The transient and steady thermal convection in microgravity is described. The approach is applicable to many three dimensional flows in containers of various shapes with various thermal gradients imposed. The method employs known analytical solutions to two dimensional thermal flows in simpler geometries, and does not require recourse to numerical calculations by computer.
The four natural boundary conditions are derived for a newly observed symmetry in Bourdon tube action. The problem is described by an eighth-order system of pde’s for a Donnell-type shell theory. These, together with boundary conditions for the geometric symmetry and for the free end with rigid plug, comprise a well-posed mathematical problem. It requires only one-fourth the integration domain of the problem posed without symmetry, hence reduces storage requirements in computer calculations to one-sixteenth the number for the original problem.
A mathematical method is derived which permits calculation, based upon linear shell theory, of all stresses and displacements in a precision welded bellows having any number of layers, when displaced due to internal fluid pressure. If the meridional shape contains large curvatures or sharp corners the shell theory is replaced by the theory of solid elastic tori, which is used also for including the effects of the welds. As computation errors would eventually invalidate solutions for bellows with too many layers, a boundary layer technique is devised to overcome this difficulty. The method may therefore be applied to a bellows with any large number of layers without increase in the overall percentage error in displacement. An exact error check is possible in the solution for the transverse shear, indicating with our present step-intervals and medium size computer, errors of less than one percent.