Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 127 (2015) 963 – 969
International Conference on Computational Heat and Mass Transfer-2015
MHD Heat And Mass Transfer Flow over a Stretching Wedge with Convective Boundary Condition and Thermophoresis NagendrammaV,Sreelakshmi K, *Sarojamma G Dept. of Applied Mathematics, SPMVV, Tirupati, India
Abstract In this study the unsteady flow of a viscous incompressible fluid past a stretching wedge under the influence of a transverse magnetic field, viscous dissipation, and wall slip is investigated. The governing partial differential equations of the flow are solved numerically. The point of flow separation is observed to occur at small times and for smaller values of wedge angle. The temperature is found to be an increasing function of Eckert and Biot numbers and a decreasing function of wedge angle. here and insert your abstractbytext . Click © 2015 The Authors. Published Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under – 2015. Peer-review under responsibility responsibilityof ofthe theorganizing organizingcommittee committeeofofICCHMT ICCHMT – 2015 Keywords: Unsteady wedge flow, thermophoresis, convective boundary condition
1.
Introduction
Fluid flows over wedge shaped bodies occur in many thermal engineering applications like geothermal systems, crude oil extraction, thermal insulation, heat exchangers and the storage of nuclear waste, etc., Yih [1] analyzed the effects of viscous dissipation and stress work on the MHD forced convection over a wedge, considering a variable wall temperature on the surface of the wedge. Michael et al [2] investigated the Falkner-Skan solution for laminar boundary layer flow over a wedge and is modified to allow for a slip boundary condition. Recently studies on aerosol deposition have become increasingly significant for engineering applications. The force experienced by a small aerosol particle in the presence of a temperature gradient is known as thermophoresis. One can see the results of thermophoresis on the glass globe of kerosene lantern or smudges on the wall of a kitchen near the stove. Thermophoresis, also called thermo diffusion or Soret effect, is the effect of a temperature gradient on particles, causing them to move from a hot plate to the cold. The first analysis of thermophoretic deposition on a surface which has engineering applications appears to be that of Hales el al. [3] * Corresponding author. E-mail address:
[email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT – 2015
doi:10.1016/j.proeng.2015.11.444
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Sattar [4] analized an unsteady two-dimensional laminar hydrodynamic boundary layer flow past a wedge generated by a potential flow velocity taken as a function of the distance x and time t. Kandaswamy et al. [5] investigated the unsteady Hiemenz flow of an incompressible viscous Cu-nanofluid past a porous wedge due to solar energy. The convective heat transfer is of prominent significance in procedures in which high temperatures are involved. Referring to numerous industrial and engineering processes the convective boundary conditions are more practical in material drying, transpiration cooling process etc. Battler [6] investigated the Blasius and Sakiadis flows in a viscous fluid with convective boundary conditions. The heat transfer of a stretching sheet with convective boundary conditions has been studied by Yao et al [7]. Malik et al [8] studied the flow and heat transfer in Sisko fluid with convective boundary condition over a non – isothermal stretching sheet. In this paper we made an effort to investigate the effect of thermophoresis on MHD unsteady flow over a stretching wedge with viscous dissipation. 2.
Mathematical Formulation
Consider the unsteady laminar two – dimensional flow of a viscous incompressible electrically conducting fluid along a heated impermeable wedge subject to a uniform magnetic field of strength B0 in the y-direction as shown in ଶ is the wedge angle parameter that fig 1. The included angle of the wedge is taken as ȳ ൌ ߚߨ . ߚ ൌ ାଵ corresponds to ȳ ൌ ߚߨ for a total angle ȳ of the wedge. The flow is assumed to be in the x-direction which is considered along a direction of the wedge and the y – axis normal to it. We assume the convective heating process and thus the temperature on the wedge surface ܶ and heat transfer coefficient h shall be determined. The concentration at the wedge surface takes the constant value ܥ௪ , while the ambient value, attained as y tends to infinity, takes the constant value ܥஶ .
