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KKU Journal of Basic and Applied Sciences
Journal homepage: jbas.kku.edu.sa Vol 2, 2 (2016) 28-36
P a g e | 28
Thermosolutal Marangoni Magnetohydrodynamic Boundary Layer Flow of a Casson Fluid over a Flat Plate under Effects of Soret and Dufour
Zehba A. Raizah*
Department of Mathematics, Faculty of science for Girls, Abha, King Khalid University, Saudia Arabia
Keywords: Casson Fluid, MHD, Soret effect, Dufour effect, Mass transfer, Boundary layer
1. INTRODUCTION
Non-Newtonian transport phenomena arise in many branches of process mechanical, chemical and materials engineering. Such fluids exhibit shear-stress-strain relationships which diverge significantly from the classical Newtonian (Navier-Stokes) model. Most non-Newtonian models involve some form of modification to the momentum conservation equations. These include power- law fluids [1], viscoelastic fluids including Maxwell upper- convected models [2], Walters-B short memory models [3], [4], Oldroyd-B models [5], differential Reiner-Rivlin models[6], [7] and Bingham plastics [8].
Transport phenomena in Casson fluids have
been studied theoretically, numerically and experimentally.
Neofytou [9] studied computationally the flow characteristics of both power-law and Casson fluids in symmetric sudden expansions, showing that the critical generalized Reynolds number of transition from symmetry to asymmetry and subsequently the inverse dimensionless wall shear stress are linearly related to the dimensionless wall shear rate. Mass transfer in a Casson flowing through an annular geometry was examined by Nagarani et al. [10]
who derived analytical solutions and also considered boundary absorption effects. Hemodynamic simulations of
Casson blood flow in complex arterial geometries were studied by Shaw et al. [11]. Attia and Sayed-Ahmed [12]
studied the unsteady hydromagnetic Couette flow and heat transfer in a Casson fluid using the Crank-Nicolson implicit method, showing that Casson number (dimensionless yield stress parameter) controls strongly the velocity overshoot and has a significant effect on the time at which the overshoot arises. Hayat et al. [13] obtained homotopic solutions for stagnation point flow and heat transfer of a Casson fluid along a stretching surface, also considering viscous heating effects.
Marangoni convection occurs around vapor bubbles during nucleation and growth due to the temperature variation along the surface. The surface tension variation resulting from the temperature gradient along the surface causes Marangoni convection. Marangoni convection is of importance in crystal growth melts and may influence other processes with liquid–vapor interfaces, in addition to boiling. The influence of Marangoni induced convection is more obvious under microgravity but also occurs in earth gravity [14]. Zhang and Zheng [15] investigated thermosolutal Marangoni convection,which can be formed with an electrically conducting fluid along a vertical surface in the presence of a magnetic field, heat generation and a first-order chemical reaction. They found that both the energy and concentration boundary layer thickness increase with the increasing of magnetic field parameter, Received: July 27, 2016 / Revised: October 13, 2016 / Accepted: October 17, 2016 / Published: December 01, 2016
Abstract: In this study, laminar thermosolutal Marangoni convection of a non-Newtonan fluid along a surface in the precense of the Soret and Dufour effects has been investigated numerically by an efficient, tri-diagonal, implicit finite-difference method. The Casson fluid model is used to characterise the non-Newtonian fluid behaviour. The governing boundary-layer equations of the current problem are formulated and transformed into a self-similar form.
The solutions are represented in fluid velocity, temperature and concentration and those are studied with different values of magnetic field parameter, Prandtl number, Schmidt number, Soret and Dufour parameters, thermosolutal surface tension ratio and Casson parameter. The results from this investigation are well validated and have favorable comparisons with previously published results. It is found that the effect of increasing values of the Casson parameter is to suppress the velocity field. Also, an increase in the thermosolutal surface tension ratio enhances the fluid velocity; however, it causes a reduction in the temperature and concentration distributions.
KKU Journal of Basic and Applied Sciences
P a g e | 29 while the momentum boundary layer thickness decreases
with the increasing of Lorentz force. Arafune and Hirata [16] experientially and analytically discussed thermal and solutal Marangoni convection in In-Ga-Sb system. Their analytical results showed that the surface velocity at the leading edge decreases drastically until about 0.5 (s), after which it decreases gradually. Magyari and Chamkha [17]
introduced exact analytical results for the thermosolutal MHD Marangoni boundary layers. They studied the steady laminar magnetohy-drodynami thermosolutal Marangoni convection in the presence of a uniform applied magnetic field in the boundary layer approximation. In addition, Magyari and Chamkha [18] introduced exact analytical solutions for thermosolutal Marangoni convection in the presence of heat and mass generation or consumption.
