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Let be a 3-dimensional vector, the velocity of the fluid, and let be the pressure of the fluid. The Navier–Stokes equations are:
where ν > 0 is the kinematic viscosity, the external force, is the gradient operator and is the Laplacian operator, which is also denoted by . Note that this is a vector equation, i.e. it has three scalar equations. If we write down the coordinates of the velocity and the external force
then for each i = 1,2,3 we have the corresponding scalar Navier–Stokes equation:
The unknowns are the velocity and the pressure