమ ௪ డ௪ డమ ௪ డ௩ 3 −3 మ − మ − మ − డ௬డ௭ డ௬ డ௭ డ௭ డ௭ డ௬ డ௬ డ௭ డమ ௩ డ௩ ] (15) డ௭ మ డ௬
ݒడ௬ + ݓడ௭ = − ோ మ డ௭ ′ + ோ ቀ2 డ௭ మ + డ௪
ଵ డ′
డ௪
ଵ
డమ ௪
య డమ ௪ డమ ௪ డయ ௪ ′ డ ௩ + ቁ − ݇ + ݒ − ቂݒ డ௬ మ డ௬డ௭ డ௬ య డ௭డ௬ మ మ య య మ డ௩ డ ௪ డ ௩ డ ௪ డ ௩ డ௪ 2 డ௬ డ௬ మ + ݓడ௬డ௭ మ + ݓడ௬ మ డ௭ − డ௭ మ డ௬ + డయ ௪ డమ ௪ డ௩ డమ ௩ డ௪ 2 ݒడ௬డ௭ మ − 3 డ௬డ௭ డ௭ − 3 డ௬డ௭ డ௭ − డమ ௩ డ௩ డమ ௪ డ௪ డమ ௪ డ௪ డమ ௩ డ௪ 3 డ௬ మ డ௭ − డ௬ మ డ௭ − 6 డ௭ మ డ௭ − డ௭ మ డ௬ + డయ ௪ డమ ௪ డ௪ 2 ݓమ − 4 ቃ (16) డ௭ డ௬డ௭ డ௬
ݒడ௬ + ݓడ௭ = ோ ቀడ௬ మ + డ௭ మ ቁ
(17)
ݒడ௬ + ݓడ௭ = ௌோ ቀ డ௬ మ + డ௭ మ ቁ
(18)
డఏ
డథ
డఏ
డథ
ଵ
ଵ
డమ ఏ
డమ థ
డమ ఏ
డమ థ
Subject to the boundary conditions: =0
(13)
ݒడ௬ + ݓడ௭ = ߠݎܩ+ ߶݉ܩ+ ோ ቀడ௬ మ + డ௨
ݒ
డ௨
ଵ
డమ ௨
డమ ௨ డయ ௨ డయ ௨ డయ ௨ ቁ − ݇[ݒ ቀ + ቁ + ݓ ቀ + మ య మ డ௭ డ௬ డ௬డ௭ డ௭ య య మ మ మ డ ௨ డ ௨ డ௩ డ ௩ డ௨ డ ௪ డ௨ ቁ − 2 డ௬ మ డ௬ − 3 డ௬ మ డ௬ − డ௬ మ డ௭ − డ௭డ௬ మ డమ ௨ డ௪ డమ ௪ డ௨ డమ ௩ డ௨ డమ ௩ డ௨ 2 డ௭ మ డ௭ − 3 డ௭ మ డ௭ − డ௭ మ డ௬ − 2 డ௬డ௭ డ௭ డమ ௨ డ௩ డమ ௪ డ௨ 2 డ௬డ௭ డ௭ − 2 డ௬డ௭ డ௬ − డ௪ డమ ௨ 2 డ௬ డ௬డ௭
= ݕ ݐܣ0 ∶ = ݑ0 , = ݒ−ሺ1 + ܿݖߨݏሻ, = ݓ0 ,ߠ = 1 ,߶ = 1 ∞ → ݕ ݐܣ: ܷ = ݑ, = ݒ−1 , (19) = ݓ0 ,ߠ = 0 , ߶ = 0 3. METHOD OF SOLUTION
−
(14)
When the amplitude (ε ≪ 1) the suction velocity is small, we assume the solution of the nonlinear partial differential equations (13) to (18) of the form
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Rita Choudhury, et al., J. Comp. & Math. Sci. Vol.5 (4), 340-350 (2014)
ݑሺݕ, ݖሻ = ݑ ሺݕሻ + ߝݑଵ (ݕ, )ݖ ݒሺݕ, ݖሻ = ݒ ሺݕሻ + ߝݒଵ (ݕ, )ݖ ݓሺݕ, ݖሻ = ݓ ሺݕሻ + ߝݓଵ ሺݕ, ݖሻ ሺݕ, ݖሻ = ሺݕሻ + ߝଵ (ݕ, )ݖ ߠሺݕ, ݖሻ = ߠ ሺݕሻ + ߝߠଵ ሺݕ, ݖሻ ߶ሺݕ, ݖሻ = ߶ ሺݕሻ + ߝ߶ଵ ሺݕ, ݖሻ
Zeroth-order equations: =0
݇ݒ
ௗ య ௨బ ௗ௬ య
+ ݒ
ௗ௨బ ௗ௬
ௗబ =0 ௗ௬ ଵ ௗమ ఏ ௗఏబ ݒ ௗ௬ = ோ ௗ௬ మబ
ݒ
ௗథబ ௗ௬
ଵ
= ோௌ
(21)
= ߠݎܩ + ߶݉ܩ + ଵ ௗ మ ௨బ ோ ௗ௬ మ
ௗమ థబ ௗ௬ మ
