Asymptotic estimates for an axisymmetric rotating fluid

A variational method is used to study the shape of a self-gravitating fluid mass rotating with a prescribed angular momentum per unit mass. The model is of primary interest in astrophysics, where it is taken as a model of a star. Sharp estimates are obtained on the diameter of the rotating body in terms of the total mass M and parameter $\text{Q =}\text{T}\text{W}\text{; T}$ and W represent, respectively, the kinetic and potential energy. The cases of Q small (slowly rotating body) and Q large (rapidly rotating body) are dealt with separately.     

Note on heat transfer in laminar,fully developed pipe flow with axial conduction

Zusammenfassung Der Wärmeübergang bei laminarer Rohrströmung mit konstanter Wandtemperatur wurde analytisch untersucht. Das Resultat ist eine Darstellung der Nusseltschen Zahl als Funktion von der Pecletschen Zahl.     

A class of parabolic quasi-variational inequalities,II

We consider a quasi-variational inequality (q.v.i.) introduced by A. Friedman and D. Kinderlehrer. A q.v.i. of this form gives rise, at least formally, to a Stefan problem of melting of water, where the relation ?v_x(x, t) = ?a(x, t)·(t) + b(x, t) holds on the free boundary x = s(t), and a > 0, b ? 0; the water temperature, v(x, t), is not necessarily nonnegative. In the standard Stefan problem a ≡ 1, b ≡ 0, and v ? 0. Friedman and Kinderlehrer proved the existence of a solution of the q.v.i. by a fixed point theorem for monotone mappings. Here we prove the existence of a solution by an entirely different method, based on finite difference approximations. The solution is shown to be smoother than that constructed by Friedman and Kinderlehrer.     

A free-boundary problem modeling loop dislocations in crystals

In this paper we consider a model of loop dislocations which arise in rapid thermal annealing of impurities into a crystal. The loop is a torus D(R(t)) generated by rotation of a disc of unit radius about an axis, a distance R(t) from the center of the circle. The function R(t) varies according to the flux of the interstitial density c across the boundary D(R(t)) of the torus. Outside D(R(t)) the function c satisfies a parabolic equation; on the boundary D(R(t)), c is a function of the space variable and of R(t). The problem is to solve the system for c and R. This is a free-boundary problem where the free boundary is D(R(t)). It is shown that a unique local solution does exist; however, for some class of data it blows up in finite time. For other classes of data we prove that a global solution exists, and we study its asymptotic behavior.