Thermal Instability of a Viscoelastic Walters’(model B') Fluid in Hydromagnetics in the Presence of Suspended Particles
Sanjeev Kumar1, Veena Sharma2
1Department of Mathematics, Government College Joginder Nagar Distt., Mandi (H.P.), India.
2Department of Mathematics, H. P. University Summer Hill Shimla, India.
*Corresponding author E-mail: sanjeev.gcm@gmail.com
ABSTRACT:
The effect of
suspended particles on thermal instability of a viscoelastic Walters’ (model 
) fluid in the presence
of a uniform horizontal magnetic field is considered. The sufficient conditions
for non-existence of overstability are found in the presence of horizontal
magnetic field. The effect of suspended particles is to postpone the onset of
convection. For stationary convection, Walters’ (model B') viscoelastic fluid
behaves like a Newtonian viscous fluid.
KEY WORDS: Walters’ (model B') fluid; magnetic field; suspended particles; viscoelasticity.
1. INTRODUCTION:
The stability derived from the character of the equilibrium of an incompressible heavy fluid of variable density (i.e. of a heterogeneous fluid) was investigated by Rayleigh [1883]. He demonstrated that the system is stable or unstable according as the density decreases everywhere or increases everywhere. An experimental demonstration of the development of the Rayleigh–Taylor instability was performed by Taylor [1950]. The effect of a vertical magnetic field on the development of Rayleigh–Taylor instability was considered by Hide [1955]. Reid [1961] studied the effect of surface tension and viscosity on the stability of two superposed fluids. The Rayleigh–Taylor instability of a Newtonian fluid has been studied by several authors accepting varying assumptions of hydrodynamics and hydromagnetics and Chandrasekhar [1961] in his celebrated monograph has given a detailed account of these investigations. Gupta [1963] again studied the stability of a horizontal layer of a perfectly conducting fluid with continuous density and viscosity stratifications in the presence of a horizontal magnetic field. The Rayleigh–Taylor instability problems arise in oceanography, limnology and engineering. Scanlon and Segel [1973] have considered the effect of suspended particles on the onset of Bénard convection and found that the critical Rayleigh number was reduced solely because the heat capacity of the pure gas was supplemented by that of the particles. The stability of shear flow of stratified fluids with fine dust has been considered by Palaniswamy and Purushotham [1981] and they have found the effect of fine dust to increase the region of instability.
Generally, the magnetic field has a stabilizing effect on the instability, but there are a few exceptions also. For example, Kent [1966] has studied the effect of a horizontal magnetic field which varies in the vertical direction on the stability of parallel flows and has shown that the system is unstable under certain conditions, while in the absence of magnetic field the system is known to be stable. In stellar atmospheres and interiors, the magnetic field may be (and quite often is) variable and may altogether alter the nature of the instability. Coriolis force also plays an important role on the stability of the system. In all the above studies the fluid has been assumed to be Newtonian.
With the growing
importance of non–Newtonian fluids in modern technology and industries, the investigations
of such fluids are desirable. Fredricksen [1964] has given a good review of
non–Newtonian fluids whereas Joseph [1976] has also considered the stability of
viscoelastic fluids. There are many viscoelastic fluids which cannot be
characterized either by Maxwell’s constitutive relations or by Oldroyd’s
constitutive relations. One of such viscoelastic fluids is Walters’ (model
) fluid. Walters [1960]
has proposed a constitutive equation for such type of elastico–viscous fluids.
Many other research workers have paid their attention towards the study of
Walters’ (model) fluid. The
mixture of polymethyl methacrylate and pyridine at 25°C containing 30.5 grams
of polymers per litre behaves very nearly as the Walters’ (model) viscoelastic fluid and
which is proposed by Walters [1962]. This class of fluids is used in the manufacture
of parts of space cafts, aeroplane, tyres, beltconveyors, ropes, cushions, seats,
foams, plastics, engineering equipments etc. Sharma and Kango [1999] have
studied the stability of two superposed Walters’ (model) viscoelastic fluids in
the presence of suspended particles and variable magnetic field in porous
medium. Sharma and Kumar [1997] have studied the stability of two superposed
Walters’ (model) viscoelastic
fluid. Sharma and Kumar [2001] have studied the Rayleigh–Taylor instability of
stratified Walters’ (model) in
the presence of suspended particles and variable horizontal magnetic field and
have found that the criteria determining stability are independent of the
effects of viscosity and viscoelasticity. The magnetic field stabilizes the
system. Sharma and Kumar [2000] have studied magnetogravitational instability
of a thermally conducting rotating Walters’ (model) fluid with Hall
current.
Motivated by
interest in fluid-particle mixtures, columnar instability and keeping in mind
the physical situations occurring in atmospheric physics, oceanography and
geophysics, we propose to study the effect of suspended particles on thermal
convection in a viscoelastic Walters’ (model) fluid in the presence
of uniform horizontal magnetic field.
REFERENCES:
Journals:
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Books:
1. S. Chandrasekhar, ( 1961): Hydrodynamic and hydromagnetic stability, Oxford: Clarendon Press.
2. A.G. Fredricksen, (1964): Principles and applications of Rheology, New Jersey: Prentice–Hall Inc.
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Received on 25.08.2016 Accepted on 10.09.2016 © EnggResearch.net All Right Reserved Int. J. Tech. 2016; 6(2): 191-198. DOI: 10.5958/2231-3915.2016.00029.8 |
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