ܤ
Fig. 1 Physical model and coordinate system We consider the viscous dissipation, heat generation/absorption, thermophoresis and convective boundary conditions. The basic governing equations describing the conservation of mass, momentum, energy and concentration respectively can be written as follows: డ௨ డ௩ ൌͲ, (1) డ௫ డ௨ డ௧
డ௬ డ௨
ݑ
ߩܿ ቀ డ డ௧
డ௫ డ்
ݑ
డ௧ డ
ݑ
ݒ
డ௫
డ௨
డ௬ డ்
ݒ
డ௫ డ
ݒ
ൌߥ
డ௬
డమ ௨
డ௬ మ డ்
డ௬
ൌܦ
െ
ቁൌ݇ డమ డ௬ మ
െ
ఙబమ ఋ ఘ௫ డమ ் డ௬ మ డ డ௬
ሺ ݑെ ܷሻଶ ,
(2)
డ௨ ߤሺ ሻଶ , డ௬
(3)
ሺ்ܸ ܥሻ
(4)
where u and v are velocity components in the x and y directions respectively. ߥ is the kinematic coefficient of viscosity, ߪ is the electrical conductivity which is assumed to have variable property, T is the fluid temperature, C is the concentration, Bo is the constant Magnetic field, ρ is the density of the fluid, k is the thermal conductivity, ߤ is dynamic viscosity, D is the mass diffusivity, ܿ is the specific heat at constant pressure and ்ܸ ൌ െ thermophoretic velocity, Here K is thermophoretic coefficient
ణ డ்
் డ௬
is the
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The boundary conditions are డ௨ డ் ݑൌ ܮሺ ሻǢ ݒൌ ͲǢെ݇ ൌ ݄௪ ሺܶ െ ܶሻ ; డ௬
ܥൌ ܥ௪ Ǣ ܽ ݕݐൌ Ͳ
డ௬
(5)
ݑൌ ܷሺݔǡ ݐሻǡܶ ՜ ܶஶ ǡ ܥ՜ ܥஶ ǡܽ ݕݏ՜ λ where k is the thermal conductivity, ݄௪ is the heat transfer parameter, ܷሺݔǡ ݐሻ ൌ
ణ௫ ఋ శభ
(6) is the potential flow
velocity for the wedge flow, m is an arbitrary constant and is related to the wedge angle and ߜ ൌ ߜሺݐሻ is the time dependent length scale. Following the lines of Kafoussias and Nanousis (Kafoussias and Nanousis, [9]) ,the following change of variables are introduced ሺଵାሻ
ߟ ൌ ݕට
ଶ
௫ షభ
ට
ఋ శభ
,Ȳൌට
ଶ
ାଵ
ణ௫ శభ
ට
ఋ శభ
݂ሺߟሻ, ߠ ൌ
்ି்ಮ ்ೢ ି்ಮ
and ߶ ൌ
ିಮ
(7)
ೢ ିಮ
The Eqs. (3), (4) and (6) reduce to the following ordinary differential equations ଶ ƍƍƍ ƍƍ െ ȕሺͳ െ ƍଶ ሻ Ȝሺʹ െ ʹ ƍ െ Ș ƍƍ ሻ െ ሺ୫ାଵሻ ሺ ƍ െ ͳሻଶ ൌ Ͳሺͺሻ șƍƍ ȜȘșƍ șƍ
ƍƍଶ ୗୡ
ࢥƍƍ
ࢥƍ Ȝ
Șࢥƍ
౪ ାș
ଶ
୫ାଵ
ሾି כȘ כșሿ ൌ Ͳሺͻሻ
ሾሺୡ ࢥሻșƍƍ șƍ ࢥƍ െ ቀ
The transformed boundary conditions are ݂ሺͲሻ ൌ Ͳǡ ݂ ᇱ ሺͲሻ ൌ ݂ ᇱᇱ ሺͲሻǡ ߠ ᇱ ሺͲሻ ൌ െ ݂
ᇱ ሺλሻ
ඥଶିఉ
ൌ ͳǡ ߠሺλሻ ൌ Ͳǡ Ԅሺλሻ ൌ Ͳ ܯൌ
ߚ Wedge angle parameter, number, ܵܿ ൌ
ఔ
parameter, ܰ ൌ
