3.3 Conservation of Momentum (Differential Form)

The right hand side of momentum conservation equation (3.8) involves surface integral for pressure. This term can be re-written as,

(3.10)

Also, because the control volume is fixed, the time derivative in Equation (3.8) can be placed inside the integral. Hence, Equation (3.8) can be written as

(3.11)

This equation is a vector equation. It would be convenient to decompose this equation in terms of components. Since we know,

V = ui + vj + wk

Therefore, the x component of Equation (3.11) is

(3.12)

Applying Gauss divergence theorem to the flux term of above equation, we get

(3.13)

Substituting equation (3.13) into Equation (3.12), we have

or

or

(3.14)

where is the representation of x component of the viscous shear stresses acting on the control volume. For the elemental volumes ,

or

(3.15)

This is the x component of the momentum equation in differential form. Similarly we can expressed the y and z component of the momentum equation respectively as

(3.16)

or

(3.17)

where the subscripts y and z on F_b and denote the y and z component of the body and viscous forces respectively.