1 D Mass and Momentum and Energy

Equations for Flow through a Variable Area Stream Tube

Consider the flow of air through a differential element of a stream tube (surface made of streamlines - no fluid can escape from the sides). The flow is (i) steady, (ii) uniform (frictionless, no effect of viscosity), (iii) stationary- the tube is not moving

Conservation of Mass (continuity) (The following two equations uses order of magnitude analysis)

Momentum Equation (Newton's Law) (pressure force must balance the change in the momentum along the flow direction as the drag is zero)

The final equation above is the differential form of the Bernoulli's equation.

This is the same form for the incompressible flow. The last term dropped is due to the continuity equation. When integrated with the density constant it leads to the well known form of the equation.

For the compressible case, you have to express the density in terms of the integration variable prior to integration

Energy Equation (first law of thermodynamics) (no work done, no heat transferred, ideal, reversible, no body forces)

Another way to incorporate the energy equation here is to assume that the compression takes place isentropically