They studied the steady laminar thermosolutal Marangoni convection in the presence of temperature-dependent volumetric heat sources/sinks as well as of a first-order chemical reaction. Al-Mudhaf and Chamkha [19]
introduced similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects. Chamkha et al. [20]
studied a steady coupled dissipative layer, called Marangoni mixed convection boundary layer, which can be formed along the interface of two immiscible fluids in surface driven flows. A numerical study of laminar magnetohydrodynamic thermosolutal Marangoni convection along a vertical surface in the presence of the Soret and Dufour effects was performed by Mahdy and Ahmed [21] . They indicated that both of temperature and concentration gradient at the wall increases as the thermosolutal surface tension ratio increases. Also, the increase in Prandtl number results in an enhancement in the heat transfer at the wall.
Moreover, the Soret and Dufour effects are encountered in many practical applications such as in the areas of geosciences, and chemical engineering. Many researchers were interested in this topic. Hsiao et al. [22] numerically discussed the combined heat and mass transfer of an electrically conducting, non-Newtonian power-law fluid in MHD free convection adjacent to a vertical plate within a porous medium in the presence of the thermophoresis particle deposition effect. They observed that the wall friction decreases as the magnetic parameter increases or the buoyancy ratio decreases. Srinivasacharya et al. [23]
reported on mixed convective heat and mass transfer flow along a wavy surface in a Darcy porous medium in the presence of cross diffusion effects. They found that the increase in temperature dependent viscosity leads to thicken the velocity boundary layer while reduce the thermal and solutal boundary layer thickness as well as Nusselt number and Sherwood number. A numerical model was presented by Pal and Chatterjee [24] to study the MHD mixed convection with the combined action of Soret and Dufour on heat and mass transfer of a power-law fluid over an inclined plate in a porous medium in the presence of variable thermal conductivity, thermal radiation, chemical reaction and Ohmic dissipation and suction/injection. They indicated that the temperature and concentration fields were influenced appreciably by the Soret and Dufour effects. Besides, interesting investigations for these topics can be found in [25]-[33].
Boundary-layer flows of non-Newtonian fluids have been a topic of investigation for a long time as it has applications
in various industries. The governing equations of non- Newtonian fluids are highly non-linear and much more complicated than that of Newtonian fluids. Due to the complexity of these fluids, no single constitutive equation exhibiting all properties of such fluids is available [34], [35]. Several models can be found in this regard. There are other non-Newtonian models as well. Among them is the Casson fluid model. In the literature, the Casson fluid model is sometimes stated to fit rheological data better than general viscoplastic models for many materials [13], [36], [37]. It becomes the preferred rheological model for blood and chocolate [38]. Casson fluid exhibits a yield stress [39], [40]. If a shear stress less than the yield stress is applied to the fluid, it behaves like a solid whereas if a shear stress greater than the yield stress is applied, it starts to move [41], [42]. Mukhopadhyay et al. [43] studied numerically the boundary-layer forced convection flow of a Casson fluid past a symmetric wedge.
Motivated by investigation mentioned above, the mean objective of the present study is to investigate the influence of Soret and Dufour parameters on MHD thermosolutal Marangoni boundary layer Casson fluid flow over a flat plate. A self-similar solution is presented and finite difference method is employed to solve the resulting system. The obtained numerical results are presented in terms of velocity profiles, temperature and concentration distributions as well as local heat and mass transfers and discussed.
2. MATHEMATICALANALYSIS
Consider two-dimension, steady, laminar boundary layer flow and heat and mass transfer of an incompressible, electrically conducting non-Newtonian Casson fluid over a flat plat. It is assumed that the Marangoni effect acts as a boundary condition on the governing equations for the flow. A uniform magnetic field is assumed to exist in the direction normal to the surface. Further, the Soret and Dufour effects are taken into account. The rheological equation of state for an isotropic and incompressible flow of a Casson fluid is:
τij= {2(μB+ Py/√2π)eij, π > πc
2(μB+ Py/√2πc)eij, π < πc (1) Here, τij is the (i,j)th component of the stress tensor,π = eijeij and eij are the (i,j)th component of the deformation rate, π is the product of the component of deformation rate with itself, πc is a critical value of this product based on the non-Newtonian model, μB is plastic dynamic viscosity of the non-Newtonian fluid, and Py is the yield stress of the fluid. Here, it should be noted that if a shear stress less than the yield stress is applied to the fluid, it behaves like a solid, whereas if a shear stress greater than yield stress is applied, it starts to move.
