Kelvin-Helmholtz instability with mass transfer through porous media: Effect of irrotational viscous pressure*

AWASTHI Mukesh Kumar

Department of Mathematics, University of Petroleum and Energy Studies, Dehradun-248007, India, E-mail:mukeshiitr.kumar@gmail.com

Kelvin-Helmholtz instability with mass transfer through porous media: Effect of irrotational viscous pressure*

AWASTHI Mukesh Kumar

Department of Mathematics, University of Petroleum and Energy Studies, Dehradun-248007, India, E-mail:mukeshiitr.kumar@gmail.com

(Received June 4, 2013, Revised August 7, 2013)

This paper studies the effect of the irrotational viscous pressure on Kelvin-Helmholtz instability of the plane interface of two viscous and incompressible fluids in a fully saturated porous media with mass and heat transfers across the interface. In the earlier work, the instability of the plane interface of two viscous and streaming miscible fluids through porous media was studied by assuming that the motion and the pressure are irrotational and the viscosity enters the normal stress balance. This theory is called the viscous potential flow theory. Here, we use another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance by considering viscous contributions of the irrotational pressure. The Darcy-Brinkman model is used in the investigation and the stability criterion is formulated in terms of a critical value of the relative velocity. It is observed that the heat and mass transfer has a destabilizing effect on the stability of the system while the irrotational shearing stresses stabilize the system.

Kelvin-Helmholtz stability, porous medium, irrotational viscous pressure, heat and mass transfer

Introduction

1. Problem formulation

Consider the parallel flow of two incompressible, viscous and thermally conducting fluids in two infinite, fully saturated, uniform, homogeneous and isotropic porous media with porosities ε(1),ε(2)and permeabilitiesk11,k12. In the formulation, the superscripts 1 and 2 denote the variables associated with the lower fluid and the upper fluid, respectively. In the equilibrium state, the lower fluid of densityρ(1)and viscosityµ(1)occupies the region -h＜y＜0and the

1 upper fluid of densityρ(2)and viscosity µ(2)occupies the region0＜y＜h2. The interface between the two fluids is assumed to be well defined and is initially flat to form the planey=0(Fig.1). Also, it is assumed that the two fluids are with uniform horizon-tal velocities U1and U2throughout the two superposed porous media. The bounding surfacesy=-h1and y=h2are assumed to be rigid. The temperatures aty=-h1,y =0and y=h2are T1,T0and T2, respectively. In the basic state, the thermodynamics equilibrium is hold and the interface temperature T0is set to be equal to the saturation temperature.

On applying the small disturbances, the interface can be expressed as

whereηis the perturbation from its equilibrium value. The unit outward normal up to the first order term is given by

where exand eyare the unit vectors along xand ydirections, respectively.

The velocity is expressed as the gradient of a potential function and the potential functions satisfy the Laplace equation as a consequence of the incompressibility. That is,

At the walls the normal velocity vanishes, hence

It is assumed that the phase-change takes place locally in such a way that the net phase-change rate at the interface is equal to zero. The interfacial condition, which is expressed as the conservation of mass across the interface, is given by the equation

where 〚x〛=x(2)-x(1)represents the difference, as the quantity of mass across the interface. Using Eqs.(1) and (5), it follows that

The interfacial condition for the energy transfer can be expressed as

whereL is the latent heat released during the phase transformation and S(η)is the net heat flux from the interface.

In the equilibrium state, the heat fluxes in the positivey-direction in the fluid Phases 1 and 2 are expressed as-K1(T1-T0)/h1and K2(T0-T2)/h2, respectively where K1and K2are the heat conductivities of the two fluids. Let us denote

Expand S(η)in a Taylor series about η=0 as

If we take S (0)=0, we have

which indicates that in the equilibrium state the heat fluxes are equal across the vapor-liquid interface.

The balance of the linear momentum for the viscous fluid through a porous media according to the Brinkman-Darcy equation is

If the fluids are miscible with the heat and mass transfer across the interface, the interfacial condition for the conservation of momentum can be expressed as

where pj(j =1,2)represent the irrotational pressures,σdenotes the surface tension coefficient andnis the unit normal vector on the interface, respectively. The surface tension is assumed to be a constant, neglecting its dependence on temperature.

2. Viscous correction for the viscous potential flow analysis

To include the effect of the irrotational shearing stresses, the formulation of the viscous correction for the viscous potential flow analysis is developed using the basic mechanical energy balance equation.

Suppose that n1=eyis the unit outward normal on the interface for the lower fluid,n2=-n1is the unit outward normal for the upper fluid,t=exis the unit tangent vector. We will use “i” for “irrotational”and “v ” for “viscous” and subscripts “1” and “2” for lower and upper fluids, respectively. The normal and shear parts of the viscous stress are represented by τnand τs, respectively.

