^{1}Department of Mathematics, Faculty of Computer and Mathematics Sciences, Universiti Teknologi MARA Pahang, Bandar Pusat Jengka, Pahang, Malaysia
^{2}Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor, Malaysia
^{3}Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Received date: April 30, 2017; Accepted date: May 09, 2017; Published date: May 20, 2017
Citation: Dzulkifli NF, Bachok N, Pop I, Yacob NA, Arifin NM, et al. (2017) Soret and Dufour Effects on Unsteady Boundary Layer Flow and Heat Transfer of Nanofluid Over a Stretching/Shrinking Sheet: A Stability Analysis. J Chem Eng Process Technol 8:336. doi:10.4172/2157-7048.1000336
Copyright: © 2017 Dzulkifli NF, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Visit for more related articles at Journal of Chemical Engineering & Process Technology
The effects of Soret and Dufour parameters on the boundary layer flow in nanofluid over stretching/ shrinking with time dependent is studied using Buongiorno model. The system of partial differential equations is transformed to the system of ordinary differential equations by applying similarity transformation. The results are obtained numerically using bvp4c in Matlab. The reduced skin friction coefficient reduced Nusselt number, velocity, temperature and concentration profiles are shown graphically with different values of Soret effect, Dufour effect, mass flux parameter, unsteadiness parameter, thermophoresis as well as Brownian motion parameter where the dual solutions are obtained. The unsteadiness parameter and mass flux parameter expand the range of solution for stretching/ shrinking parameter. Meanwhile, the Soret and Dufour parameters are found to affect the heat transfer rate at the surface. In order to determine the stability of the solutions, stability analysis is performed.
Unsteady boundary layer; Nanofluid; Soret effect; Dufour effect; Heat transfer; Stability analysis
A unsteadiness parameter
C nanoparticle volume fraction
C_{f} skin friction coefficient
C_{∞} ambient fluid concentration
C_{s} concentration susceptibility
D_{B} Brownian diffusion coefficient
D_{f} Dufour number
D_{m} coefficient of mass diffusivity
f dimensionless stream function
k thermal conductivity
Le Lewis number
Nb Brownian motion parameter
Nt thermophoresis parameter
Nu_{x} local Nusselt number
p fluid pressure
Pr Prandtl number
q_{w} surface heat flux
Re_{x} local Reynolds number
s mass flux parameter
Sr Soret number
t time
T fluid temperature
T_{w} plate temperature
T_{∞} plate temperature
T_{m} mean fluid temperature
u,v velocity components along the x and y directions, respectively
u_{w} velocity of the plate
v_{w} velocity of mass flux
x,y Cartesian coordinates along the surface and normal to it, respectively
Greek symbols
α thermal diffusivity
γ eigenvalue
η similarity variable
θ dimensionless temperature
ε stretching/ shrinking parameter
μ dynamic viscosity
v kinematic viscosity
ϕ nanoparticle volume fraction parameter
ρ fluid density
(ρc_{p})_{nf} heat capacity of the nanofluid
(ρc_{p})_{p} heat capacity of the nanoparticle
τ_{w} surface shear stress.
ψ stream function
Subscripts
w condition at the surface of the surface
∞ ambient condition
n_{f} nanofluid
p nanoparticle
f basefluid
Superscript
Differentiation with respect to η.
Generally, it was known that heat and mass fluxes were created from temperature and concentration gradient, respectively. However, heat flux is actually can existed due to the concentration gradient which is known as Soret effect. Same goes to the mass flux where the flux occurred by the temperature gradient and is called Dufour effect. Configuration involving the heat and mass transfer with Soret and Dufour effects is an important subject due to a wide range of applications such as the solidification of binary alloys, groundwater pollutant migration, chemical reactors, geosciences multi-component melts, oil-reservoirs, isotope separation, and in mixture between gases. Generally, the effects of diffusion of matter caused by temperature gradients (Soret effect) and diffusion of heat caused by concentration gradients (Dufour effect) can be become influential when the temperature and concentration gradients are very large. Joly et al. [1] analyzed the thermal and solutal effects on natural convection in a vertical enclosure. Mansour et al. [2] have obtained a multiplicity of solutions induced by thermosolutal convection in a square porous cavity with horizontal concentration gradient in the presence of Soret effect. Chamkha and Rashad [3] studied the Soret and Dufour effects on unsteady double-diffusive convection flow from a rotating vertical cone with chemical reaction effects. Kafoussias and Williams [4] investigated the effects of Soret and Duffour on mixed convection and mass transfer laminar boundary layer flow over a vertical flat plate. Besides Soret and Dufour effects, Alam et al. [5] has considered suction variable on mixed convection flow over a semi-infinite vertical porous flat plat and they found that the wall suction stabilized the boundary layer growth of velocity, temperature and concentration. Similar to Alam et al. [5], El-Kabeir [6] also considered the porous medium in the study but he considered stretching cylinder rather than flat plate. In addition, the effect of chemical reaction was also investigated in the work.