Moreover, viscous dissipation, Joule heating and radiation effects are neglected. Under the above assumptions, the laminar boundary layer equations of a viscous and incompressible fluid describing mass, linear momentum, energy and concentration can be written in the usual dimensional form as:
P a g e | 30
∂u
∂x+∂v
∂y= 0, (2)
u∂u
∂x+ v∂u
∂y= ν (1 +1
β)∂2u
∂y2−σB02
ρ u, (3)
u∂T
∂x+ v∂T
∂y= α∂2T
∂y2+ Dk
cscp
∂2C
∂y2, (4)
u∂C
∂x+ v∂C
∂y= D∂2C
∂y2+Dk
Tm
∂2T
∂y2, (5)
where (u, v) are the velocity components, T is the temperature, C is the concentration of species, ν is the kinematic viscosity, β = μB√2πc/Py is the non- Newtonian (Casson) parameter, 𝜌 density of the fluid, α is the thermal diffusivity, D is the diffusion coefficient, cs, cp are the specific heat at constant pressure and concentration susceptibility, k is the thermal diffusion ratio, Tm is the mean fluid temperature. σ and Bo are the electrical conductivity the applied magnetics flux density.
The dependence of surface tension on temperature and concentration can be expressed as:
δ = δ0− γ(T − T∞) − γ̆(C − C∞) (6) with, γ = − ∂δ/δT|C, γ̆ = − ∂δ/δC|T
The physical boundary conditions of the problem are:
μ (1 +1 β)∂u
∂y|
y=0
= −∂δ
∂T|
C
∂T
∂x|
y=0
−∂δ
∂C|
T
∂C
∂x|
y=0
,
v(x, 0) = 0, T(x, 0) = T∞+ ax2,C(x, 0) = C∞+ bx2, u(x, ∞) → 0, T(x, ∞) → T∞, C(x, ∞) → C∞
(7)
Introducing the following dimensionless variables:
ψ = (aγν
ρ )1/3 x f(η) , η = (aγ
μν)1/3y , θ(η) =T−T∞
ax2 , ϕ(η) =C−C∞
bx2 ,
(8)
Where a and b are positive constant, 𝜇 dynamicviscosity.
Inserting equation (8) into equations (2)-(7) yields:
(1 +1
β) f′′′+ ff′′− f′2− M2f′= 0 (9) 1
Prθ′′+ fθ′− 2f′θ + Du ϕ′′= 0, (10) 1
Scϕ′′+ fϕ′− 2f′ϕ + Srθ′′ = 0 (11) Subject to the following boundary conditions
f(0) = 0, (1 +1
β) f′′(0) = −2(1 + R), θ(0) = 1, ϕ(0) = 1,
(12a)
f(∞) → 0, θ(∞) → 0, ϕ(∞) → 0 (12b) where, Pr =ν
α is the Prandtl number, Sc =ν
D is the Schmidt number, Du = Dkb
cscpaν denotes Dufour parameter and Sr = Dka
Tmbν denotes the Soret parameter. R = bγ̆/aγ is the ratio of the solutal and thermal Marangoni numbers.
