# Laval nozzle for throat smaller than critical area

Consider a Laval Nozzle with inlet area $$A_{inlet}$$ in a subsonic flow with velocity $$V$$ and density $$\rho$$ (and total pressure $$p_0$$, total temperature $$T_0$$ and total density $$\rho_0$$).

The mass flow $$\dot{m}$$ is then given by $$\dot{m}=\rho A_{inlet} V$$. Using Bernoulli equations for mach number $$M=1$$ the critical pressure $$p^*$$, critical density $$\rho^*$$ and critical sound velocity (temperature) $$c^*$$ can now be determined, such that also the critical area $$A^*$$ at which $$M=1$$ is reached can be calculated using $$A^*=\frac{\dot{m}}{\rho^*c^*}$$.

Now, I wonder what will happen if the Laval nozzle has a throat area $$A_{throat}$$ smaller than $$A^*$$. Certainly $$M=1$$ must be attained at the throat, but this seems to violate the analysis above. And what happens at cross section $$A=A^*$$, which here occurs before the throat?

In this case, the nozzle will overflow similar to how a normal funnel can overflow, and part of the flow will be directed around the nozzle, such that the mass flux reduces. Ultimately, the mass flux will reduce to $$\dot{m}=A_{throat}\rho^*c^*$$, such that the condition $$A^*=A_{throat}$$ is met again.