DBE: Navier-Stokes Equations (Momentum Balance Equations)

For our balance, we will choose an arbitrary fixed (in space) control volume. Thus, it is an open system (i.e., one through which mass may flow). Accounting for momentum within that volume (and transfered with surroundings) we get:

rate of accumulation of momentum in CV = net rate of momentum transport into CV + sum of the forces acting on the CV

Writing this mathematically, we get:

where the sum of the forces includes two classes of force:

The body forces could include magnetic and/or electrical forces, but generally only include gravitational:

The surface forces include shear and normal stresses. For a Newtonian fluid, the only normal stresses we will consider are pressure stresses so we get:


The pressure must be dotted into the normal vector because the pressure has no direction; however, by convention we do give the pressure a sign convention so that it is positive when compressive. This is why we include the negative sign here (compression is opposite the normal vector).

Combining these expressions gives us:

Now using the same bit of math wizardry that we used before (i.e., the Gauss divergence theorem) and rearranging things

With our arbitrary control volume, the integrand must be zero. So, we now have a differential momentum balance equation:

The left hand side of this equation is often called the substantial or lagrangian time derivative as this is the time derivative that "follows" the fluid (because of the inclusion of the advective second part):

If we are interested in only incompressible, Newtonian fluids we get the final form of the Navier-Stokes equations as:

Simplifying the Navier-Stokes Equations

The first thing to note about the N-S equations, is that there are three of them!

Unlike the previous equations, there are only two assumptions/conditions that we can make here:


The equation that we derived required that we also assume that we are dealing with an incompressible and Newtonian fluid.


Simplify the general material energy balance equation (identifying assumptions)