On the Characterization of Nonoscillatory Motions in Maxwell Fluid in a Porous Medium Heated from Below

Jyoti Prakash* and Renu Bala

Department of Mathematics and Statistics, Himachal Pradesh University, Shimla – 171005, India

*Corresponding Author: jpsmaths67@gmail.com

ABSTRACT:

In the present paper condition for characterizing nonoscillatory motions which may be neutral or unstable in a horizontal layer of Maxwell fluid in a porous medium (modified Darcy-Brinkman-Maxwell Model) heated from below is obtained. It is proved that for a horizontal layer of Maxwell fluid in a porous medium heated from below an arbitrary neutral or unstable mode of the system is definitely nonoscillatory in character and in particular the ‘ principle of the exchange of stabilities’ is valid if (RλP_r)/〖4π〗^2 ≤1 . The result is uniformly valid for all combinations of free and rigid boundaries.

KEY WORDS: Maxwell fluid; Oscillatory motions; Thermal Convection; Porous Media; Modified – Darcy - Brinkman – Maxwell Model.

INTRODUCTION:

The problem of the onset of thermal instability in a horizontal layer of viscous fluid heated from below was first of all investigated experimentally by Benard(1900) and theoretically by Rayleigh(1916).The contributions of the subsequent researchers towards this phenomenon may be found in Chandrasekhar (1961), Drazin and Reid (1981) and Bejan (2004). This problem (also known as Rayleigh-Benard thermal instability problem) has been extended to a porous medium by Horton and Rogers (1945), Lapwood (1948), Katto and Masuoka (1967). Mckibbin and O’Sullivan (1980), Chen and Chen (1988), Straughan (2006), Malashetty et al (2007), Saravanan (2009) etc. The thermal convection in porous media has been widely investigated up to now.

Recently, interest in the flows of viscoelastic fluids through porous and nonporous media has grown considerably due to their importance in various areas in science, engineering and technology such as material processing, petroleum, chemical and nuclear industries, geophysics, bioengineering and reservoir engineering (Malashetty and Swamy (2007)).

Vest and Arpaci (1969) determined critical Rayleigh numbers, wave numbers and frequencies for overstability of a Viscoelastic fluid layer for both free and rigid boundaries. Sokolov and Tanner (1972) investigated the stability problem for a plane layer of a general viscoelastic fluid heated from below.

Yoon et al (2004) studied the onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid. Li and Khayat(2005) investigated the influence of inertia and elasticity on the onset and stability of Rayleigh-Benard thermal convection for highly elastic polymeric solutions with constant viscosity. Laroze et al (2007) studied thermal convection in a rotating binary viscoelastic liquid mixture. Tan and Masuoka (2007) studied the modified Darcy-Brinkman-Maxwell model based thermal convection of a horizontal layer of Maxwell fluid in a porous medium heated from below and determine the critical Rayleigh number, wave number and frequency for overstability. Fu et al (2007) investigated numerically the thermal convection phenomena of viscoelastic fluids in a closed top porous square box heated from below. Malashetty and Swamy (2007) analyzed the linear stability of a viscoelastic liquid saturated horizontal anisotropic porous layer heated from below and cooled from above by considering Oldroyd type liquid. Zhang et al (2008) made linear and non linear stability analyses of thermal convection for Oldroyd- B fluids in porous media heated from below. Yin et al (2012) investigated the effects of hydrodynamic boundary and constant flux heating conditions on a Maxwell fluid in a horizontal porous layer heated from below.

The establishment of the non-occurrence of any slow oscillatory motions which may be neutral or unstable implies the validity of the principle of the exchange of stabilities (PES). Pellew and Southwell (1940) proved the validity of PES for the classical Rayleigh-Benard convection problem. However no such result existed for the case of a Maxwell fluid in a porous medium heated from below. Tan and Masuoka (2007) obtain one such result (for the case of free boundaries only) which is a wave number dependent result. The aim of the present paper is to derive a sufficient condition for the occurrence of non oscillatory motions which may be neutral or unstable. It is shown that for a horizontal layer of Maxwell fluid in a porous medium heated from below, if then an arbitrary neutral or unstable mode of the system is definitely nonoscillatory in character and in particular PES is valid, where R, and respectively denote the Rayleigh number, the dimensionless relaxation time and the Prandtl number. Further the result obtained herein is uniformly valid for the quite general nature of the boundary surfaces.

