# Indeterminism of Bernoulli Equation

Suppose I have a have a pump of inlet area $$A_1$$, outlet area of $$A_2$$ and pressure head $$h_p$$. Ambient pressure is atmospheric pressure. The inlet and outlet are at the same elevation $$z_1 = z_2 = 0$$. Inlet and outlet velocities are denoted by $$v_1$$ and $$v_2$$ respectively. The flow is assumed to be incompressible.

The Bernoulli equation is given by $$\frac{P_1}{\gamma}+\frac{{v_1}^2}{2g}+z_1 + h_p = \frac{P_2}{\gamma}+\frac{{v_2}^2}{2g}+z_2$$ And mass conservation demands $$A_1 v_1 = A_2 v_2$$. Since ambient pressure is the same at outlet and inlet $$P_1 = P_2 = P_{atm}$$. Plugging in and rearranging, we obtain $$v_2 = \sqrt{\frac{2gh_p}{1-(A_2/A_1)^2}}$$ Now if the areas $$A_1$$ and $$A_2$$ are the same, the solution breaks down. For very close values of $$A_1 \approx A_2$$, the value of $$v_2$$ becomes extremely large. In fact, for exactly equal areas we find $$v_1 = v_2$$, just by mass continuity, but with no prediction of the value of $$v_1$$ or $$v_2$$ from the Bernoulli equation. So predicting the velocity very sensitively depends on the exact area. This makes zero sense. I have probably made a false assumption, but where? I doubt that engineers select different values of $$A_1$$ and $$A_2$$ over concerns of predictivity of the Bernoulli equation.

• The Bernoulli equation omits viscous frictional drag in the pipe. Commented Jan 30, 2023 at 12:16

The pressure head $$h_p$$ must be zero otherwise the fluid indeed accelerates without limit (assuming no frictional losses).
Mathematically speaking, one is not allowed to solve explicitly for $$v_2$$ when $$A_1=A_2$$ since there is division by zero. One must leave the equation as: $$2gh_p=v_2^2(1-A_2^2/A_1^2)$$ And see that $$h_p=0$$ for flow to have constant velocity.