M2= (σB02)(μ)1/3/(aγρ)2/3 or M = (√σ Bo) √μ6 / √aγρ3
The local heat and mass flux may be written as:
qw= −k∂T
∂y|
y=0
= −kax2 3√aγμν θ′(0), (13)
qm= −k∂C
∂y|
y=0
= −Dbx2 √aγ μν
3 ϕ′(0)
(14)
The local heat and mass transfer coefficients are given by:
h = qw
ax2, j = qm bx2
(15)
In practical applications, the quantities of the physical interest in our case are the local Nusselt Nu and Sherwood numbers Sh, which can be defined as:
Nu =hx
k = − √3 aγμν x θ′(0), (16) Sh =jx
D= − √3 aγμν x ϕ′(0), (17)
3. NUMERICALMETHOD
The solutions of the similar non-linear differential eqs. (9)–
(11) subjected to the boundary conditions (12) are obtained by using an implicit finite-difference method discussed by Blottner [44]. At the outset, f′ is replaced by another variable V then a three-point central difference formula is used to approximate the first and second derivatives of the dependent variables. The obtained algebraic system is solved using the Thomas algorithm. The initial step size is
∆η1= 0.001 and the growth factor is considered to be K = 1.037 such that∆ηi = K∆ηi−1. The edge of the boundary layer at infinity is represented by ηmax= 35. As convergence criteria, the dependent variables were calculated iteratively until the relative difference between the current and the previous iterations reaches 10−5. The description of the current numerical method has been discussed in details by Chamkha et al. [45, 30]. In order to check the accuracy of the present method, the obtained results are compared in special cases of the present study with previously published data. This comparison is presented in table I that shows comparison values of f ′(0) for various values of M with Mahdy and Ahmed [21] and Zhang and Zheng [15]. It is noted that a good agreements are found between the present results and those reported by Mahdy and Ahmed [21] and Zhang and Zheng [15]. It can
KKU Journal of Basic and Applied Sciences
P a g e | 31 be concluded that the present method is suitable for the
solution of the present system.
4. RSULTSANDDISCUSSIONS
In this section, computations were carried out for various values of physical parameters such as magnetic field parameter, Prandtl number, Schmidt number, Soret and Dufour parameters, thermosolutal surface tension ratio and Casson parameter.
TABLE I: COMPARISION VALUES OF 𝐟 ′(𝟎) FOR VARIOUS VALUES OF M
M=0 M=1 M=2
Mahdy and Ahmed [21] (shooting)
2.5199 2.2268 1.6786 Zhang and Zheng
[15] (shooting)
2.4569 2.1572 1.624
Present (finite difference)
2.2717 2.0289 1.5855
The effects of Casson parameter β on the velocity, temperature and concentration profiles, respectively are illustrated in fig. 1, 2 and 3. In fig. 1, in the initial, the velocity increases as Casson parameter β increases and after reverse region (η ≈ 0.8) the velocity decreases as Casson parameter β increases. Thickness is decreasing as Casson parameter β increases. In fig. 2 and 3, both of temperature and concentration profiles are decreasing slightly as Casson parameter β increases. The effects of thermosolutal surface tension ratio R on the velocity, temperature and concentration profiles are illustrated in fig.
4-6. It is seen that, as the thermosolutal surface tension ratio R increases, the velocity profiles are increasing while the temperature and concentration profiles are decreasing.
Fig. 1. Velocity profiles for several values of Casson parameter 𝛃.
Fig. 2. Temperature profiles for several values of Casson parameter 𝜷.
Fig. 7-9 present the effects of the Hartmann number on the fluid velocity, temperature and concentration distributions.
Application of a magnetic field normal to an electrically conducting fluid has the tendency to produce a drag-like force called the Lorentz force which acts in the direction opposite to that of the flow, causing a flow retardation effect. This causes the decreasing in the velocity of the fluid. However, this decrease in flow speed is accompanied by corresponding increases in the fluid thermal state level.
These behaviors are clearly depicted in the decrease in the fluid velocity and an increase in the fluid temperature and concentration distributions as observed in fig. 7-9.
Fig. 3. Concentration profiles for several values of Casson parameter 𝜷.
Fig. 4. Velocity profiles for several values of thermosolutal surface tension ratio 𝑹.
Fig. 5. Temperature profiles for several values of thermosolutal surface tension ratio 𝑹.
0 1 2 3 4 5 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Du=0.02, M=1.0, Pr=0.71, R=1.0, Sc=0.62, Sr=0.2
f'
0 1 2 3 4 5 6 7
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Du=0.02, M=1.0, Pr=0.71, R=1.0, Sc=0.62, Sr=0.2
0 1 2 3 4 5 6 7 8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Du=0.02, M=1.0, Pr=0.71, R=1.0, Sc=0.62, Sr=0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Du=0.02, M=1.0,
Pr=0.71, Sc=0.62, Sr=0.2
f'
R=0,0.1,0.5,1.0,1.5,2,3,4,5
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Du=0.02, M=1.0,
Pr=0.71, Sc=0.62, Sr=0.2
R=0,0.1,0.5,1.0,1.5,2,3,4,5
P a g e | 32 Fig. 6. Concentration profiles for several values of thermosolutal
surface tension ratio 𝑹.
