Dynamics of Fluid Flow

Abstract

In flows material does not vanish, nor does new material appear. The velocity fields therefore have to satisfy the law of conservation of mass. This law is easiest to formulate for steady flows if the shape of the streamlines is already known. We consider a stream filament through every cross-section of which the same amount of mass flows per unit time. If this mass were not the same in two cross-sections, the mass content of the stream filament between two crosssections would have to decrease or increase, contradicting the idea of a steady state. If A is the cross-section of the stream filament at a certain position, w the mean velocity in this cross-section, and ρ the associated density, then per unit time, the fluid volume $A \cdot w$ flows through the cross-section. The fluid mass flowing through the cross-section per unit time is $\rho \cdot A \cdot w$. Continuity requires that $\rho \cdot A \cdot w$ must have the same value in all cross-sections of a stream filament. This implies that a stream filament of a steady flow cannot terminate in the interior of the fluid. It may extend from one boundary of the fluid space under consideration to the other boundary of the space, or it can turn back on itself.