Derivation of fundamental equation

Derivation

We have,

Actually, here p is a tensor (3x3 matrix) and so has 9 components as-

However, as there are only normal stresses acting in the fluid on the body, we have -

(i)

Now, invoking Gauss Divergence Theorem, Equation( 3.10) gives -

In Tensorial form -

Invoke Divergence Theorem =>

Now,

For an orthogonal coordinate system (x₁,x₂,x₃).

 

For unit vectors 

Relation (i) =>

And We know,

Further as the fluid is in steady state and equilibrium =>

(say)

Thus, Relation (i) =>

Hence the relation is derived.