First order equations: డ௩భ డ௪ + డ௭భ = 0 డ௬
(22) (23) (24) (25)
(26)
ݒଵ
డ௨బ డ௨ + ݒ డ௬భ = ߠݎܩଵ + ߶݉ܩଵ + డ௬ ଵ డ మ ௨భ డమ ௨ డయ ௨ డయ ௨ ቀ మ + మభ ቁ − ݇(ݒ యభ + యబ ோ డ௬ డ௭ డ௬ డ௬ డ య ௨భ డమ ௨బ డ௩భ డమ ௩భ డ௨బ ݒ − 2 డ௬ మ డ௬ − 3 డ௬ మ డ௬ − డ௬డ௭ మ మ డ௨బ డ ௩భ డ௨బ డమ ௪భ − 2 ) డ௬ డ௭ మ డ௬ డ௬డ௭
ݒ
డ௩భ డ௬
+
(27)
ଵ డభ ଵ డమ ௩ డమ ௩ + ோ ቀ2 డ௬ మభ + డ௭ మభ + డ௬ డమ ௪భ డయ ௩భ డయ ௩భ ቁ − ݇ ቀ2ݒ + ݒ + య డ௬డ௭ డ௬ డ௬డ௭ మ డయ ௪భ ݒ డ௬ మ డ௭ ቁ (28)
= − ோ మ
డ௪భ డ௬
ݒଵ
డఏబ డ௬
+ ݒ
డఏభ డ௬
ݒ
డథభ డ௬
+ ݒଵ
డథబ డ௬
(20)
Substituting (20) in the equation (13) to (18) and equating the co-efficient of like powers of ߝ and neglecting ߝ ଶ and higher power of ߝ, we get the following sets of differential equations: ௗ௩బ ௗ௬
ݒ
+ ோ ቀ డ௬ మభ + 2
డమ ௪భ + డ௭ మ మ య య డ ௪ డ ௩భ డ ௩భ ቁ − ݇ ቀݒ డ௬ యభ + ݒ డ௬ మ డ௭ + డ௬డ௭ డయ ௪భ (29) 2ݒ డ௬డ௭ మ ቁ ଵ డభ డ௭
= − ோ మ
=
ଵ
డమ ௪
ଵ డమ ఏ ቀ మభ ோ డ௬
+
డమ ఏభ ቁ డ௭ మ
= ோௌ ቀ డ௬ మభ + ଵ
డమ థ
డమ థభ ቁ డ௭ మ
(30) (31)
The corresponding boundary conditions are
= ݕ ݐܣ0: ݑ = 0 , ݒ = −1, ߠ = 1 , ߶ = 1 ݑଵ = 0, ݒଵ = −ܿ ݖߨݏ, ݓଵ = 0, ଵ = 0 , ߠଵ = 0 , ߶ଵ = 0 ∞ → ݕ ݐܣ: ݑ = ܷ , ݒ = −1, ߠ = 0 ݑଵ = 0, ݒଵ = 0 , ݓଵ = 0, ଵ = 0 , ߠଵ , ߶ଵ = 0
(32)
The solutions of the equations (21), (24) and (25) subject to the boundary conditions (32) are ݒ = −1 , ߠ = ݁ ିோ௬ ߶ = ݁ ିோௌ௬ To solve equation (22) , we adopt multiparameter perturbation scheme ( Nowinski and Ismail, 1965)13 and we write ݑ = ݑ + ݇ݑଵ
(33)
(as ݇ ≪ 1, for small shear rate). Also we remember that the parameters ߝ and ݇ must be independent of each other and are of the same order. Using (33) in (22) and equating the coefficient of like powers of ݇ and neglecting the higher powers of ݇, we get
Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)
345 ௗమ ௨బబ ௗ௬ మ