Fig. 7. Velocity profiles for several values of Hartman number 𝑴.
Fig. 8. Temperature profiles for several values of Hartman number 𝑴.
In order to demonstrate the effect of Soret and Dufour parameters on temperature and concentration profiles, we presented such effects in figures 10 and 11. From these figures, the Soret number increases (and the Dufour number decreases) the fluid temperature profile decreases.
This relevant to if two chemically different non reacting gases or liquids, which were initially at the same temperature, are allowed to diffuse into each other, and then there arises a difference of temperatures in the system.
In fact, this is, exactly, the definition of the inverse phenomenon of thermal diffusion (Dufour effects). The increase in this temperature difference leads to increase the buoyancy force which brings strong natural convection.
Fig. 9. Concentration profiles for several values of Hartman number 𝑴.
Fig. 10. Temperature profiles for several values of Soret and Dufour parameters.
Fig. 11. Concentration profiles for several values of Soret and Dufour parameters.
Fig. 12 presents the behavior of the temperature distributions for the variation of Prandtl number. Prandtl number signifies the ratio of momentum diffusivity to thermal diffusivity. It is clear that, the temperature decreases with increasing Prandtl number. Moreover, the thermal boundary layer thickness decreases by increasing
Prandtl number.
0 1 2 3 4 5 6 7 8 9 10 11 12
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Du=0.02, M=1.0,
Pr=0.71, Sc=0.62, Sr=0.2
R=0,0.1,0.5,1.0,1.5,2,3,4,5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Du=0.02, Pr=0.71, R=1.0, Sc=0.62, Sr=0.2
f'
M=0,0.5,1,1.5,2,2.5,3,3.5,4,5,10
0 5 10 15 20 25 30 35
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Du=0.02, Pr=0.71, R=1.0, Sc=0.62, Sr=0.2
M=0,0.5,1,1.5,2,2.5,3,3.5,4,5
0 5 10 15 20 25 30 35
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Du=0.02, Pr=0.71, R=1.0, Sc=0.62, Sr=0.2
M=0,0.5,1,1.5,2,2.5,3,3.5,4,5
0 1 2 3 4 5 6 7 8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M=1.0, Pr=0.71, R=1.0, Sc=0.62
Sr Du 0.1 0.6 0.2 0.3 0.4 0.15 0.8 0.075 1.2 0.05 2.0 0.03
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M=1.0, Pr=0.71, R=1.0, Sc=0.62
Sr Du 0.1 0.6 0.2 0.3 0.4 0.15 0.8 0.075 1.2 0.05 2.0 0.03
0 1 2 3 4 5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Du=0.02, M=1.0, , R=1.0, Sc=0.62, Sr=0.2
Pr=1.0,1.5,2.0,3.0,5.0,6.2,7.0,10.0
KKU Journal of Basic and Applied Sciences
P a g e | 33 Fig. 12. Temperature profiles for several values of Prandtl number.
Fig. 13. Concentration profiles for several values of Schmidt number.
Fig. 13 displays the effects of the Schmidt number Sc on the concentration profiles. As the Schmidt number increases, the concentration profiles are decreasing.
Moreover, the concentration boundary layer thickness decreases by increasing Schmidt number.
Fig. 14. Effects of thermosolutal surface tension ratio 𝑹 with Casson parameter 𝜷 on the temperature gradient.
Fig. 15. Effects of thermosolutal surface tension ratio 𝑹 with Casson parameter 𝜷 on the concentration gradient.
The effects of thermosolutal surface tension ratio 𝑅 with Casson parameter 𝛽 on the temperature and concentration gradients are shown in fig. 14 and 15. As the thermosolutal surface tension ratio 𝑅 increases leads to an increase in both of temperature and concentration gradients. In general, the temperature and concentration gradients are increase slightly as the Casson parameter 𝛽 increases. The variations of the temperature and concentration gradients under the effects of Soret and Dufour parameters with Casson parameter have been introduced in fig. 16 and 17.
