Abstract
In this paper, we investigate the effect of vertical throughflow on the onset of convection in a horizontal layer of a nonDarcy porous medium saturated with a nanofluid. A normal mode analysis is used to find solutions for the fluid layer confined between parallel plates with freerigid boundaries. The criterion for the onset of stationary and oscillatory convection is derived. The analysis incorporates the effects of Brownian motion, thermophoresis and a convective boundary condition. The effects of the concentration Rayleigh number, Lewis number, Darcy number and modified diffusivity ratio on the stability of the system are investigated.
Keywords:
throughflow; convective boundary condition; oscillatory convection; linear stability analysis1 Introduction
In the last several years, an innovative technique for improving the heat transfer characteristics by adding ultra fine metallic particles in common fluids such as water and oil has been investigated. The term nanofluid refers to these kinds of fluids which have applications in automotive industries, energy saving devices, nuclear rectors, etc., Choi [1]. Nanoparticles also have medical applications including cancer therapy and nanodrug delivery (Shaw and Murthy [2]). Buongiorno [3] noted that the absolute velocity of nanoparticles can be viewed as the sum of the base fluid velocity and a relative (slip) velocity. He concluded that in the absence of turbulence, Brownian diffusion and thermophoresis would be important. A lot of work has been done on nanofluids; see, for instance, Nadeem et al.[4], Malik et al.[5], Nadeem et al.[6] and the review by Das et al.[7].
The effect of a magnetic field on flow and heat transfer problems is important in industrial applications, such as in the buoyant upward gasliquid flow in packed bed electrodes (Takahashi and Alkire [8]), sodium oxidesilicon dioxide glass melt flows (Guloyan [9]), reactive polymer flows in heterogeneous porous media [10], electrochemical generation of elemental bromine in porous electrode systems (Qi and Savinell [11]).
The convective boundary condition is more general and realistic in engineering and industrial processes such as transportation cooling processes, material drying, etc. It therefore seems appropriate to use the convective boundary condition to study other boundary layer flow situations. Aziz [12] studied the Blasius flow over a flat plate with a convection thermal boundary condition. Ishak [13] investigated the effects of suction and injection on a flat surface with convective boundary condition. The study of the boundary layer flows over flat surfaces under convective surface boundary condition has attracted the attention of many researchers, such as Makinde and Aziz [14], Yao et al.[15].
Nanofluids have great potential as coolants due to their enhanced thermal conductivities. The enhancement of effective thermal conductivity was confirmed by experiments conducted by many researchers (Masuda et al.[16]). Instability of nanofluids in natural convection was studied by Tzou [17]. Tzou [18] studied thermal instability of nanofluids in natural convection. Thermal instability in nanofluids in a porous medium has been a topic of interest due to potential applications of such flows in food and chemical processes, petroleum industry, biomechanics and geophysical problems. Buongiorno’s model was applied to the problem of the onset of instability in a porous medium layer saturated with a nanofluid by Nield and Kuznetsov [19,20]. Kuznetsov and Nield [21] used the Brinkman model to study thermal instability in a horizontal porous layer saturated with a nanofluid. Other related studies of thermal instability in a porous medium saturated with a nanofluid include those by Kuznetsov and Nield [22] and Nield and Kuznetsov [2326].
In this study, we extend the work by Nield and Kuznetsov [26] to a nonDarcy porous medium saturated with a nanofluid. We have considered a convective boundary condition in place of an isothermal condition. The effect of the Biot number, magnetic parameter, BrinkmanDarcy parameter on thermal instability has been studied.
2 Mathematical formulation
We consider a horizontal layer of porous medium confined between the planes
where
We note that in equation (2),
is the convective derivative. We introduce nondimensional variables as
where
while the boundary conditions for velocity and nanoparticle concentration are
After substituting equations (5) into (1)(4), the resulting nondimensional equations are written as follows (ignoring primes):
The boundary conditions are written as
The dimensionless parameters in equations (8)(12) are the Prandtl number Pr, the Darcy number Da, the Vadasz number Va, the density Rayleigh number Rm, the RayleighDarcy number Ra, the concentration Rayleigh number Rn, the BrinkmanDarcy number
We note here that the parameter Rm is a measure of the basic static pressure gradient.
2.1 Basic solution
A timeindependent quiescent solution is obtained in the z direction only and has the form
Assumptions that Le is very large (of order 10^{2} to 10^{3}, see Buongiorno [3]), equations (9)(11) now reduce to
Solving equations (15) and (16) with boundary condition (12) gives
where
As
the same results were obtained by Nield and Kuznetsov [26].