ௗమ ௨బభ ௗ௬ మ
Rita Choudhury, et al., J. Comp. & Math. Sci. Vol.5 (4), 340-350 (2014)
+ ܴ݁
ௗ௨బబ ௗ௬
+ ܴ݁
ௗ௨బభ ௗ௬
= −ܴ݁ሾି ݁ݎܩோ௬ + ି ݁݉ܩோௌ௬ ሿ = −ܴ݁
ௗయ ௨బబ ௗ௬ య
Modified boundary condition are = ݕ ݐܣ0: ݑ = 0 , ݑଵ = 0 ∞ → ݕ ݐܣ: ݑ = ܷ , ݑଵ = 0
(34) (35)
(36)
The solution of the equation (34) and (35) subject to boundary conditions (36) we get, ݑ = ሺ݅ − ݁ ିோ௬ ሻܷ + +
ீ൫ షೃ ି షುೝೃ ൯ ோሺ మ ିሻ
ீ൫ షೃ ି షೃೄ ൯
and
ோሺௌ మ ିௌሻ
ݑଵ = ܯଵ ݁ ோ௬ + ܤଵ ି ݁ݕோ௬ −ܯଶ ݁ ିோ௬ − ܯଷ ݁ ିோௌ௬
(37)
(38)
To solve (26) , (28), (29) we assume that, ݒଵ = −ߨݒଵଵ ሺݕሻܿݖߨݏ ′ ሺݕሻݖߨ݊݅ݏ (39) ݓଵ = ݒଵଵ ଵ = ܴ݁ ଶ ଵଵ ሺݕሻܿݖߨݏ Substituting these in equations (28) and (29), ′ eliminating pressure term ଵଵ , we get the following differential equation in ݒଵଵ as ூ ′′′ ′′ ′ ݒଵଵ + ܴ݁ݒଵଵ − 2ߨ ଶ ݒଵଵ − ߨ ଶ ܴ݁ݒଵଵ + ߨ ସ ݒଵଵ + ଶ ′′′ ସ ′ ሻ ܴ݁݇ሺݒଵଵ − 2ߨ ݒଵଵ + ߨ ݒଵଵ = 0 (40)
The boundary conditions ଵ ′ = ݕ ݐܣ0 ∶ ݒଵଵ = గ , ݒଵଵ =0 ∶ ∞ → ݕ ݐܣ
′ ݒଵଵ = 0 , ݒଵଵ =0
(41)
To solve (40) , we take ݇ ≪ 1 for small shear , and so we can assume that ݒଵଵ = ݒଵଵ ሺݕሻ + ݇ݒଵଵଵ ሺݕሻ
(42)
Substituting (42) in equation (40) and equating the coefficient of like powers of ݇ and neglecting the higher powers ݇, we get ூ ′′′ ′′ ′ ݒଵଵ + ܴ݁ݒଵଵ − 2ߨ ଶ ݒଵଵ − ߨ ଶ ܴ݁ݒଵଵ + ସ ߨ ݒଵଵ = 0 (43)
ூ ′′′ ′′ ′ ݒଵଵଵ + ܴ݁ݒଵଵଵ − 2ߨ ଶ ݒଵଵଵ − ߨ ଶ ܴ݁ݒଵଵଵ + ′′′ ସ ଶ ߨ ݒଵଵଵ +ܴ݁ݒଵଵ − 2ߨ ܴ݁ݒଵଵ + ସ ′ ܴ݁ߨ ݒଵଵ = 0 (44)
with relevant boundary conditions: ଵ = ݕ ݐܣ0 ∶ ݒଵଵ = గ , ݒଵଵଵ = 0 ′ ′ ݒଵଵ = 0 , ݒଵଵଵ =0 ݒ ∶ ∞ → ݕ ݐܣଵଵ = 0 , ݒଵଵଵ = 0 ′ ′ ݒଵଵ =0, ݒଵଵଵ =0
(45)
Solving equations (43) and (44), using boundary conditions (45) we get, ଵ ఒ ݒଵଵ = గିఒ ቀ− గమ ݁ ିగ௬ + ݁ ିఒమ ௬ ቁ (46) మ
ݒଵଵଵ = గିఒ ݁ ିగ௬ − గିఒ ݁ ିఒమ ௬ + ߦି ݁ݕఒమ ௬ క
క
మ
మ
(47)
