In order to analytically find the stability of fluid flows, it is useful to note that hydrodynamic stability has a lot in common with stability in other fields, such as magnetohydrodynamics, plasma physics and elasticity; although the physics is different in each case, the mathematics and the techniques used are similar. The essential problem is modeled by nonlinear partial differential equations and the stability of known steady and unsteady solutions are examined. The governing equations for almost all hydrodynamic stability problems are the Navier–Stokes equation and the continuity equation. The Navier–Stokes equation is given by:
where
* is the velocity field of fluid
* is the pressure of fluid
* is the body force acting on fluid e.g., gravity
* is the kinematic viscosity
* partial derivative of the velocity field with respect to time
* is the gradient operator
Here  is being used as an operator acting on the velocity field on the left hand side of the equation and then acting on the pressure on the right hand side.
and the continuity equation is given by:
where  is the material derivative of the density.
Once again  is being used as an operator on  and is calculating the divergence of the velocity.
But if the fluid being considered is incompressible, which means the density is constant, then  and hence:
The assumption that a flow is incompressible is a good one and applies to most fluids travelling at most speeds. It is assumptions of this form that will help to simplify the Navier–Stokes equation into differential equations, like Euler's equation, which are easier to work with.