3
Interfacial Instabilities: The Saman-Taylor instability
3.1
Introduction
In this chapter, we turn to instabilities that arise because of the presence of an interface,
which is intrinsically nonlinear. We choose to focus on viscous ngering in a Hele-Shaw
7
Local bifurcation theory
In our linear stability analysis of the Eckhaus equation we saw that a neutral curve is
generated, as sketched again below.
k
=0
<0
n=1
>0
R
As the control parameter R is smoothly varied, the point at which = 0 (at any xed k)
de
9
The Ginzburg-Landau equation
In Sec. 8, we derived the Stuart-Landau equation (8.26) for the weakly nonlinear dynamics
of the amplitude A( ) in the vicinity of bifurcation. In the context of the Eckhaus equation,
we expanded R = Rcm +R1 about the minimu
6
A review of linear stability analysis
Before turning to our study of nonlinear eects in Chapter 7 below, rst we will review what
we already know about linear stability analysis from part I of the course. We do this in the
context of a partial dierential
5
Introduction
In part I of the course, we considered perturbations of small amplitude 1 to an initial
base state. (Note the change of notation. In part I we denoted the amplitude by . In
part II we will reserve to instead denote a control parameter, intr
2
Thermal instabilities: RayleighBnard convection
e
2.1
Introduction
- Broad range of applications, because in practice, convection occurs for very small
temperature gradients. Important in environmental science e.g. climate, building
ventilation; geophys
4
Shear ow instabilities
4.1
Introduction
There are many ows that have regions of parallel, or nearly parallel, streamlines where
signicant shear takes place.
- Channel ows
- Boundary layers
- Unbounded shear layers
In non-dimensional form, the Navier-Sto
Stability Theory
Prof A. Juel
Spring Semester 2014
Preliminaries notation
Three dierent notations for partial derivatives have been used in these notes. Make sure
that you are familiar with all of them, and can recognise them interchangeably.
As you will
Stability theory
Answers to problem sheet 4: Bifurcation theory.
Q1. Steady states exist for a > 0 with xB = a. Using the results of Q1 above,
with F (x; a) = a x2 and dF/dx = 2x, we see that
For the state x = + a, dF/dx < 0, and the state is linearly st
MATH45132: Stability Theory
Problem Sheet 3.
Q1. Consider two innite parallel planes separated by a distance L, the gap lled
with an incompressible viscous uid of kinematic viscosity . Suppose the uid has
a basic dimensional velocity eld of the form
u (x
MATH45132: Stability Theory
Problem Sheet 2.
Q1. Consider the Eckhaus equation
R1 [ + ] t = ,
where R is a real positive parameter and ( = 0, , t) = 0, ( = 1, , t) = 1.
(i) By considering the linear stability of the basic state B (, , t) , show that
the e
MATH45132: Stability Theory
Problem Sheet 1.
Q1. Consider a physical system for a quantity x(t) governed by the equation
dx
= F (x; a),
dt
where a is a real parameter. Suppose that a steady equilibrium solution x = xB
exists. By introducing x = xB + x(t)