A First Course in Fluid Dynamics

The fundamental form

We use the same method to derive the equation for momentum as we used for the continuity (or mass) equation. The rate of increase of momentum inside any volume V, as sketched in fig. VIII. 1, is equal to the sum of the

Now flow rates through S are all controlled by v · dS, a volume per time, which is proportional to local area and to local outward velocity; since the ith component of momentum per volume is just ρvi, we must have a first term

where the minus converts it to an inflow.

The second term arises from the body force, and is therefore

where Fi is the (ith component of) the body force per mass.

The interaction across a fluid surface is given by the stress tensor, and since we want the force on V due to fluid outside, and have the normal direction taken as out of V, this is

Thus the equation of motion for the fluid inside S is

As before we put all terms as volume integrals, by using the (generalised) divergence theorem, and noting that

This leads us to

where we have used the fact that V is time independent in the first term. Finally, because V is arbitrary and the integrand is assumed to be smooth, we have

This is the equation of motion. We may rearrange it by using the mass equation

into the form

Note the occurrence of the term ρv· ∇v, and that the left side is just

i.e. density times acceleration following the particle.

Stress and rate of strain

(a) Isotropic and deviatoric parts of the stress tensor

This equation of motion is of no help at all until we have a form for σij.