Theory of Amplitude Equations
Previously we have discussed the evolution of infinitesimal perturbations of a uniform state into
saturated, stationary, spatially periodic solutions. By restricting attention to these simple
solutions, it was straightforward
Problem 1
(a) Linearization is trivial (just like that of the standard Swift-Hohenberg equation) and we get
2
tu = ru + s xu (1 + 2 ) 2 u 3ub u ,
x
so we immediately get the growth rate
2
q = (r 3ub ) (1 q 2 ) 2 + isq
of an infinitesimal solution u = ei
Phys 7268
Midterm exam
Due: 3/10/11
Problem 1
Generalizations of Swift-Hohenberg equation. The form of the Swift-Hohenberg equation is determined by
the symmetries of the system, such as translational and rotational symmetries. For instance, breaking the
Rayleigh-Bnard Convection
Introduction
The Rayleigh-Bnard (henceforth referred to as RB) system has been studied by researchers for a
complete century by now and is still a topic of interest to them. It would be interesting to find out
why? The R-B proble
Rayleigh-Bnard Convection
Introduction
The Rayleigh-Bnard (henceforth referred to as RB) system has been studied by researchers for a
complete century by now and is still a topic of interest to them. It would be interesting to find out
why? The R-B proble
Secondary Instabilities and Stability Balloons
To determine the physical relevance of the steady, nonlinear, spatially-periodic states formed
above the instability of the spatially uniform state we must in turn test their stability against small
perturbat
7224 Homework Solution 1
January 30, 2003
1
Problem 1
(a) The dimensions of the various quantities concerned are
l
l2
[v ] = ; [L] = l; [ ] = .
t
t
The dimensions of Reynolds number R is:
[R ] =
vL
lt
= l 2 = 1.
tl
So R is dimensionless.
(b) The airow ove
7224 Homework Solution 1 Problem 1 tt u(x, y, t) = ru (1 + Rewrite second-order PDE as t u = u t u = ru (1 + The Jacobian matrix: f J= | = u u=ub
2 2 2 2
) u u3 .
) u u3 . 1 ) 3u2 0 b 1 0 u u
0 r (1 +
2 2
For ub = 0, the 3u2 term is dropped o. We obtain:
7224 Homework Solution 3
Problem 1
(a)
For a 2 2 real matrix
A=
a11 a12
a21 a22
The eigenvalues of the matrix A are given by
2 4
=
2
where = trace(A) and = det(A) = a11 a22 a12 a21 .
If > 0, < 0 and 2 4 0 then there are two real eigenvalues, which
are ne
7224 Homework Solution 4
Problem 1
(a)
m + + C sin = A sin 0 t
Let t = ct t, the equation changes into
d
c2
A
d2
+ ct + C t sin = c2 sin(0 ct t)
2
dt
m dt
m
mt
(b) To eliminate a second coecient, we have three possible ways
i) Let ct /m = 1 and ct = m/,
7224 Homework Solution 7 Problem 1 We need to nd the saturated nonlinear states of the system described by the following PDE: t u = ru (1 +
2 2
) u+
[( u)2 u].
(a) Linearizing about the uniform state u = 0 (we can drop the last term since it is cubic in
Phys. 7268
Assignment 1
Due: 2/3/11
Problem 1
Applications of the Reynolds Number. For problems in which an isothermal uid ows through a pipe or past
an object like a cylinder, an analysis of the Navier-Stokes equations reveals a dimensionless stress para
Phys. 7268
Assignment 2
Due: 2/10/11
Problem 1
Type of Linear Instability for a Swift-Hohenberg Equation with a Second-Order Time Derivative.
Consider a two-dimensional Swift-Hohenberg equation with a second-order time derivative:
2
t u(x, y, t) = ru (1 +
Phys. 7268
Assignment 3
Due: 2/17/11
Problem 1
Stability Criteria. Let us derive the necessary and sucient conditions for a 2 2 real matrix to have eigenvalues
with negative real parts.
(a) Consider a 2 2 real matrix A with matrix elements aij . Show that
Phys. 7268
Assignment 4
Due: 2/24/11
Problem 1
Identication of a Dimensionless Stress Parameter. To see how a dimensionless stress parameter like the
Rayleigh number might be discovered from known dynamical equations, consider the evolution equation for a
Phys. 7268
Assignment 5
Due: 3/3/11
Problem 1
Hexagonal patterns. Consider the superposition of modes
u(x, y ) = ei(q1 x+1 ) + ei(q2 x+2 ) + ei(q3 x+3 ) + c.c.
with x = (x, y ) and the wave vectors q1 , q2 , and q3 forming an equilateral triangle
q1 + q2
Phys. 7268
Assignment 6
Due: 3/10/11
Problem 1
Nonlinear saturation in the Swift-Hohenberg equation.
Consider the Swift-Hohenberg equation
2
t u = ru (1 + x )2 u u3
(1)
(a) Repeat the calculation for the nonlinear saturated stripe state using Galerkin exp
Phys. 7268
Assignment 7
Due: 3/17/11
Problem 1
Rayleigh-Bnard convection with poorly conducting boundaries. Consider the equation (which actually
e
describes free uid convection when the top and bottom plates are poor thermal conductors)
t u = ru (1 +
22
Phys. 7268
Assignment 8
Due: 3/31/11
You are highly encouraged to use Maple or Mathematica to do these problems.
You are welcome (and encouraged) to use and modify the Maple program(s) posted on the class homepage.
Problem 1
Quintic nonlinearity. Consid
Phys. 7268
Assignment 9
Due: 4/7/11
Problem 1
Long wavelength instabilities of stripe state in the amplitude equation formalism. It turns out that
many calculations that could, in principle, be performed using the Galerkin expansion, simplify considerably