ఙబమ ఋ ఘ
ఊ ඥଶିఉ
ౙ ାࢥ ౪ ାș
ଶ
ቁ șƍƍ ሿ ൌ ͲሺͳͲሻ
ሺͳ െ ߠሻǡ ߶ሺͲሻ ൌ ͳ
(11) (12)
Magnetic field parameter, ܲ ݎൌ ஔౣ
ఘ
Prandtl number, ܿܧൌ
మ ಮ
ο்
Eckert
ୢஔ ்ಮ is the unsteadiness parameter, ܰ௧ ൌ Thermophoresis ሺ்ೢ ି்ಮ ሻ ୶ౣషభ ୢ୲ can be compared with the well established scaling parameter for the unsteady boundary
Schimdt number, ಮ
ߣ ൌ
ሺೢ ିಮ ሻ
layer parameter (Schlichting, [10]). The major physical quantities of engineering interest are the local skin friction, the local Nusselt number and the local Sherwood number, thermophoretic velocity and wall thermophoretic deposition velocity are defined respectively as follows భ
డ௨
ܥ ܴ݁௫మ ඥʹ െ ߚ ൌ ʹ݂ ᇱᇱ ሺͲሻǡwhere ߬௪ ൌ ߤሺ ሻ௬ୀ
(13)
డ௬
భ ି మ
డ்
ܴܰ݁ݑ௫ ඥʹ െ ߚ ൌ െߠ ᇱ ሺͲሻǡwhere ݍ௪ ൌ െ݇ሺ ሻ௬ୀ డ௬ ିଵȀଶ ݁௫ ඥʹ
ܸௗ ൌ ሺ ࢂ ࢊכൌ 3.
௫ ఔ
ࢂࢊ ࢞ ࣇ
డ
െ ߚ ൌ െᇱ ሺͲሻǡ where ܯ௪ ൌ െܦሺ ሻ௬ୀ
ሻ௬ୀ ൌ െට
ଵ
ଶିఉ
ൌ െට
ିࢼ
ࡾࢋ࢞
భ మ
ܴ݁௫
ࡿࢉ
డ௬
ଵାே
ߠ ᇱ ሺͲሻ
(14) (15) (16)
ᇱ ሺሻ
Results and Discussion
In this study the unsteady flow of a viscous incompressible fluid past a stretching wedge under the influence of a transverse magnetic field in the presence of viscous dissipation, convective boundary condition and wall slip is investigated. The governing partial differential equations of the flow are solved numerically using fourth order RK method with shooting technique. To validate the accuracy of the numerical scheme, the values ݂ሺͲሻǡ ݂ ᇱ ሺͲሻǡ ݂ ᇱᇱ ሺͲሻ are compared with the results of White [11] in the absence of ߚ ൌ Ͳǡ ܯൌ Ͳǡ ߣ ൌ Ͳ and with those of Rehman et al (2013). The computational results are graphically presented and discussed for different variations in the magnetic field (M), wedge angle ሺߚሻ, the unsteady parameter ሺߣሻ and thermophoresis parameter ሺܰ௧ ሻ. Table 2 illustrates the thermophoretic particle deposition velocity at the wall for different values of the wedge angle and unsteady parameter. The tabulated values of ܸௗ ܴ݁ כଵȀଶ indicate that for all values of ߚ the thermophoretic deposition velocity
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increases with unsteady parameter. Increasing values of ߚ increases the thermophoretic deposition velocity.