To solve (27) , (30) and (31) we assume the following form ݑଵ = ݑଵଵ ሺݕሻܿݖߨݏ ߠଵ = ߠଵଵ ሺݕሻܿݖߨݏ (48)
߶ଵ = ߶ଵଵ ሺݕሻܿݖߨݏ On using (48) into (27) , (30) and (31), we get ′′ ′ ݑଵଵ − ߨ ଶ ݑଵଵ + ܴ݁ሺݑଵଵ + ߨݒଵଵ ݑ′ ሻ ′ +ܴ݁ሺߠݎܩଵଵ + ߶݉ܩଵଵ ሻ − ܴ݁݇ሺ2ߨݑ′′ ݒଵଵ − ′′′ ′′′ ′′ ′ ଷ ′ ݑଵଵ − ߨݒଵଵ ݑ + ߨݒଵଵ ݑ − ߨ ݑ ݒଵଵ + ′ ሻ ߨ ଶ ݑଵଵ =0 (49)
′′ ′ ߠଵଵ − ߨ ଶ ߠଵଵ + ܴܲ݁ݎ൫ߠଵଵ + ߨݒଵଵ ߠ′ ൯ = 0 (50)
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After solving (50) and (51) , using boundary condition (52) we get, ߠଵଵ = ℎ݁ ିఈమ ௬ + ℎଵ ݁ ିሺగାோሻ௬ + ℎଶ ݁ ିሺఒమ ାோሻ௬ + ݇ൣି ݁ܤఈమ ௬ + ܤଽ ݁ ିሺగାோሻ௬ + ܤଵଷ ݁ ିሺఒమ ାோሻ௬ + (53) ܤଵଶ ି ݁ݕሺఒమ ାோሻ௬ ሿ ߶ଵଵ = ܤଶଵ ݁ ିఈర ௬ + ܤଵଽ ݁ ିሺగାோௌሻ௬ + ܤଶ ݁ ିሺఒమ ାோௌሻ௬ + ݇ൣܤଶ ݁ ିఈర ௬ + ܤଶଶ ݁ ିሺగାோௌሻ௬ + ܤଶ ݁ ିሺఒమ ାோௌሻ௬ + ܤଶହ ି ݁ݕሺఒమ ାோௌሻ௬ ሿ (54)
In order to solve (49), we assume that ݑଵଵ = ݑଵଵ ሺݕሻ + ݇ݑଵଵଵ ሺݕሻ
(55)
Substituting (55) into (49) and using boundary condition (52), equating the coefficient of like powers of ݇ we get, ′′ ′ ݑଵଵ + ܴ݁ݑଵଵ − ߨ ଶ ݑଵଵ = −൫ܣଵ ݁ ିሺగାோሻ௬ + ܣଶ ݁ ିሺఒమ ାோሻ௬ + ܣଷ ݁ ିሺగାோሻ௬ + ܣସ ݁ ିሺగାோௌሻ௬ + ܣହ ݁ ିሺఒమ ାோሻ௬ + ି ݁ ܣሺఒమ ାோௌሻ௬ + ି ݁ ܣఈమ ௬ + ି ݁ ଼ܣఈర ௬ ሻ
ܣଵ ି ݁ݕሺఒమ ାோௌሻ௬ + ܣଵ଼ ି ݁ݕሺఒమ ାோሻ௬ + ܣଵଽ ି ݁ݕሺఒమ ାோሻ௬ + ܣଶ ି ݁ݕሺగାோሻ௬ + ′ ′′′ ܴ݁ߨ ଶ ݑଵଵ − ܴ݁ݑଵଵ (57) subject to boundary conditions = ݕ ݐܣ0 ∶ ݑଵଵ = 0 , ݑଵଵଵ = 0 ݑ ∶ ∞ → ݕ ݐܣଵଵ = 0 , ݑଵଵଵ = 0 (58) Solving equations (56) and (57), using boundary condition (58) we get ݑଵଵ = ି ݁ܮఒమ ௬ + ܮଵ ݁ ିሺగାோሻ௬ + ܮଶ ݁ ିሺఒమ ାோሻ௬ + ܮଷ ݁ ିሺగାோሻ௬ + ܮସ ݁ ିሺగାோௌሻ௬ + ܮହ ݁ ିሺఒమ ାோሻ௬ + ି ݁ ܮሺఒమ ାோௌሻ௬ + ݈ ݁ ିఈమ ௬ + ି ݁ ଼ܮఈర ௬ (59)
ݑଵଵଵ = ܳ݁ ିఒమ ௬ + ܳଵ ݁ ିሺగାାோሻ௬ + ܳଶ ݁ ିሺగାோሻ௬ + ܳଷ ݁ ିሺగାோௌሻ௬ + ܳସ ݁ ିሺఒమ ାோሻ௬ + ܳହ ݁ ିሺఒమ ାோሻ௬ + ܳ ݁ ିሺఒమ ାோௌሻ௬ + ܳ ݁ ିఈమ ௬ + ଼ܳ ݁ ିఈర ௬ + ܳଽ ݁ ିఒమ ௬ + ܳଵ ି ݁ݕሺఒమ ାோௌሻ௬ + ܳଵଵ ݁ ିሺఒమ ାோௌሻ௬ + ܳଵଶ ି ݁ݕሺఒమ ାோሻ௬ + ܳଵଷ ݁ ିሺఒమ ାோሻ௬ + ܳଵସ ି ݁ݕሺఒమ ାோሻ௬ + ܳଵହ ݁ ିሺఒమ ାோሻ + ܳଵ ି ݁ݕሺగାோሻ௬ + (60) ܳଵ ݁ ିሺగାோሻ௬ 4. RESULT AND DISCUSSION The velocity profile is given by ݑ = ݑ + ݇ݑଵ + ߝݑଵଵ ܿݖߨݏ
