Theory of flows of ideal fluids not only provides us the basis of study of fluid flows, but also gives us fundamental physical ideas of continuous fields in Newtonian mechanics. Governing equations of flows of an ideal fluid are summarized as

Equation of continuity: $$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 $$ Euler’s equation of motion: $$ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho} \nabla p + \mathbf{f} $$ Adiabatic motion of a fluid particle: $$ \frac{\partial s}{\partial t} + \mathbf{u} \cdot \nabla s = 0 $$

Boundary condition for an ideal fluid flow is that the normal component of the velocity vanishes on a solid boundary surface at rest $$ u_n = \mathbf{u} \cdot \mathbf{n} = 0 $$ where $\mathbf{n}$ is the unit normal to the boundary surface

If the boundary is in motion, the normal component $u_n$ of the fluid velocity should coincide with the normal velocity component $U_n$ of the boundary $$ u_n = U_n $$ Tangential components of both of the fluid and moving boundary do not necessarily coincide with each other in an ideal fluid

¹ Tsutomu Kambe. (2007). Elementary Fluid Mechanics. World Scientific, Singapore.