Here, it is clear that the temperature gradient has opposite trend compared to concentration gradient under the effects
of Soret and Dufour parameters. As the Soret parameter increases with decrease in Dufour parameter leads to an increase in temperature gradient and decrease in concentration gradient. The effects of Hartman parameter 𝑀 with Casson parameter 𝛽 on the temperature and concentration gradients are shown in fig. 18 and 19. In these figures, both of temperature and concentration gradients are decrease as the Hartman parameter 𝑀 increases.
Fig. 16. Effects of Soret and Dufour parameters with Casson parameter 𝜷 on the temperature gradient.
Fig. 17. Effects of Soret and Dufour parameters with Casson parameter 𝜷 on the concentration gradient.
Fig. 18. Effects of Hartman parameter 𝑴 with Casson parameter 𝜷 on the temperature gradient.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Du=0.02, M=1.0,
Pr=0.71, R=1.0, Sr=0.2
Sc= 0.3, 0.4, 0.63, 1.0, 1.3, 2.0, 3.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
R=5.0 R=4.0 R=3.0 R=2.0
R=1.0 R=0.5
-'
R=0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
R=5.0 R=4.0 R=3.0 R=2.0
R=1.0 R=0.5
-'
R=0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60
-'
Sr=0.4 & Du=0.15 Sr=1.2 & Du=0.05 Sr=2.0 & Du=0.03
Sr=0.1 & Du=0.6
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Sr=1.2 & Du=0.05
Sr=2.0 & Du=0.03 Sr=0.4 & Du=0.15
-'
Sr=0.1 & Du=0.6
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
-'
M=10.0 M=5.0 M=0.0
M=1.0
P a g e | 34 Fig. 19. Effects of Hartman parameter 𝑴 with Casson parameter 𝜷
on the concentration gradient.
5. CONCLUSION
Laminar thermosolutal maragonai convection of a non- Newtonan fluid along a surface in the precense of the Soret and Dufour effects has been invistigated numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. The Casson fluid model is used to characterise the non-Newtonian fluid behaviour. The governing boundary- layer equations of the current problem are formulated and transformed into a self-similar form. Comparison with
previously published work was performed and the results were found to be in excellent agreement. From this study, it was found that:
1. An increase of the Casson parameter leads to suppress the velocity field. Boundary layer thickness decreases as Casson parameter increases.
2. An increase in the thermosolutal surface tension ratio enhances the fluid velocity; however, it causes a reduction in the temperature and concentration distributions.
3. An increase in the Hartmann number leads to increase both of the temperature and concentration distributions, whereas, it decreases the velocity features.
4. An increase in the Dufour number with decrease in the Soret number leads to an increase in the temperature profiles and a decrease in the concentration profiles.
5. With increasing Prandtl number the temperature decreases and the thermal boundary layer thickness decreases.
6. As the Schmidt number increases, the concentration profiles are decreasing and the concentration boundary layer thickness decreases.
Nomenclature 𝑎, 𝑏 positive constant
𝐵 applied magnetic field intensity cp specific heat at constant pressure cs specific heat at constant concentration 𝐷 the diffusion coefficient
𝑓 dimensionless stream function ℎ heat transfer coefficient 𝐶 concentration
𝐵0 strength of magnetic field 𝐷𝑢 Dufour parameter 𝑗 mass transfer coefficient 𝑘 reaction rate of solute 𝑀 Hartmann number 𝑃𝑟 Prandtl number 𝑞𝑚 surface mass transfer 𝑞𝑤 surface heat transfer 𝑁𝑢 Nusselt number
𝑅 thermosolutal surface tension ratio 𝑆𝑐 Schmidt number
𝑆ℎ Sherwood number 𝑆𝑟 Soret parameter 𝑇 dimensional temperature 𝑇𝑚 mean fluid temperature
(𝑢, 𝑣) velocity components of the fluid
(𝑥, 𝑦) coordinate axes Greek symbols 𝛼 thermal diffusivity
ɸ dimensionless concentration 𝜃 dimensionless Temperature 𝜈 kinematic viscosity
β parameter of the Casson fluid 𝜂 similarity variable
𝜌 density of the fluid 𝜇 dynamic viscosity
𝛾 surface tension on the first derivative of the temperature
𝛾̆ surface tension on the first derivative of the concentration
𝛿 surface tension
𝛿0 minimum value of the surface tension, a positiveconstant
𝜎 electrical conductivity 𝜓 stream function Subscripts Subscripts
𝑤 conditions at the surface
∞ conditions in the free stream
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0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
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-'
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