Table. 1. Comparison table for the Falkner – Skan boundary layer equation for the case of Ȝ ൌ ȕ ൌ ܯൌ ܿܧൌ כܣൌ Ͳ כܤǢ ൌ ͲǤͳǢ
ൌ ͵Ǥ Ǣ ൌ ʹǤ Ǣ
ൌ ͲǤͻͶǢ ݉ ൌ ͳǢ ߟ Present work
0.0 0.5 1.0 1.5 2.0 3.0 4.0 5.0
0.00000 0.05864 0.23301 0.51508 0.88687 1.79572 2.78410 3.78352
݂ሺߟሻ Rehman et White al [12] 0.00000 0.0000 0.05872 0.05864 0.23332 0.23299 0.51575 0.51503 0.88800 0.88680 1.79780 1.79557 2.78709 2.78388 3.78738 3.78323
Present
݂ƍሺߟሻ Rehman et White
Present
work
al
work
0.00000 0.23425 0.46067 0.66153 0.81676 0.96912 0.99783 1.00000
0.00000 0.23456 0.46127 0.66235 0.81770 0.97006 0.99872 1.00087
0.00000 0.23423 0.46063 0.66147 0.81669 0.96905 0.99777 0.99994
0.469645 0.465075 0.434418 0.361831 0.255679 0.067705 0.006872 0.001507
݂ƍƍሺߟሻ Rehman et al White 0.47027089 0.46568757 0.43494906 0.36218408 0.25581418 0.06763291 0.00684790 0.00025589
0.46960 0.46503 0.43438 0.36180 0.25567 0.06771 0.00687 0.00026
Table. 2. Variations of thermophoretic particle deposition velocity at the wall for different values of ȕ and Ȝ when ൌ
ൌ כൌ כൌ ൌ ൌ ͲǢ ൌ ͲǤͳǢ
ൌ ͵Ǥ Ǣ ൌ ʹǤ Ǣ
ൌ ͲǤͻͶǢ ߚ
ߣ
0.5
1.0
1.6
ି
భ
݄ܵ௫ ܴ݁௫ మ ሺെI’ሺͲሻሻ
0.0 0.5 1.0 1.5
0.557634 0.685835 0.791505 0.874996
0.0 0.5 1.0 1.5
0.714183 0.877392 1.016094 1.135401
0.0 0.5 1.0 1.5
1.163862 1.422359 1.644545 1.839384
Figs.2 and 3 are the plots of velocity for different values of ߚ (or equivalently m). ߚ ൌ Ͳ and ݉ ൌ Ͳ correspond to ଵ ଵ horizontal plate. ߚ ൌ and ݉ ൌ correspond to vertical plate. ߚ ൌ ͳ and ݉ ൌ ͳ correspond to stagnation point ଶ
ଷ
flow. Physically, ߚ Ͳ and ݉ Ͳ imply an accelerated flow (favourable pressure gradient) and for accelerated flow the velocity profiles are with no point of inflection. For decelerated flow ݉ ൏ Ͳǡ ߚ ൏ Ͳ (corresponding to an adverse pressure gradient) the velocity profiles contain a point of inflection. For decelerated flows the flow separation occurs for ߚ< -1.3. The flow separation is found to occur from the wall at ݉ ൌ ͲǤͳͳǡ ߚ ൌ ͳǤͻͻͻ. White [11] observed flow separation at ߚ ൌ െͲǤͳͻͺͺ͵ͺ.