Low thermal conductivity of working fluids such as water, mineral oils, ethylene glycol is a primary limitations of enhancing the heat transfer performance. To avoid this situation to add high thermal conductivity of nanoparticles in the working fluid. Due to the excellent thermal property, nanofluids are used in the areas of building heating, heat exchangers, and the automotive cooling process. The term of nanofluid first introduced by Choi [7] in 1995 is one of the mean to improve or enhance the thermal conductivity, which is proportional to heat transfer of the conventional regular fluid such as water, ethylene glycol and mineral oil. Basically, these regular fluids have low thermal conductivity and by dispersing nanoparticles in the base fluid, the thermal conductivity of the fluid can be enhanced. Several works are concerned with natural convection filled with nanoparticles, Khanafer et al. [8], Tiwari and Das [9], Oztop and Abu-Nada [10], who concluded that the Cu-water based nanofluid has the higher heat transfer rate compared with other nanofluids. Lai and Yang [11], Aminossadati and Ghasemi [12] reported that increase the solid volume fraction of the nanoparticles yields a increasing value of heat transfer.
Buongiorno [13] proposed a model for momentum, heat and mass transfer in nanofluid where this model focussed on two slip mechanism that can produced relative velocity between nanoparticles and regular fluid which were Brownion diffusion and thermophoresis. Buongiorno model has been used by Kuznetsov and Nield [14] to study the natural convective boundary layer flow passing through a vertical plate in nanofluid. Meanwhile, different boundary layer condition has been applied in the model by Mansur and Ishak [15], Kuznetsov and Nield [16]. Some of the authors studied the effect of nanouids for the case of mixed convection Nemati et al. [17], Cimpean and Pop [18], etc. Very good literature reviews on convective flow and applications of nanofluids have been done by Jou and Tzeng [19], Das et al. [20], Kleinstreuer et al. [21], Kakaç and Pramuanjaroenkij [22], Wong and Leon [23], Wen et al. [24], Jaluria et al. [25], Mahian et al. [26], Nield and Bejan [27], Sheikholeslami and Ganjii [28], and Shenoy et al. [29] have given reviews of nanofluid on natural convection heat transfer.
Recently, the stability analysis is important to be performed when there are more than one solution exists in order to determine the stability of the solutions. The analysis can be obtained by adopting the work proposed by Merkin [30]. According to According to Ishak [31] the implementation of the analysis was to investigate the growth of disturbances for first and second solutions where the solution with initial decay of disturbance represented the stable solution, while the solution with initial growth of disturbance indicated the unstable solution. There were some authors who performed the analysis in their studies over various cases such as Weidman et al. [32]. Aleng et al. [33], Nazar et al. [34], Hafidzuddin et al. [35], Bachok et al. [36] and Najib et al. [37] studied several problems on stretching/shrinking sheet in a viscous and incompressible fluids and they found that the first solution was stable while the second solution was unstable. It is worth mentioning that the flow caused by a stretching/shrinking surface gets up mostly in the field of chemical engineering and metallurgy such as aerodynamic and polymer extrusion, cooling of metallic plate, drawing of paper films, glass blowing, and paper production.
The present study is an extension of Bachok et al. [38] with new boundary condition proposed by Kuznetsov and Nield [16]. The main purpose of this present work is to investigate the boundary layer flow and heat transfer characteristics over a stretching/ shrinking sheet in nanofluid when the Soret and Dufour effects are taken into consideration for unsteady problem. The governing equations are transformed to ordinary differential equations using dimensionless similarity transformation parameter and are solved numerically by Matlab. The stability analysis is performed in order to determine the stability of the numerical solutions.
