We all are well aware with the equation of continuity (I guess) which is given by:
$$A_1V_1= A_2V_2$$
Where $A_1$ and $A_2$ are any two cross sections of a pipe and $V_1$ and $V_2$ are the speeds of the fluid passing through those cross sections.
Suppose $A_1$ is $1\ m^2$ and $V_1$ is $2\ m/s$ and $A_2$ is $10^{-7}\ m^2$. This will mean that $V_2$ will be $2×10^{7}\ m/s$ which is much closer to the speed of light.
But my teacher just said it is not possible. He didn't give a reason.
Is he saying this because density of the liquid will change (given mass increases and length contracts with higher speed)?
Why can't we use this equation to push fluids to a higher speed?
If there is a limit on the maximum speed we can get to with this equation, what is it?
For simplicity of calculations, you may take water.
Best Answer
That simplified form of continuity equation assumes that the fluid is incompressible. That is only a valid assumption at low Mach numbers. I think a typical “rule of thumb” is that a Mach number less than 0.3 is required for the assumption to hold. So for the continuity equation to hold in that form requires a speed which is much less than the speed of sound which in turn is much less than the speed of light. You cannot use the continuity equation to achieve supersonic flow, let alone relativistic flow.
Note, that is not to say that supersonic flow is impossible, but rather that it is not possible simply by application of that form of the continuity equation. You need a form that accounts for compressibility. Superluminal flow is fundamentally impossible as no massive particle can reach c with finite energy.