For a two-dimensional Cartesian grid, the equation can be expanded to
On a staggered grid, the x-momentum equation is
and the y-momentum equation is
The goal at this point is to determine expressions for the face-values for u, v, and P and to approximate the derivatives using finite difference approximations. For this example we will use backward difference for the time derivative and central difference for the spatial derivatives. For both momentum equations, the time derivative becomes
where n is the current time index and Δt is the time-step. As an example for the spatial derivatives, derivative in the west-face diffusion term in the x-momentum equation becomes
where I and J are the indices of the x-momentum cell of interest.