Main Question or Discussion Point
Consider the following flow: x = (1+ct)x0. Let the density rho(t) = rho0/(1+ct) so that the flow conserves mass. Physically, this is just a bunch of fluid elements on the positive x0-axis each given initial velocities that are proportional to their initial positions. Each fluid element should therefore continue with its initial speed, not accelerate, and have no forces acting on it while the density decays asymptotically to zero. Plugging this solution into Euler's equation, however, gives a pressure (1/2)rho0c2x2/(1+ct)3 which is proportional to the non-zero partial derivative of the density with respect to time times the fluid speed. Is this pressure to be regarded as fictitious/ non-physical?