Figs 4 – 7 reveal the points of flow separation corresponding to the unsteady parameter ߣ for four specific values of ߚ. For accelerated flows it is observed that the velocity decreases near the boundary and later it is enhanced and ultimately attaining the free stream value.. In the case of horizontal plate ሺߚ ൌ Ͳሻ the flow separation occurs at ߣ ൌ ͲǤͶ and in the case of vertical flat plate i.e. ߚ ൌ ͳȀʹ, the flow separation occurs faster at ߣ ൌ ͳ and in the case of stagnation point flow ሺߚ ൌ ͳሻ the flow separation takes place at ߣ ൌ ͳǤ͵ and in the case of wedge flow ሺߚ ൌ ͳǤሻ the flow separation is much delayed that is when ߣ ൌ ʹǤʹ. The variation of temperature with Eckert number is plotted in Fig. 8. It is evident that the temperature increases from the wall and attains a peak value at ߟ ൌ ͲǤͷ and then decreases in the rest of the region. As Ec
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increases there is predominant rise in the temperature. When Ec is as small as 0.2 the peak value of temperature is twice that of the value corresponding to that of ܿܧൌ ͲǤͳ. For higher values of Ec say 0.7 there is a six fold increase in the peak value of temperature when compared to the case ܿܧൌ ͲǤͳ. The effect of the Biot number on temperature is plotted in Fig.9. Increasing Biot number amounts to stronger convection and produces higher temperature on the wedge surface and consequently the thermal effects perforate deep into the quiescent fluid.
Fig. 10 shows the influence of thermophoresis on particle deposition onto a wedge surface. It is evident that the concentration decreases as the thermophoresis parameter ܰ௧ increases owing to the smaller temperature differences between the hot fluid and the free stream. The effect of thermophoretic coefficient k on the concentration is plotted in Fig. 11. It is evident that the concentration increases with increase in k. The enhancement may be attributed to the favourable temperature gradients.
1
1
0.9
β =-1.3 β =-1.0
0.8
f ' (η )
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0.5
1
1.5
2
2.5
3
3.5
4
β =1/2 λ =0.5;M=0.5;m=1;Pr=0.71;Ec=0.6;A*=o.o1;B*=0.01; Nr=1;Sc=0.94;k=0.5;Nt=2.0;Nc=3.0;h=0.1;γ =0.1;
0.5
0.4
0
β =1/3
0.6
0.5
0
β =1/4
0.7
λ =0.5;M=0.5;m=1;Pr=0.71;Ec=0.6;A*=o.o1;B*=0.01; Nr=1;Sc=0.94;k=0.5;Nt=2.0;Nc=3.0;h=0.1; γ =0.1;
0.6
β =1/6
0.8
β =-0.5 β =0
0.7
f ' (η )
β =0
β =-1.6
0.9
4.5
0
5
0
0.5
1
1.5
2
η
3
3.5
4
4.5
5
η
Fig. 2: velocity profiles for different Values of ߚ
Fig. 3: velocity profiles for different Values of ߚ for accelerated flows
for decelerated flows
1
1.2 λ =0 λ =0.1
0.9
λ =0 λ =0.2
1
λ =0.2
0.8
λ =0.5 , λ =1.0 λ =1.3 λ =1.5
λ =0.3 λ =0.4 λ =0.5
0.7
0.8
0.6
0.6 β =0;M=0.5;m=1;Pr=0.71;Ec=0.6;A*=o.o1;B*=0.01; Sc=0.94;k=0.5;Nt=2.0;Nc=3.0;h=0.1;γ =0.1;
0.5
f ' (η )
f ' (η )
2.5
0.4
0.4 0.3
β =1/2;M=0.5;m=1;Pr=0.71;Ec=0.6;A*=o.o1;B*=0.01; Sc=0.94;k=0.5;Nt=2.0;Nc=3.0;h=0.1;γ =0.1;
0.2
0.2
0 0.1 0
0
0.5
1
1.5
2
2.5
3
3.5
4
η
Fig.4. Velocity profiles for different Values of ߣ when ߚ ൌ Ͳ
-0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
η
Fig. 5. Velocity profiles for different Values of ߣ when ߚ ൌ ͲǤͷ
