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152 3. Further applications

leading-order equation for [This result can only give a flavour of how
things might proceed because, with strong shock waves most cer-
tainly appear and then a potential function does not exist. To investigate this
properly, we need to return to the original governing equations, without the
isentropic assumption.]
Q3.12 Asymmetrical bending of a pre-stressed annular plate. The lateral displacement,
of a plate is described by the equation (written in non-dimensional
variables)

and corresponds to weak bending rigidity. The annular plate is defined
by with the boundary conditions

where is a constant independent of Seek a solution
find the equation for u and then find the first two terms in each of
the asymptotic expansions of as valid away from the boundaries
of the region, and in the two boundary layers (near Match your
expansions as necessary. [For more details, see Nayfeh, 1973.]
Q3.13 A nonlinear elliptic equation. The function satisfies the equation

and it is defined in The boundary conditions are

Use the asymptotic sequence and hence obtain the first two terms
in the asymptotic expansion valid away from the boundaries x = 0 and y = 0;
this solution is valid on y = 1. Now find the first term only in the asymptotic
expansion valid in the boundary layer near y = 0, having first found the size
of the layer; match as necessary. Repeat this procedure for the boundary layer
near x = 0 and show that, to leading order, no such layer is required. However,
deduce that one is needed to accommodate the boundary condition at
Write the solution in this boundary layer as

(written in appropriate variables) and formulate the problem for the leading
term in the asymptotic expansion for V, but do not solve for V.