Applications of Fluid Dynamics

When fluid flows through a narrowing pipe, the continuity equation $A_1 v_1 = A_2 v_2$ requires the speed to increase. Bernoulli's equation then demands that pressure decrease to conserve energy density. This inverse relationship between speed and pressure is the Venturi effect.

For a horizontal pipe, Bernoulli's equation gives:

$$P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$$

The pressure drop is:

$$\Delta P = P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2)$$