Unsteady boundary layer flow over a stretching/ shrinking surface immersed in nanofluid is considered. At t<0, it is assumed that the surface is in stationary state with velocity μ_{w=0}. As t>0, the surface begin to stretch or shrink where the velocity of the sheet is μ_{w}=Ax / t which A>0 is dimensionless acceleration parameter. The velocity of mass flux is represented by v_{w} where v_{w}>0 is for injection and v_{w}<0 is for suction. Let the uniform temperature of the plate is T_{w} T_{∞ }and C_{∞} are the temperature and the nanoparticle volume fraction of the ambient fluid. Following the assumptions above, the governing equations of the problem are, see Bachok et al. [38], and Alam et al. [5],
(1)
(2)
(3)
(4)
(5)
subject to boundary conditions (see Kuznetsov and Nield [16])
t<0, v=0, u=0, T=T_{∞}, C=C_{∞} for all x and y,
(6)
u →0, T → T_{∞}, C →C_{∞} as y → ∞,
Where x and y are the Cartesian coordinate along and perpendicular to the plate with u and v are the velocity component in x and y directions, respectively, T is the temperature of the nanofluid, C is the nanoparticle fraction, v is the kinematic viscosity of the nanofluid, α is the thermal diffusivity of the nanofluid, ρ is the fluid pressure, D_{B} is the Brownian diffusion coefficient, D_{T} is the thermophoretic diffusion coefficient, D_{m} is the coefficient of mass diffusivity, C_{p} is the specific heat at constant pressure, T_{m} is the mean fluid temperature, k_{T} is the thermal diffusion ratio, Cs is the concentration susceptibility, τ=(ρ c_{p})_{f} / (ρ c_{p})_{p} where(ρc_{p})_{p} is the heat capacity of the nanofluid and(ρc_{p})_{p} is the heat capacity of the nanoparticle, respectively.
In order to find the similarity solution of Equations 1- 6, the similarity transformation parameters are introduced as follows
(7)
where η is the dimensionless similarity variable, the prime denotes differentiation with respect to η , ψ is the stream function which defines ) and ϕ(��) are dimensionless stream, temperature and concentration functions of the fluid in the boundary layer, respectively and v_{w} is represented as v_{w}=-A(v/t)^{1/2}s, where s is the constant mass flux which s>0 for suction and s<0 for injection.
Using similarity transformation Equation 7, the partial differential Equation 1- 5 are transformed to ordinary differential Equation 8-10 subject to boundary conditions
(8)
(9)
(10)
(11)
Where Pr is Prandtl number, is Brownian motion parameter, Nt is the thermophoresis parameter, Le is Lewis number, Df is Dufour number and Sr is the Soret number which are defined as
(12)
The skin friction coefficient and the local Nusselt number are the quantities of physical interest in this problem and defined as
(13)
Where τ_{w} and q_{w} are the shear stress and heat flux at the surface, respectively as given by
(14)
Where μ is the dynamic viscosity of the fluid and k is the thermal conductivity of the nanofluid. Equations 7, 13, 14 we obtain
(15)
Where Re_{x}=u_{w}/v represents local Reynold number.
Weidman et al. [32], and Roşca and Pop [39,40] have shown that the lower branch solutions are unstable (not physically realizable), while the upper branch solutions are stable (physically realizable). We test these features by considering Equations 1-5. Thus, we introduce the new dimensionless time variable τ=ln(t/t_{o}), where t_{o} is a characteristic time. The use of τ is associated with an initial value problem and is consistent with the question of which solution will be obtained in practice (physically realizable). Using the variables τ and Equation 7, we have
(16)
As in Weidman et al. [32], we introduce the new variable τ=ln(t/t_{o}) as it is associated with initial value problem, where to is a characteristic time and we take t_{o}=1 so we have τ=ln(t) Thus, Equations 1-5 can be written as
(17)
(18)
(19)
subject to the initial and boundary conditions
(20)
To determine the stability of the solution and satisfying the boundary-value problem Equation 8-11, we write Weidman et al. [32], and Roşca and Pop [39,40])
(21)
Where γ is an unknown eigenvalue parameter, and and are small relative to Substituting Equation 21 into Equation 17-19, and take τ=0, we obtain the following linear eigenvalue problem
(22)
(23)
(24)
subject to the boundary conditions
Solving the eigenvalue problem Equations 22-25 we obtain an infinite number of eigenvalues γ_{1}<γ_{2}<γ_{3}<…. If the smallest eigenvalue is positive the flow is stable and if the smallest eigenvalue is negative the flow is unstable.