The mechanical energy equations for upper and lower fluids are, respectively:

where Dj(j=1,2)is the symmetric part of the rate of strain tensor for lower and upper fluids, respectively. The normal velocity is continuous across the interface, so

and summing the respective sides of Eqs.(13) and (14), we obtain

Here two viscous pressures pvand pvare introdu-

1

2 ced for lower and upper potential flows, respectively. It is assumed that these two pressure corrections can resolve the discontinuity of the shear stress and the tangential velocity at the interface, so

Taking the above conditions into account, Eq.(15) takes the form

If we compare Eqs.(15) and (16), we have

The viscous pressure is governed by the following equation

Including the viscous pressure along with the irrotational pressure, the equation of conservation of momentum (12), will take the form

Here the irrotational pressure pifor (j=1,2)can

j be obtained by solving the Bernoulli’s equation.

3. Linearized equations

The small disturbances are imposed on Eqs.(6), (7) and (22) and retaining the linear terms, we can obtain the following equations.

4. Normal mode analysis and dispersion relation

The normal mode technique is used to find the solution of the linearized governing equations.

Let the interface elevation be represented by

wherekandωdenote the wave number and the complex growth rate, respectively andC denotes the complex constant.

The solution of Eq.(3) by using the normal mode analysis and the boundary conditions can be expressed as

On solving Eq.(21) along with Eq.(17), the contribution of the viscous pressure can be written as

Substituting the values ofη,φ(2),φ(1)and Eq.(29) into Eq.(25), we obtain the following dispersion relation

5. Dimensionless form of the dispersion relation

Equation (40) contains the growth rate parameter θ= µ(1)/[ρ(1)hQ], which depends linearly on the kinematic viscosityν(1)=µ(1)/ρ(1)of the lower fluid.

6. Comparison with previous results

The dispersion relation for the pressure corrections for the potential flow analysis of KHI with consideration of the heat and mass transfer is quadratic in the growth rate and the instability occurs due to the positive values of the disturbance growth rate (i.e., ωI＞0). If ωIis negative, the perturbation decays with time, while if ωI＞0, the system is unstable as the perturbation grows exponentially with time. The caseωI=0 is the marginal stability case.

Figure 3 shows the comparison between the relative velocity curves obtained in the VPF analysis by the Brinkman model (without condideration of gravity) with those obtained in the present (VCVPF) analysis forh=0.0015mand α=1000kg/m3s. It may be

1observed that the absence of gravity makes the system more unstable but the VCVPF solution is still more stable than the VPF solution. The critical values of the relative velocity and the wave number for different vapour fractions for the VCVPF solution as well as the VPF solution are given in Table 1. Figure 4 shows a comparison between the growth rates obtained from the VPF solution with those obtained from the VCVPF solution. It can be observed that the growthrates in the VCVPF solution are lower in comparison with those in the VPF solution, which indicates that the VCVPF solution is more stable than the VPF solution.

The effect of the irrotational shearing stresses on the VPF analysis of KHI with consideration of the heat and mass transfer across the interface in the homogeneous media was studied by Awasthi et al.[9]. To study the effect of the porous medium on the stability of the system, our results are compared with the results obtained by Awasthi et al.[9]in Fig.5. The following parameters are considered for the system of interest containing water in the lower region and vapour in the upper region.

7. Results and discussions

In this section, the numerical computation is made by using the expressions presented in the previous sections. The water and the vapor are taken as the working fluids identified with Phase 1 and Phase 2, respectively, such that T2＞T0＞T1. The steam is treated as incompressible since the Mach number is expected to be small. In the vaporization case, the watervapor interface is in the saturation condition and the temperatureT0is equal to the saturation temperature.

For different values of the vapor fraction, the neutral curves for the relative velocity are shown in Fig.6 with3the heat transfer coefficientα= 1 000 kg/ms. As the vapor fraction increases, the vapor pressure at crest will fall below the equilibrium vapor pressure and the evaporation will take place. As a consequence of this, the amplitudes of the disturbance wave will be diminished, which stabilizes the system as observed from Fig.6. Figure 7 shows the neutral curves for the relative velocity for different values of the heat transfer coefficientα. It can be observed that the heat transfer has a destabilizing effect on the stability of the system. The critical values of the relative velocity and the corresponding wave number for different values of the heat transfer coefficient αare given in Table 2. The Table confirms that the critical value of the relative velocity decreases asα increases. As the heat transfer increases across the interface, the disturbance waves will grow faster and the system will be destabilized.

