Studies in Applied Mathematics

Benney, D. J.; Maslowe, S. A.

doi: 10.1002/sapm1975543181pmid: N/A

It is shown, using a quite general formulation, that the amplitude evolution equation for slowly varying finite amplitude waves is usually first order in both space and time. One advantage of the present formulation is that it becomes possible to easily identify, from their linear eigensolutions, interesting exceptional cases in which the amplitude evolves according to a partial differential equation that is second order in either space or time. The theory is applied to a number of specific problems, including flows with broken line profiles, and inviscid shear flows having nonlinear critical layers.

Ulrich, Lionel Yves

doi: 10.1002/sapm1975543207pmid: N/A

This thesis presents a multicommodity flow problem on a telephone network, consisting in allocating telephone lines from one switching center to another to physical routes represented by edges. The criterion is the minimum of the total investment cost, in which the cost of each edge is a step function. Representing each group of lines between two vertices as a flow gives a very large model; decomposing these flows along paths permits the use of generalized programming; defining the total cost from the lines themselves exhibits a solution method by approximation. Three heuristic methods are presented. Solving by successive approximations to the actual costs of the lines gives a good solution. Rerouting lines from one path to another presents several pathologies which can be corrected, and shows the possibility of improving the solution yielded by the approximation method.

Lebovitz, N. R.; Schaar, R. J.

doi: 10.1002/sapm1975543229pmid: N/A

Equilibrium solutions y = ϕ(x) of an autonomous system of differential equations, depending on a parameter x, are considered. Bifurcation of a second family of solutions y = ψ(x) and exchange of stabilities is supposed to occur at (x,y) = (0,0). Considering x as slowly varying leads to a singularly perturbed initial‐value problem whose reduced path encounters a point of bifurcation. Rigorous asymptotic estimates are found for the difference between the (unique) solution of the full problem and that solution of the reduced problem which proceeds along stable segments of the reduced path.

Saffman, P. G.

doi: 10.1002/sapm1975543261pmid: N/A

The production of a vortex ring formed by using a piston to drive fluid through an orifice is considered. A cylindrical vortex sheet is supposed to be formed initially which rolls up into a vortex ring. Energy and momentum are conserved during rollup and determine the speed and size of the ring. It is shown that these quantities are independent of the vorticity distribution in the core of the ring. Reasonable agreement with experimental observations is found. A speculation is made about the criterion for the rings to be laminar or turbulent.

Schaeffer, David G.

doi: 10.1002/sapm1975543269pmid: N/A

We give a necessary and sufficient condtion for there to exist a discrete frequency of oscillation in an inviscid, axially symmetric rotating ring of fluid, a problem introduced by Barcilon. It turns out that for most cross sections of the ring there are no discrete frequencies, only a continuous spectrum.