A Review on Stability and Instability in Fluids and Solids

Jagjit Singh Patial*

Associate Professor Mathematics. G.P.G.C. Bilaspur H.P.

*Corresponding Author E-mail:

ABSTRACT:

A criteria for the stability/instability of solid bodies in contact is given here. On the basis of criteria for stability/instability, the coefficient of stability/instability is derived. By using the equation (U-c)(D^2-α^2 )ϕ-U^'' ϕ=0 governing parallel two-dimensional inviscid parallel shear flows, if the basic study flow is of the form U=U(z) (z_1≤z≤z_2). Raleigh had proved that if c_i≠0 then c_r must lie in the range U_min<c_r<U_max, Howard (1961) proved that the value of c=c_r+ic_i must lie in the semicircle 〖0<{c_r-□(1/2)(U_max+U_min)}〗^2+c_i^2≤{□(1/2)(U_max-U_min)}^2. Further results for criteria of instability on the basis of Raleigh and Howard’s results are given in this paper

1 INTRODUCTION:

A phenomenon may satisfy all conservation law of nature exactly, but it may be unobservable. For a phenomenon to occur in nature, it has to satisfy one more condition, namely that it must be stable to small disturbances. A smooth ball resting on the surface of a hemi sphere which is concave upwards is stable to small displacements and but it is unstable to small displacements if the surface is convex upwards (Figure-1)

6. REFERENCES:

1. Drazin, P. G., W. H. Reid (1981) “Hydrodynamic Stability” Cambridge University Press London.

2. Howard L. N.,(1961) Note on a Paper of John W. Miles, Journal of Fluid Mechanics 13, 158-160.

5. Received on 04.01.2012 Accepted on 17.02.2012

7. Research J. Engineering and Tech. 3(2): April-June 2012 page171-175