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1.4 λ =0
0.9
λ =0
λ =0.5 λ =1.0 , λ =1.3 λ =1.6 λ =1.9
0.8 0.7
λ =1.0 , λ =1.5 λ =1.9 λ =2.2
1
0.6
0.8 f ' (η )
f ' (η )
λ =0.5
1.2
β =1;M=0.5;m=1;Pr=0.71;Ec=0.6;A*=o.o1;B*=0.01; Sc=0.94;k=0.5;Nt=2.0;Nc=3.0;h=0.1;γ =0.1;
0.5
0.6
0.4 0.3
0.4
β =1/2;M=0.5;m=1;Pr=0.71;Ec=0.6;A*=o.o1;B*=0.01; Sc=0.94;k=0.5;Nt=2.0;Nc=3.0;h=0.1;γ =0.1;
0.2 0.2 0.1 0
0
0.5
1
1.5
2
2.5
3
3.5
0
4
0
0.5
1
1.5
2
η
2.5
3
3.5
4
η
Fig. 6. Velocity profiles for different Values of ߣ when ߚ ൌ ͳ
Fig. 7. Velocity profiles for different Values of ߣ when ߚ ൌ ͳǤ 0.7
0.12 Ec=0.1 Ec=0.2
γ =0.05
γ =0.2
0.5
β =1.6;λ =0.5;m=1;Pr=0.71;M=0.5;A*=o.o1;B*=0.01; M=0.5;Sc=0.94;k=0.5;Nt=2.0;Nc=3.0;h=0.1;γ =0.1;
0.08
γ =0.1 γ =0.15
0.6
Ec=0.5 Ec=0.7
0.1
θ (η )
θ (η )
0.4 0.06
β =1.6;λ =0.5;m=1;Pr=0.71;Ec=0.6;A*=0.01;B*=0.01;
0.3 0.04
0.02
0
0.1
0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
η
Fig. 9. Temperature profiles for different values of ߛ
Fig. 8. Temperature profiles for different values of Ec 1
1 Nt=1.5 Nt=2.5 Nt=3.5 β =1.6;λ =0.5;m=1;M=0.5;Pr=0.71;Ec=0.6;A*=0.01; B*=0.01;Sc=0.94;k=0.5;Nc=3.0;h=0.1;γ =0.1; Nt=4.5
0.9 0.8
k=0.2 k=0.4 k=0.7 k=1.2
0.9 0.8
0.7
0.7
0.6
β =1.6;λ =0.5;m=1;Pr=0.71;Ec=0.6;A*=0.01;B*=0.01; M=0.5;Sc=0.94;Nt=2.0;Nc=3.0;h=0.1;γ =0.1;
0.6
0.5
φ(η )
φ(η )
M=0.5;Nt=2.0;Nc=3.0;Sc=0.94;h=0.1;
0.2
0.5
0.4
0.4 0.3
0.3 0.2
0.2 0.1
0.1 0
0
0.5
1
1.5
2
2.5
3
3.5
η
Fig. 10. Concentration profiles for different values of Nt
4
0
0
1
2
3
4
5
Fig. 11. Concentration profiles for different values of k
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NagendrammaV et al. / Procedia Engineering 127 (2015) 963 – 969
4.
Conclusions:
The analysis presents the effects of thermophoretic particle deposition and viscous dissipation on the unsteady flow of a viscous incompressible fluid over a wedge with slip velocity and convective heat at the boundary. From the computational results the following conclusions are given: Flow separation is observed to occur at smaller values of ߣ and for smaller values of ߚ. The unsteady parameter reduces the thickness of the thermal boundary layer. Eckert number and Biot number enhance the temperature while wedge shows a reduction. The Schmidt’s number, thermophoresis parameter, Eckert number, wedge angle, unsteady parameter and slip parameter have a reducing influence on concentration resulting in thinner solutal boundary layers. The thermophoresis coefficient, concentration ratio, Prandtl number, heat generation/absorption and Biot number enhance the species concentration. The skin friction coefficient is reduced with ߣ while it experiences an increase with increasing values of ߚ. The rate of heat transfer decreases with increasing ߚ and Eckert number. The mass transfer rate is enhanced with thermophoresis coefficient and Biot number Acknowledgement The authors are thankful to the reviewers for their suggestions to improve the quality of the paper References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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