Stream function formulation Navier–Stokes equation

Taking the curl of the Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (w = 0 and no dependence of anything on z), where the equations reduce to:

\rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}\right) = -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + \rho g_x

\rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}\right) = -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}\right) + \rho g_y.

Differentiating the first with respect to y, the second with respect to x and subtracting the resulting equations will eliminate pressure and any potential [disambiguation needed] force. Defining the stream function ψ through

u = \frac{\partial \psi}{\partial y} \quad ; \quad v = -\frac{\partial \psi}{\partial x}

results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation degrade into one equation:

\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial \psi}{\partial y} \frac{\partial}{\partial x}\left(\nabla^2 \psi\right) - \frac{\partial \psi}{\partial x} \frac{\partial}{\partial y}\left(\nabla^2 \psi\right) = \nu \nabla^4 \psi

where \nabla^4 is the (2D) biharmonic operator and ν is the kinematic viscosity, \nu=\frac{\mu}{\rho}. This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero.

In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function.