A Control Volume Approach for the Derivation of Euler’s Equation

Euler’s equations of motion can also be derived by the use of the momentum theorem for a control volume.

Derivation
In a fixed x, y, z axes (the rectangular cartesian coordinate system), the parallelopiped which was taken earlier as a control mass system is now considered as a control volume (Fig. 12.4).

Fig 12.4   A Control Volume used for the derivation of Euler's Equation

We can define the velocity vector and the body force per unit volumeas

The rate of x momentum influx to the control volume through the face ABCD is equal to ρu² dy dz. The rate of x momentum efflux from the control volume through the face EFGH equals

Therefore the rate of net efflux of x momentum from the control volume due to the faces perpendicular to the x direction (faces ABCD and EFGH) = where, , the elemental volume = dx dy dz.

Similarly,

The rate of net efflux of x momentum due to the faces perpendicular to the y direction (face BCGF and ADHE) =

The rate of net efflux of x momentum due to the faces perpendicular to the z direction (faces DCGH and ABFE) =

Hence, the net rate of x momentum efflux from the control volume becomes

The time rate of increase in x momentum in the control volume can be written as

  (Since, , by the definition of control volume, is invariant with time)
Applying the principle of momentum conservation to a control volume (Eq. 4.28b), we get

(12.11a)

The equations in other directions y and z can be obtained in a similar way by considering the y momentum and z momentum fluxes through the control volume as

	(12.11b)
	(12.11c)

The typical form of Euler’s equations given by Eqs (12.11a), (12.11b) and (12.11c) are known as the conservative forms.