Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid

Abstract

The stability of the conduction regime of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and Oldroyd-B fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for power-law fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter \(\Lambda _2 \) portrays stabilizing influence on the system while increasing stress relaxation parameter \(\Lambda _1\) displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to Oldroyd-B fluid.

This is a preview of subscription content, log in to check access.

Abbreviations

a :

Vertical wave number

c :

Wave speed

\(c_r\) :

Phase velocity

\(c_i\) :

Growth rate

2d :

Thickness of the porous layer

\(\vec {g}\) :

Acceleration due to gravity

\(\hat{{k}}\) :

Unit vector in z-direction

K :

Permeability

p :

Pressure

P :

Modified pressure

\(\vec {q}=(u,v,w)\) :

Velocity vector

\(R_\mathrm{D}\) :

Darcy–Rayleigh number

t :

Time

T :

Temperature

\(T_1\) :

Temperature of the left vertical wall

\(T_2\) :

Temperature of the right vertical wall

\(\left( {x,y,z} \right) \) :

Cartesian coordinates

\(\alpha \) :

Ratio of heat capacities

\(\beta \) :

Thermal expansion coefficient

\(\theta \) :

Fluid temperature

\(\Theta \) :

Disturbance fluid temperature

\(\lambda _1 \) :

Stress relaxation time constant

\(\lambda _2 \) :

Strain retardation time constant

\(\Lambda _1 \) :

Relaxation parameter

\(\Lambda _2 \) :

Retardation parameter

\(\mu \) :

Fluid viscosity

\(\kappa \) :

Effective thermal diffusivity

\(\rho \) :

Fluid density

\(\rho _0 \) :

Reference density at \(T_0 \)

\(\psi \) :

Stream function

\(\Psi \) :

Disturbance stream function

References

  1. 1.

    Gill, A.E.: A proof that convection in a porous vertical slab is stable. J. Fluid Mech. 35, 545–547 (1969)

  2. 2.

    Barletta, A., Alves, L.S.B.: On Gill’s stability problem for non-Newtonian Darcy’s flow. Int. J. Heat Mass Transf. 79, 759–768 (2014)

  3. 3.

    Rees, D.A.S.: The stability of Prandtl–Darcy convection in a vertical porous slot. Int. J. Heat Mass Transf. 31, 1529–1534 (1988)

  4. 4.

    Lewis, S., Bassom, A.P., Rees, D.A.S.: The stability of vertical thermal boundary layer flow in a porous medium. Eur. J. Mech. B 14, 395–408 (1995)

  5. 5.

    Kwok, L.P., Chen, C.F.: Stability of thermal convection in a vertical porous layer. ASME J. Heat Transf. 109, 889–893 (1987)

  6. 6.

    Qin, Y., Kaloni, P.N.: A nonlinear stability problem of convection in a porous vertical slab. Phys. Fluids A 5, 2067–2069 (1993)

  7. 7.

    Rees, D.A.S.: The effect of local thermal nonequilibrium on the stability of convection in a vertical porous channel. Transp. Porous Med. 87, 459–464 (2011)

  8. 8.

    Shankar, B.M., Kumar, J., Shivakumara, I.S.: Stability of natural convection in a vertical couple stress fluid layer. Int. J. Heat Mass Transf. 78, 447–459 (2014)

  9. 9.

    Shankar, B.M., Kumar, J., Shivakumara, I.S.: Effect of horizontal alternating current electric field on the stability of natural convection in a dielectric fluid saturated vertical porous layer. J. Heat Transf. 137, 042501 (2015)

  10. 10.

    Makinde, O.D., Mhone, P.Y.: Temporal stability of small disturbances in MHD Jeffery–Hamel flows. Comput. Math. Appl. 53, 128–136 (2007)

  11. 11.

    Beg, O.A., Makinde, O.D.: Viscoelastic flow and species transfer in a Darcian high-permeability channel. J. Petrol. Sci. Eng. 76, 93–99 (2011)

  12. 12.

    Gozum, D., Arpaci, V.S.: Natural convection of viscoelastic fluids in a vertical slot. J. Fluid Mech. 64, 439–448 (1974)

  13. 13.

    Takashima, M.: The stability of natural convection in a vertical layer of viscoelastic liquid. Fluid Dyn. Res. 11, 139–152 (1993)

  14. 14.

    Straughan, B.: Stability and Wave Motion in Porous Media. Springer, New York (2008)

  15. 15.

    Alishaev, M.G., Mirzadjanzade, AKh: For the calculation of delay phenomenon in filtration theory. Izvestya Vuzov, Neft I Gaz 6, 71–78 (1975)

  16. 16.

    Khuzhayorov, B., Auriault, J.L., Royer, P.: Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media. Int. J. Eng. Sci. 38, 487–504 (2000)

  17. 17.

    Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. Wiley, Hoboken (2007)

  18. 18.

    Hirata, S.C., de Alves, L.S.B., Delenda, N., Ouarzazi, M.N.: Convective and absolute instabilities in Rayleigh-Bénard-Poiseuille mixed convection for viscoelastic fluids. J. Fluid Mech. 765, 167–210 (2015)

Download references

Author information

Correspondence to B. M. Shankar.

Additional information

Communicated by Patrick Jenny.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shankar, B.M., Shivakumara, I.S. On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid. Theor. Comput. Fluid Dyn. 31, 221–231 (2017). https://doi.org/10.1007/s00162-016-0415-8

Download citation

Keywords

  • Natural convection
  • Vertical porous layer
  • Oldroyd-B fluid
  • Viscoelastic fluid