4. CONCLUSIONS:

Thermal instability of a horizontal layer of Maxwell fluid (modified Darcy-Brinkman-Maxwell Model) in a porous medium heated from below is studied. A sufficient condition for the occurrence of nonoscillatory motions is derived which is uniformly valid for all combinations of free and rigid boundaries. Further, result obtained herein is wave number independent, thus a definite improvement over the existing results.

5. REFERENCES:

1. Benard H 1900 Les tourbillions cellulaires dans une nappe liquid. Revenue generale des Sciences pures et appliqués 11 1261-71 and 1309- 28

2. Bezan A 2004 Convection Heat Transfer third ed. John Wiley and Sons New Jersey

3. Chandrasekhar S 1961 Hydrodynamic and Hydromagnetic stability Clarendon Oxford

4. Chen F and Chen C F 1988 Onset of finger Convection in a horizontal porous layer underlying a fluid layer J. Heat Transf. 110(2) 403 – 09

5. Drazin P and Reid W 1981 Hydrodynamic Stability Cambridge University Press Cambridge

6. Fu C Zhang Z and Tan W 2007 Numerical simulation of thermal convection of a viscoelastic fluid in a porous square box heated from below Physics of Fluids 19 104107(1 – 12)

7. Horton C and Rogers F 1945 Convection currents in a porous medium J. Appl. Phys. 16(6) 367-70

8. Katto Y and Masuoka T 1967 Criterion for the onset of convective flow in a fluid in a porous medium Int. J. Heat mass Transf. 10(3) 297-309

9. Laroze D, Martinez-Mardones J and Bragard J 2007 Thermal convection in a rotating binary viscoelastic liquid mixture Eur. Phys. J. Special Topics 146 291-300

10. Li Z and Khayat R E 2005 Finite amplitude Rayleigh-Benard Convection and pattern selection for viscoelastic fluids J. Fluid Mech. 529 221- 51 M. H. Schultz, Spline Analysis, Prentice Hall, Englewood Cliffs, NJ, (1973).

11. M. H. Schultz (1973) Spline Analysis Prentice Hall Englewood Cliffs NJ

12. Malashetty M S and Swamy M 2007 The onset of convection in a viscoelastic liquid saturated anisotropic porous layer, Trans. Por. Med. 67 203 – 18

13. Malashetty M S Swamy M and Kulkarni S 2007 Thermal convection in a rotating porous layer using a thermal non- equilibrium model Phys. Fluids 19 054102 (1-16)

14. Mckibbin R and O’Sullivan M J 1980 Onset of convection in a layered porous medium from below J. Fluid Mech. 96(2) 375-93

15. Pellew A and Southwell R V 1940 On the maintained convective motion in a fluid heated from below Proc. Roy Soc. A 176 312 –43

16. Rayleigh L 1916 On the convective currents in a horizontal layer of fluid when the higher temperature is on the upper side Phil. Mag. 32 529- 46

17. Saravanan S 2009 Centrifugal acceleration induced convection in a magnetic fluid saturated anisotropic rotating porous medium Trans. Por. Med. 77 79 – 86

18. Sokolov M and Tanner R I 1972 Convective stability of a general viscoelastic fluid heated from below The Phys. Fluids 15(4) 534 – 39

19. Straughan B 2006 Global nonlinear stability in porous convection with a thermal non – equilibrium model Proc. Roy. Soc. A 462 409 – 18

20. Tan W and Masuoka T 2007 Stability analysis of a Maxwell Fluid in a Porous medium heated from below Physics Letters A 360 454-60

21. Vest C M and Arpaci V S 1969 Overstability of a viscoelastic layer heated from below J. Fluid Mech. 36 (3) 613 – 23

22. Yin C Fu C and Tan W 2012 Onset of thermal convection in a Maxwell fluid saturated porous medium. The effects of hydrodynamic boundary and constant flux heating conditions Trans. Porous. Med. 91 777 – 90

23. Yoon D Y Kim M C and Choi C K 2004 The onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid Trans. Por. Med. 55 275 – 84

24. Zhang Z Fu C and Tan W 2008 Linear and non linear stability analysis of thermal convection for Oldroyd-B fluids in porous media heated from below Phys. Fluids 20 084103(1 – 12)