3. 1 The acceleration field of a fluid
The cartesian vector form of velocity field that varies in space and time is

To write Newton’s second law for an infinitesimal fluid system, we need to calculate the acceleration vector field () of the flow which can be given as,

Since each scalar component (u,v,w) is a function of four variables (x,yz,t) ,then we can use the chain rule to obtain each scalar time derivative.

The term is called the local acceleration. The other terms are collectively called the convective acceleration, which arises when the particle moves through regions of spatially varying velocity.

3.1 The Differential equation of linear momentum

Let consider an infinitely small elemental control volume having dimensions dx,dy,dz in X ,Y ,Z directions respectively. Schematic of such as

Fig. 3.1. Schematic of the elemental control volume

Considering the control volume as shown in Fig. 3.2 for  momentum equation we get

3.1

Here ∀ is the volume of the elemental control volume.

As the control volume is considered to be so small the integral is reduced to a derivative

3.2

The momentum fluxes at all six faces are

Using the the inlet momentum flux and outlet momentum flux in the above equation we get

3.3