2.2 Perturbation solution
To study the stability of the system, we superimpose infinitesimal perturbations on the basic state solution,
Substituting (21) in equations (8)(11), and linearizing by neglecting products of primed quantities, we get the following equations:
with the boundary conditions
The derivatives of
For regular fluids, the parameters Rn,
3 Normal modes and stability analysis
Differential equations (24), (25), (29) and boundary conditions (26) constitute a linear boundary value problem that can be solved using the method of normal modes. The perturbation quantities are of the form
where l and m are wave numbers in the x and y directions and n is the growth rate of the disturbances. Substituting into the differential equations, we obtain
subject to the boundary conditions
Here
Substituting equation (35) into equations (31)(33) gives a system of 3N algebraic equations in 3N unknowns. The vanishing of the determinant of coefficients produces an eigenvalue equation for the system. Regarding Ra as the eigenvalue, we find Ra in terms of the other parameters. It is interesting to note that the convective boundary condition is applicable for rigidfree and rigidrigid boundary conditions. In this study we mainly focus on rigidfree boundary and discuss the case of stationary and oscillatory convection. The vanishing of the shearstresses tangent to the surface and continuity equation gives the boundary conditions
We assume that the solutions to W, Θ and Φ are of the form
which satisfies the boundary conditions. Using the above transformation in equations
(31)(33) and integrating with respect to z from
where
For the nontrivial solution, the determinant of the augmented matrix is equal to
zero, s is a dimensionless growth factor. We put
3.1 Stationary convection
In the case of nonoscillatory convection,
From equation (39), we get the stationary Rayleigh number as
where
The critical cell size at the onset of instability is obtained when
Solving the above equation, we get a polynomial in
where the values of
We calculate the corresponding critical Rayleigh number
3.2 Oscillatory convection
In case of the oscillatory convection,
In order for ω to be real, it is necessary that
From equations (43) and (44), Ra (independent of ω) and ω (independent of Ra) are written as
The critical cell size at the onset of instability is obtained by solving
The critical Rayleigh number
4 Results and discussion
The critical Rayleigh numbers for stationary and oscillatory convection are calculated from equations (40) and (46), respectively. The influence of the Biot number, magnetic parameter, Darcy number, porosity of the medium on the stationary Rayleigh number and oscillatory Rayleigh number are shown in Figures 16, where we have also shown the critical stationary and oscillatory Rayleigh numbers for different Darcy and Biot numbers. In the present problem, we mainly focused on the influence of Biot number on stationary and oscillatory critical Rayleigh numbers.
Figure 1. Variation of the stationary Rayleigh number with wave number for different value of
(a)Bi,M(
The effects of the Biot number, magnetic parameter, Darcy number and porosity on the
stationary Rayleigh number are shown in Figure 1, where it is evident that
Figure 2 shows the influence of the modified diffusivity ratio λ, the modified particledensity increment and the Lewis number on the stationary Rayleigh
number. The stationary Rayleigh number increases with
Figure 2. Variation of the stationary Rayleigh number with wave number for different value of
(a)λ,
The influence of the Biot number, magnetic parameter, Peclet number and Vadasz number on the oscillatory Rayleigh number is shown in Figure 3. The critical oscillatory Rayleigh number increases with the magnetic parameter while decreasing with parameters Bi, Va and Q. Hence the magnetic parameter stabilizes the oscillatory convection, while the other parameters are destabilizing to the oscillatory regime.
Figure 3. Variation of the oscillatory Rayleigh number with wave number for different value
of (a)Bi,M(
The critical stationary and oscillatory Rayleigh number as a function of the Darcy
number is shown in Figure 4. The critical stationary Rayleigh number decreases with the Biot number. For
Figure 4. Influence of the Biot number on (a)
Figure 5. Effect of the Biot number on critical
We have defined Rn so that it is positive when the particle density increases upwards (the destabilizing
situation). From Figure 6, we note that Ra takes a negative value when Rn is sufficiently large. In this case, the destabilizing effect of nanoparticle concentration
is so large that the bottom of the fluid layer must be cooled relative to the top
to produce a state of neutral stability as earlier found by Kuznetsov and Nield [21] in the absence of a magnetic field and higher Biot numbers. In the present problem,
a state of neutral stability appeared when
Figure 6. Effect of the Biot number on critical
5 Conclusion
In this study we used linear stability to investigate the onset of thermal instability in a nonDarcy porous medium saturated with a nanofluid, and with a convective boundary condition. We have determined the effects of various embedded parameters such as the Biot number, the magnetic parameter and the Darcy number on the critical Rayleigh number for the onset of both oscillatory and stationary thermal instabilities. We have shown that increasing the Darcy number and the magnetic parameter has the effect of increasing the critical Rayleigh number for the onset of thermal instabilities in the case of stationary convection, while increasing the Biot number and the porosity is destabilizing to the stationary regime. The modified diffusivity ratio, particle density increment and the Lewis number help to stabilize stationary convection. Oscillatory convection was found not to be as sensitive to the fluid and physical parameters as stationary convection.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors were involved in carrying out this study.
Acknowledgements
The authors wish to thank the University of KwaZuluNatal for financial support.
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