What boundary condition must the velocity potential satisfy \((a)\) at a rigid boundary; \((b)\) at a free surface of a fluid; \((c)\) at a plane interface between two different fluids?

For a rigid boundary, the fluid particles at the boundary cannot move or cross the boundary. As a consequence, the flow velocity along the normal direction to the boundary must be zero. Let \(\phi\) denote the velocity potential of the flow. We know that the flow velocity \(\mathbf{u}\) is related to the velocity potential by $$\mathbf{u} = \nabla \phi,$$ where \(\nabla\) is the gradient operator. Since the flow velocity along the normal direction to the boundary is zero, we must have $$\frac{\partial \phi}{\partial n} = 0,$$ where \(\frac{\partial \phi}{\partial n}\) denotes the derivative of \(\phi\) along the normal direction to the boundary. This is the boundary condition for the velocity potential at a rigid boundary.

At a free surface, the pressure on the surface equals the pressure of the atmosphere. As the fluid is incompressible, the Bernoulli equation states that: $$\frac{1}{2} |\mathbf{u}|^2 + \frac{p}{\rho} = \text{constant},$$ where \(p\) is the pressure, \(\rho\) is the fluid density, and \(|\mathbf{u}|^2 = (\nabla \phi)^2\). At the free surface, the constant equals the atmospheric pressure \(p_\text{atm}\) divided by \(\rho\). Therefore, at the free surface, we must have $$\frac{1}{2} (\nabla \phi)^2 + \frac{p}{\rho} = \frac{p_\text{atm}}{\rho}.$$ This is the boundary condition for the velocity potential at a free surface.

At a plane interface between two different fluids, the pressure and velocity are continuous across the interface. Let \(\phi_1\) and \(\phi_2\) denote the velocity potentials for fluids 1 and 2, respectively. The flow velocities are \(\mathbf{u}_1 = \nabla \phi_1\) and \(\mathbf{u}_2 = \nabla \phi_2\). Since the velocity is continuous across the interface, we must have $$\nabla \phi_1 = \nabla \phi_2$$ at the interface. This is the continuity of the flow velocity condition. Regarding the pressure, we know that the pressures \(p_1\) and \(p_2\) in the fluids 1 and 2 are related to the velocities \(|\mathbf{u}_1|\) and \(|\mathbf{u}_2|\) by the Bernoulli equation. We can write this condition as $$\frac{1}{2} (\nabla \phi_1)^2 + \frac{p_1}{\rho_1} = \frac{1}{2} (\nabla \phi_2)^2 + \frac{p_2}{\rho_2},$$ where \(\rho_1\) and \(\rho_2\) are the densities of fluids 1 and 2, respectively. As the flow velocities are continuous across the interface, we can set \(|\mathbf{u}_1| = |\mathbf{u}_2|\) and simplify this equation to $$\frac{p_1}{\rho_1} = \frac{p_2}{\rho_2}.$$ This is the continuity of pressure condition. In conclusion, at a plane interface between two different fluids, the boundary conditions for the velocity potential should satisfy the continuity of the flow velocity and pressure.