Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 101

3.4.1 Governing Rayleigh equation.
For simplicity, we consider the Navier-Stokes equations given by
Equations 3.16 to 3.18 in the constant density, zero viscosity, two-dimensional
flow limit. Thus, we have
( u/ t + u u/ x + v u/ y) = - p/ x (3.89)
( v/ t + u v/ x + v v/ y) = - p/ y (3.90)
where u(x,y,t) and v(x,y,t) are Eulerian velocities at the fixed point (x,y), in the
x and y directions, and t is time. Also, p(x,y,t) and represent pressure and
mass density . Our velocities conserve mass via Equation 3.19, so that
u/ x + v/ y = 0 (3.91)
We consider the stability of a “parallel shear flow,” with a velocity U(y) in
the x direction, to wave-like disturbances. This “hydrodynamic stability”
problem deals with “aeroelastic” applications since it combines elements of fluid
dynamics and elasticity.
If we set u(x,y,t) = U(y) + u'(x,y,t), v(x,y,t) = v'(x,y,t), and p(x,y,t) = P(y) +
p'(x,y,t), where U and P are mean quantities, and (primed) disturbances are
smaller, substitution in Equations 3.89 to 3.91, and linearization, lead to
( u'/ t + U(y) u'/ x + U'(y) v ) = - p'/ x (3.92)
( v'/ t + U v'/ x) = - p'/ y (3.93)
u'/ x + v'/ y = 0 (3.94)
where U'(y) denotes dU(y)/dy. We wish to simplify the analysis by dealing with
one dependent variable as opposed to two. Now, Equation 3.94 suggests that we
can write
u'(x,y,t) =

y (x,y,t)

(3.95)

v'(x,y,t) = - x (x,y,t) (3.96)
without loss of generality, since substitution in Equation 3.94 leads to a trivial
identity. Here, (x,y,t) is the “disturbance streamfunction,” and it is possible to
formulate the entire problem using this single variable. To do this, let us
differentiate Equations 3.92 and 3.93 with respect to y and x, that is,

{u' yt + U(y) u' yx + U'(y) v y + U'(y) u' x + U"(y) v} = - p' yx (3.97)

{v' xt + Uv ' xx } = - p' xy (3.98)

Subtraction gives
u' yt + U(y) u' yx + U'(y) v' y + U'(y) u' x + U"(y) v' - v' xt - Uv' xx = 0 (3.99)
Using Equations 3.94 to 3.96, we obtain
yyt + U(y) yyx - U"(y) x + xxt + U xxx = 0 (3.100)
We simplify Equation 3.100 further by introducing the separation of variables