(61)
The non –dimensional shearing stress at the plate = ݕ0 (56)
′′ ′ ݑଵଵଵ + ܴ݁ݑଵଵଵ − ߨ ଶ ݑଵଵଵ = ܰଵ ݁ ିሺగାோሻ௬ + ܰଶ ݁ ିሺగାோሻ௬ + ܰଷ ݁ ିሺగାோௌሻ௬ + ܰସ ݁ ିሺఒమ ାோሻ௬ + ܰହ ݁ ିሺఒమ ାோሻ௬ + ܰ ݁ ିሺఒమ ାோௌሻ௬ + ܣଵହ ݁ ିఈమ ௬ + ܣଵ ݁ ିఈర ௬ +
ߪ = ቀோ డ௬ ቁ ଵ డ௨
డ௨ డ௩ 3 డ௬ డ௬
− ݇ ቂ ݒడ௬ మ + ݓడ௬డ௭ −
௬ୀ డ௨ డ௩ − 2 డ௭ డ௭
డమ ௨
− డ௭ డ௬ ቃ
డమ ௨
డ௨ డ௪
௬ୀ
(62)
The non-dimensional heat flux at the plate ݕᇱ = 0 in terms of Nusselt number ܰ௨ is given by
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ܰ௨ = −
డ் ′ ′ ି் ′ ൯ ቀడ௬ ′ ቁ ఘబ ು ൫்ೢ ∞ ଵ ′ ሺ0ሻ ′ ሺ0ሻܿݖߨݏ൧ + ߝߠଵଵ − ൣߠ ோ
(63) = The non-dimensional mass transfer at the wall ݕ′ = 0 in terms of Sherwood number ܵ is given by ܦ ߲ ܥ′ ܵ = − ቆ ቇ ܸ ሺܥ௪′ − ∞ܥ′ሻ ߲ ݕ′ ଵ ′ ሺ0ሻܿݖߨݏ൧ = (64) ൣ߶′ ሺ0ሻ + ߝ߶ଵଵ ௌோ The constants of the differential equations are obtained but not presented here for the sake of brevity. The objective of this paper is to bring out the effects of visco-elastic parameter on the three dimensional free convective flow with heat and mass transfer of a visco-elastic fluid along a vertical porous plate. The visco-elastic effect is exhibited through the non-dimensional parameter ݇.The corresponding results can be obtained for non-Newtonian fluid by setting ݇ = 0. Figures 1 to 7 represent the fluid velocity ݑagainst ݕto observe the effects visco-elastic parameter with the combination of other flow parameters viz. Prndtl number Pr, Grashof number for heat transfer Gr, Grashof number for mass transfer Gm, Reynolds number Re, and Schmidt number Sc.
