- #1

member 428835

I was reading about Bernoulli's equation for steady, inviscid, incompressible flow. Now it's my understanding this equation is derived from the Navier-Stokes (momentum balance); then these two equations are identical regarding information offered. However, while thinking about applications of Bernoulli I came to a problem I couldn't reason: suppose you have a tug boat that's pulling a cone underwater, and that the process is steady and the velocity of the boat (and the cone) is constant. Let the entrance of the cone have pressure ##P_1## and velocity ##V_1## and let pressure and velocity at exit be ##P_2## and ##A_2##. Bernoulli's equation gives $$P_1+\frac{1}{2}\rho V_1^2=P_2+\frac{1}{2}\rho V_2^2$$. However, Navier-Stokes is expressed $$

\partial_t\iiint_v \rho \vec{V} \, dv+ \iint_{\partial v} \rho \vec{V} (\vec{V} \cdot \vec{n}) \, dS = \sum \vec{F}\implies \\

\rho(V_2^2 A_2-V_1^2 A_1) = P_1A_1-P_2A_2+F_{boat}$$

But clearly the final expression is not the same, namely the force of the boat is present. Can someone explain why I am having a discrepancy?