Can Bernoulli’s equation be used to describe the flow of water through a rapid in a river ? Explain.

Question

Two vessels A and B are of the same size and are at the same temperature. A contains 1 g of hydrogen and B contains 1 g of oxygen. $P_A$ and $P_B$ are the pressures of the gases in A and B respectively, then $\frac{P_A}{P_B}$ is:

Answer 1

Short Answer

No. Bernoulli's equation cannot be used for water flow through river rapids.

Bernoulli's Assumptions

Bernoulli's equation requires:

$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant (along streamline)}$$

Steady flow - no time variation
Streamline (laminar) flow - smooth, parallel paths
Inviscid flow - negligible viscosity
Incompressible flow - constant density
No turbulence - no eddies or mixing

River Rapids Reality

Rapids Characteristics

Turbulent flow - chaotic eddies, white water [web:33]
High velocity gradients - different speeds at different points
Air entrainment - not purely incompressible
Viscous effects - energy dissipation in turbulence

Why It Fails

In rapids:

Streamlines break apart - no continuous path for equation
Energy dissipates as heat in turbulence, violating conservation
Pressure and velocity fluctuate wildly point-to-point

Conclusion

Bernoulli's equation requires streamline flow. Rapids are turbulent, so the equation is invalid [web:33][web:35].

Appropriate Analysis

For rapids, use:

Navier-Stokes equations (full viscous flow)
CFD simulations (computational fluid dynamics)
Energy balance with losses