According to Harris et al. [41], the range of possible eigenvalues can be determined by relaxing a boundary condition on or For the present problem, we relax the boundary condition as η →∞ and for a fixed value of γ we solve the system of Equations 22- 24 subject to [25] along with the new boundary condition F_{0} ηη(0)=1.
In order to obtain numerical results of the reduced skin friction coefficient F_{0} ηη (0) and reduced Nusselt number -θ=(0) with the effect of various physical parameters, we fixed some parameter values such as Prandtl number Pr=0.71 (air), Lewis number Le=1, Brownian motion parameter Nb=0.5 and thermophoresis parameter Nt=0.5 and solve the system of ordinary differential Equation 8-10 subject to boundary conditions Equation 11 by applying bvp4c in Matlab. Further, the presence of Soret and Dufour parameters are now considered and the values of Soret, Sr and Dufour, Df are chosen in such way so that the mean temperature Tm remains as a constant (see El-Kabeir [6]). A comparison with previous results (see Bachok et al. [38]) with the absence of Soret and Dufour effects is made in order to validate the numerical computation where the comparison shows a good agreement as can be seen in Table 1. Apart from that, the presence of Soret and Dufour effects doesn’t affect the range of solution of ε since the value of ε_{c} are identical with the absence of both effects.
Df | Sr | s | Bachok et al. [38] ε_{c} |
Present results ε_{c} |
---|---|---|---|---|
0 | 0 | 0 | 0.0000 | 0.0000 |
1 | -0.2138 | -0.2139 | ||
2 | -0.9259 | -0.9260 | ||
0.15 | 0.4 | 0 | - | 0.0000 |
1 | - | -0.2139 | ||
2 | - | -0.9260 |
Table 1: Comparison numerical values of critical epsilon, ε_{c} for different s when Nt=Nb=0.5, Le=2, A=1 and Pr=0.71.
The effects of mass suction/ injection parameters on the reduced skin friction coefficient F_{0} ηη (0) and reduced Nusselt number -θ=η(0) with stretching/ shrinking parameter ε, when A=1 are shown in Figures 1 and 2. Moreover, Figures 3 and 4 present the effects of unsteadiness parameter, A on those two quantities physical of interest. Based on the Figures 3 and 4 the dual solutions are found when ε ≤ ε_{c} and no solution can be obtained when ε >ε_{c}. From these figures, the range of solutions, ∣ε_{c}∣ are expanded as the value of A and s increased.
Figures 1 and 4 show that the reduced skin friction coefficient increases when the ε decreases whereas the reduced Nusselt number slightly decreases for the same decreasing parameter. Besides, the range of solutions for higher value of mass flux parameter are wider compared to smaller value as can be noticed in Figures 1 and 2. Likewise Figures 3 and 4 demonstrates the ranges of solution for larger value of unsteadiness parameter are broader than lower value of A with ε.
Figure 1 illustrates that as mass flux increases, the values of F_{0} ηη (0) are found decelerated for ε >0 (stretching sheet) and opposite trend can be observed when ε (shrinking sheet). Moreover, the values -θ(0) are larger for higher values of mass flux parameter s as depicts in Figures 2 and 3 shows that a hike in unsteadiness parameter increases the skin friction at the surface for shrinking sheet but for stretching sheet ε >0, the skin friction declines. It is also can be mentioned from Figure 4, the ε <0 accelerating unsteadiness parameter increases the reduced Nusselt number which proportional to the heat transfer rate at the surface.
Figure 5 represents the effect of different values of Soret Sr and Dufour D_{f} on reduced Nusselt number with. From the figure, it is found that the increase in Sr (or decreasing D_{f} ) doesn’t expand the range of solution where the dual solutions are found when ε >ε_{c} =-0.2139 while no solution is reported when ε >ε_{c} =-0.2139. Apart from that, the increase in Sr (or decreasing D_{f})) decreases the reduced Nusselt number and the decrease in ε decreases the reduced Nusselt number. This shows that the heat transfer rate at the surface decreases for higher Soret (or lower Dufour) parameter.
