| Abstract: | In a classic paper 1] of 1907, W. M'Farr Orr discovered, among other things, the “infinitesimal” instability of inviscid plane Couette flow. Surprisingly, although Orr's paper remains a standard reference in the field, later investigators 2, 3] have been able to call inviscid plane Couette flow stable without finding it necessary to controvert Orr's result. What has happened is that, at least in problems governed by linear (or linearized) equations with time-independent coefficients, the term “instability” has come to be identified with the presence of solutions exhibiting exponential time-growth. Orr found instability indeed: a class of solutions certain members of which grow in time by more than each preassigned factor. Unlike the exponential instabilities, however, Orr's solutions die away like 1/t after achieving their greatest growth. This ephemerality probably accounts for the discounting of Orr's result. Orr did not look into the general initial value problem. This is done in the sequel, with the result that the situation becomes clear. Under general disturbances, Couette flow turns out to be neither stable nor quasi-asymptotically stable*. The rate of growth depends on the smoothness of the initial data: classical solutions grow no faster than t, but sufficiently rough distribution-valued initial data leads to growth matching any power of t. Before presenting detailed results, we briefly review Orr's fundamental work on the problem. |