Figure 1: variation of u against y for Pr=2, Re=0.7 , Sc=2 ,Gr=6 , Gm=2.
Figure 2: variation of u against y for Pr=2, Re=0.9, Sc=2, Gr=6 , Gm=2
Figure 3: variation of u against y Pr=2, Re=0.7, Sc=4,Gr=6,Gm=2
Figure 4: variation of u against y for Pr=2 ,Re=0.7 , Sc=2 , Gr=8 , Gm=2
Figure 5: variation of u against y for Pr=2 , Re=0.7 , Sc=2 , Gr=6 , Gm=5
Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)
Rita Choudhury, et al., J. Comp. & Math. Sci. Vol.5 (4), 340-350 (2014)
Figure 6: variation of u against y for Pr=5 , Re=0.7 , Sc=2 , Gr=6 , Gm=2
The figures 1 to 6 depict the growth of the fluid velocity for the cooled plate ( > ݎܩ0)but a reverse trend is noticed for the heated plate ሺ < ݎܩ0ሻ in figure 7 for both Newtonian and non-Newtonian fluid flow phenomenon, for > ݎܩ0, the fluid velocity enhances but for < ݎܩ0 the same diminishes with the rise of visco-elastic parameter in combination of other flow parameters.
Figure 7: variation of u against y for Pr=2 , Re=0.7 ,Sc=2 ,Gr=-6 ,Gm=2
Figure 8: variation of σ against Pr
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Figure 9: variation of σ against Gr
Figure 10: variation of σ against Gm
Figure 11: variation σ against Sc
The figures 8 to 11 exhibit the variations of the shearing stress ߪ against the Prandtl number ܲݎ, the Grashof number for heat transfer ݎܩ, the Grashof number for mass transfer Gm and the Schmidt number ܵܿ respectively. The shearing stress ߪ attains increasing trend with the growth of the visco-elasticity when the Prandtl number ܲݎ and the Grashof number for mass transfer
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݉ܩincreases (figures 8 and 10) but a reverse pattern is observed with the enhancement of the Grashof number for heat transfer and the Schmidt number ܵܿ (figure 9, 11). From the expressions (63) and (64), it can be seen that the rate of heat transfer in the form of Nusselt number and the mass transfer in the form Sherwood number are not sufficiently affected by the visco-elastic parameter. 5. CONCLUSIONS The steady three dimensional Walters liquid (Model ܤ′ ) past an vertical porous plate in presence of heat and mass transfer has been investigated.This study leads to following conclusions: 1. The velocity field is considerably affected by the visco-elastic parameter in combination of other flow parameters. 2. The fluid velocity depicts on accelerating trend with the growing effect of visco-elasticity for cooled plate > ݎܩ0 but decelerating trend is noticed for heated plate( < ݎܩ0) 3. The shearing stress enhances with the rising effect of visco-elastic parameter against the Prandtl number and Grashof number for mass transfer. 4. The shearing stress diminishes with the growth of visco-elasticity against the Grashof number for heat transfer and Schmidt number. 5. The effects of visco-elasticity is insufficient in the temperature and the concentration fields.
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