Notes on the Unsteady Rectilinear Motion of a Perfect Gas. 10: On the Influenceof Gravity Upon the Expansion of an Ideal Gas with gamma = 3

摘要

A vertical column of an ideal homentropic gas is at rest in a gravitationalfield, with constant acceleration of gravity g. At the upper end the column is bounded by a piston, while downwards it extends indefinitely. The gas pressure p at the piston has the value p sub 0, while above the piston there is vacuum extended to infinity. At time t = 0 the piston is suddenly removed, permitting the gas to expand and to move into the vacuum. The motion of the gas for t is greater than 0 is studied. Burgers considered this problem for gamma equals 5/3 and found that the gas starts to move upwards, but after some time, due to the action of the gravity the velocity changes direction and the gas begins to fall back. In this phase of the motion the solution shows some singularities, which indicate that physically a shock wave, moving in downwards direction, makes its appearance. The flow beyond the shock wave can only be analyzed tentatively and the analysis retains a preliminary character. Also a final state of rest is only approximately obtained for large t. Burgers' problem is considered once more for a fictitious gas with gamma = 3. This assumption simplifies the analysis and the singularities found for gamma equals 5/3 are drastically reduced. Also since the answers are simpler it is possible to a large extent to invert the solutions. Distinct from the case with gamma = 5/3 there is no overshoot, followed by a falling back, accompanied by a shock wave. For gamma = 3 the gas begins at t = 0 to move upwards but slows down and reaches a maximum height with speed zero, and density zero.