Laval nozzle for throat smaller than critical area

Question

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?

Answer 1

So, I talked to a professor today, as he explained it, what will happen is as follows:

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.