Abstract:
Transition phenomena commonly occur in nature. These arise
either due to structural or behavioral changes in the medium. Examples
for these abound in all applied sciences and to mention a few
of these, we have, boundary layer, elastic-plastic deformation, and
shocks. The present work is devoted to the study of laminar
boundary layer transition. In this case, transition from the region
near the surface of the body to the main stream takes place within
a thin layer called the boundary layer. Although the basic properties
of the fluid remain the same its behavior changes appreciably fro:
the surface of the body to the main stream. Owing to the presence of
spin, rotation or vorticity effects, the transition phenomenon is nonlinear,
irreversible and non-conservative and hence it cannot be
treated satisfactorily by superposition or perturbation techniques. In
this thesis an attempt is made to study the transition as an asymptotic
phenomenon from the boundary layer. The flow in the presence of any body is divided into two regions,
(a) boundary layer, (b) all the region excepting the boundary layer,
called the transition region. The classical boundary layer theory due
to Prandtl, is based on his main assumptions that (a) in the boundary
layer, the viscous and inertial forces are of the same order, (b) the
transverse velocity in the case of a flat plate is taken of the same
order as that of the transverse coordinate, (c) the variation of pressure
in the boundary layer is negligible. On careful examination, it
becomes clear that the above assumptions are not quite reasonable.
In the present investigation the boundary layer thickness is estimated
without making any of these assumptions since the ratio of the viscous
to the inertial forces varies continuously from infinity near the boundary
to zero at the outer edge of the boundary layer. Also the order of
the transverse velocity need not be the same as that of the transverse
coordinate and the continuity of pressure across the boundary layer
comes out from the transition analysis and therefore it is not necessary
to assume it.
By making an order of magnitude analysis, the boundary layer
thickness for two dimensional flow is estimated in terms of two parameters.
One of these parameters depends upon the relative order
of magnitude of the viscous and inertia forces at the outer edge of the
boundary layer and the other depends upon the order of vorticity allowable
at the outer edge of the boundary layer. The transition phenomenon in boundary flow is treated as an
asymptotic phenomenon from the boundary layer. In order to study
the transition region, a limiting form of the Navier-Stokes equations
in three dimensions is obtained, which is called the transition equation.
Owing to the importance of vorticity in the transition region,
the transition equation is solved for the vorticity. The form of vorticity
shows that in general the functions which govern the transition
region are either subharmonic or superharmonic functions.
In classical two dimensional flow the study of cylindrical vortex
is made by employing matching techniques. There does not exist any
mathematical treatment of the spiral formation which exists in case of
flow past a body at large Reynolds number. In the present thesis a
study of two dimensional flow past a body at large Reynolds number
is undertaken on the basis of transition analysis, thus obtaining a satisfactory
mathematical treatment of various phenomena that occur in
the boundary layer flow. The transition equations for axisymmetric
and two dimensional flow are also obtained. Besides other known results,
transition equation in two dimensions gives the stagnation
points and the formation of spirals which is noticed in the flow of a
real fluid past any body at large Reynolds number. Transition equation
also gives the formation of cylindrical vortices. These vortices
are given by the limiting form of the stream function and come out
from the transition equation itself without the use of any matching
process as is done in current literature. Hence it can be concluded
that the transition equation is a global representation of different phenomena
which exist in fluid flow past a body at large Reynolds number.
The transition concept is also extended to magnetohydrodynamics.
A formula for the magnetohydrodynamic boundary layer
thickness is obtained in terms of two parameters on the basis of a
magnitude analysis. The transition equation for two dimensional magnetohydrodynamic
case is also obtained, and its solution gives the
spiral formations.