The effects of Soret and Dufour with mass flux parameter s on reduced Nusselt number, - θη(0) is shown in Figure 6. The increasing Soret (or decreasing Dufour) decreases the reduced Nusselt number which means the rate of heat transfer decreases for higher Soret (or lower Dufour) parameter. While for increasing mass flux parameter s the reduced Nusselt number is increasing for both solutions. The presence of Soret and Dufour effects is found doesn’t change the range of the solutions where the dual solutions exist when s>sc=0.9688 and no solution can be obtained when s<sc=0.9688. In addition, it is revealed that the solutions exist only for s>0 which represents mass suction parameter. Hence, the dual solutions can be obtained when the mass suction parameter is considered the study and no solution is existed as mass injection parameter is taken into account.
The variations of local Nusselt number with thermophoresis parameter Nt and Brownion motion parameter Nb for several values of Soret and Dufour are shown in Figures 7 and 8 respectively. Figure 7 shows that the local Nusselt number decreases with increasing Soret (or decreasing Dufour) which indicates the heat transfer rate at the surface decreases when the heat flux due to the concentration gradient is increased or mass flux generated by temperature gradient is decreased. In addition, the local Nusselt number is increased when the thermophoresis parameter increases. Physically, the rate of heat transfer is found increased as the migration of the particles increase. Figure 7 demonstrates the local Nusselt number which represents rate of heat transfer decreases when the Soret (Dufour) parameter increases (decreases) as well as the Brownion motion parameter Nb increases.
Figures 8-21 present the effect of unsteadiness, mass flux, stretching/ shrinking velocity, and Sufor and Dufour parameters on velocity, temperature and concentration profiles. As can be seen in these profiles, the boundary conditions Equation 11 are satisfied asymptotically. Besides that, the dual solutions are obtained which support the variations of reduced skin friction coefficient as well as reduced Nusselt number as presented in Figures 1-8. These profiles also show that the boundary layer thicknesses for first solution are smaller than second solution in each profile. The effect of increasing Sr (or decreasing Df) on temperature and concentration profiles can be observed in Figures 18 and 19 for shrinking parameter and Figures 20 and 21 for stretching parameter. It is found that increasing Sr (or decreasing D_{f}) decreases the temperature and the concentration at the surface for both stretching and shrinking cases.
The stability of dual solutions can be determined by performing the stability analysis and has been solved using bvp4c to obtain the smallest eigenvalue γ for some values of mass suction parameter s and shrinking parameter γ as can be seen in Table 2. According to the table, the smallest eigenvalues for upper branch solution is positive while for lower branch solution is negative. Thus, the first solutions with positive smallest eigenvalues indicate that the solution is stable and physically reliable while second solution is unstable and physically unreliable.
s | ε_{c} | ε | γ (First solution) | γ (Second solution) |
---|---|---|---|---|
1 | -0.2139 | -0.2135 | 0.0314 | -0.0311 |
-0.213 | 0.0472 | -0.0466 | ||
-0.21 | 0.0991 | -0.0964 | ||
2 | -0.9260 | -0.926 | 0.0024 | -0.0007 |
-0.92 | 0.1509 | -0.1468 | ||
-0.91 | 0.2482 | -0.2373 |
Table 2: The smallest eigen values γ for some values of s and when Nt=Nb=0.5, Df=0.15, Le=1, Sr=0.4, A=1 and Pr=0.71.
The effect of Soret and Dufour parameters as well as the unsteadiness, mass flux, thermophoresis, Brownian motion parameters on heat transfer characteristics for unsteady boundary layer flow over stretching/ shrinking sheet in nanofluid is investigated numerically. It is found that the dual solutions are obtained in this study and presented graphically for reduced skin friction coefficient, reduced Nusselt number as well as the velocity, temperature and nanoparticle volume fraction profiles. The different values of Soret and Dufour parameters are found doesn’t affect the range of ε and s parameters since the critical value of ε (ε_{c}=-0.2139) and s (s_{c}=0.9688) are unchanged even though the values of Soret and Dufour parameter are different. In contrast, the higher values of unsteadiness and mass flux parameters expand range of where the dual solutions can be obtained when ε >-22 for different mass flux parameter and ε >- 88 for various unsteadiness parameter. The stability of the solutions is performed where the first solution is found stable while the second solution is unstable.