Page 129 -
P. 129
112 2. Introductory applications
where and are constants, with the density, satisfy-
ing (This example is based on the more general
equation given in Christodoulou & Narayan, 1992.) For (the narrow
annulus approximation), introduce and then write
the density as find the first three terms in an asymptotic ex-
pansion for P. On the basis of this information, deduce that your expansion
would appear to be uniformly valid for
Q2.32 An elastic displacement problem. A simplified version of an equation which de-
scribes the displacement of a (weakly) nonlinear string, in the presence of
forcing, which rests on an elastic bed, is
where is a constant, with Find the first two terms in an
asymptotic expansion, for and use this evidence to deduce that this
expansion would appear to be uniformly valid for
Q2.33 Laminar flow through a channel. A model for laminar flow through a channel
which has porous walls, through which suction occurs, can be reduced to
where is an arbitrary constant of integration, with
(This is taken from Proudman,
1960; see also Terrill & Shrestha, 1965, and McLeod in Segur, et al., 1991;
here, the stream function is proportional to the function and
1/(Reynolds’ Number).) Assume that A(0) exists and is non-zero, and then
find the first term in an asymptotic expansion for and for valid
for x = O(1), and then the first two terms valid in the boundary layer (the first
being simply the boundary value there).
Q2.34 Slider bearing. The pressure, p, within the fluid film of a slider bearing, based
on Reynolds’ equation, can be reduced to the equation
written in non-dimensional form; here, is a constant and is the
given (smooth) film thickness, with (and
Find the first two terms in an asymptotic expansion, for valid for
x = O(1), and then the first term only in the boundary layer, matching as
necessary. (The first term in the boundary layer can be written only in implicit
form, but this is sufficient to allow matching.)
Q2.35 An enzyme reaction. The concentration, of oxygen in an enzyme reaction
can be modelled by the equation