The effect of the porosity of the lower phase ε(1)on the neutral curves of the relative velocity is shown in Fig.8. The stable region reduces as the porosity of the lower phase inc reases andsothat it has a destabilizingeffectonthestabilityofthesystem.Theeffectof the upper phase porosity ε(2)on the critical values of the relative velocity is shown in Table 3. It can be observed that the critical value of the relative velocity increases asε(2)increases and hence, the stable region increases and so the upper phase porosity has a stabilizing effect. The effect of the lower phase permeability on the neutral curves of the relative velocity is shown in Fig.9. As the permeability of the lower phase increases, the stable region reduces and this indicates that the lower phase permeability has a destabilizing effect on the stability of the system. The values of the critical relative velocity and the corresponding wave number for different values of the upper phase permeability are shown in Table 4. It can be observed that the upper phase permeability has a small stabilizing effect on the stability of the system.

8. Conclusion

The effect of irrotational shearing stresses on the viscous potential flow analysis of Kelvin-Helmholtz instability in the presence of the heat and mass transfer through a porous medium is investigated. The viscous pressure is included in the normal stress balance and it is assumed that this viscous pressure will resolve the discontinuity of tangential stresses, as the case in the viscous potential flow theory. This viscous pre-ssure is obtained by the mechanical energy balance equation. A dispersion relation is derived and the stability is discussed theoretically as well as numerically. The stability criterion is given in terms of a critical value of the relative velocity. It is observed that the heat and mass transfer has a destabilizing effect on the stability of the system while the vapor fraction plays a stabilizing role. It is also observed that the irrotational shearing stresses stabilize the system in the presence of the heat and mass transfer while the porous medium has a destabilizing effect. The lower phase porosity destabilizes the system while the upper phase porosity has a stabilizing effect.

[1] FUNADA T., JOSEPH D. D. Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel[J]. Journal of Fluid Mechanics, 2001, 445: 263-283.

[2] DONG Yu-hong. The nonlinear behavior of interface between two-phase shear flows with large density ratios[J]. Journal of Hydrodynamics, Ser. B, 2006, 18(5): 587-592.

[3] ASTHANA R., AGRAWAL G. S. Viscous potential flow analysis of Kelvin-Helmholtz instability with mass transfer and vaporization[J]. Physica A, 2007, 382(2): 389-404.

[4] AWASTHI M. K., AGRAWAL G. S. Nonlinear analysis of capillary instability with heat and mass transfer[J]. Communications Nonlinear Science in and Numerical Simulation, 2012, 17(6): 2463-2475.

[5] WANG J., JOSEPH D. D. and FUNADA T. Pressure corrections for potential flow analysis of capillary instability of viscous fluids[J]. Journal of Fluid Mechanics, 2005, 522: 383-394.

[6] AWASTHI M. K., ASTHANA R. and AGRAWAL G. S. Pressure corrections for the potential flow analysis of Kelvin-Helmholtz instability[J]. Applied Mechanics and Material, 2011, 110-116: 4628-4635.

[7] WANG J., JOSEPH D. D. and FUNADA T. Viscous contributions to the pressure for potential flow analysis of capillary instability of viscous fluids[J]. Physics of Fluids, 2005, 17: 052105.

[8] AWASTHI M. K., AGRAWAL G. S. Viscous contributions to the pressure for the potential flow analysis of magnetohydrodynamic Kelvin-Helmholtz instability[J]. International Journal of Applied Mechanics, 2012, 4(1): 1-16.

[9] AWASTHI M. K., ASTHANA R. and AGRAWAL G. S. Pressure corrections for the potential flow analysis of Kelvin-Helmholtz instability with heat and mass transfer[J]. International Journal of Heat and Mass Transfer, 2012, 55(9-10): 2345-2352.

[10] EL-SAYED M. F. Effect of normal electric fields on Kelvin-Helmholtz instability for porous media with Darcian and Forchheimer flows[J]. Physica A, 1998, 255(1): 1-14.

[11] EL-SAYED M. F., Electrohydrodynamic Kelvin-Helmholtz instability of two superposed Rivlin-Ericksen viscoelastic dielectric fluid-particle mixtures in porous medium[J]. Chaos, Solitons and Fractals, 2002, 14(8): 1137-1150.

[12] HOU Jian, ZHANG Shun-kang and DU Qing-jun et al. A streamline based predictive model for enhanced oil recovery potentiality[J]. Journal of Hydrodynamics, 2008, 20(3): 314-322.

[13] ASTHANA R., AWASTHI M. K. and AGRAWAL G. S. Kelvin-Helmholtz instability of two viscous fluids in porous medium[J]. International Journal of Applied Mathematics and Mechanics, 2012, 8(14): 1-13.

[14] ALLAH M. H. O. Viscous potential flow analysis of Interfacial instability with mass transfer through porous media[J]. Applied Mathematics and Computation, 2011, 217(20): 7920-7931.

[15] AWASTHI M. K., ASTHANA R. Viscous potential flow analysis of capillary instability with heat and mass transfer through porous media[J]. International Communications and Heat Mass Transfer, 2013 40: 7-11.

10.1016/S1001-6058(14)60069-X

* Biograpgy: AWASTHI Mukesh Kumar (1986-), Male, Ph. D., Assistant Professor