## Bernoulli's Equation for a Steadily Flowing Liquid

The Navier - Stokes equation is - a complicated equation which is difficult to solve, except in situation allowing much simplification. If we make the following assumptions, many of important features of the flow are retained.

1. The fluid is inviscid, so and the last term of the Navier – Stokes equation disappears.

2. The fluid is incompressible. This means we can divide each term of the Navier – Stokes equation by The first term on the right becomes 3. The flow is steady, so With these assumptions, the Navier – Stokes equation becomes We now use the identity to obtain If the force field is conservative we can write hence (1).

We can integrate this equation along a streamline. Suppose that s is a parameter for the streamline, then is a tangent vector to the streamline. Take the dot product of with both sides of (1) to give (2). is perpendicular to both and but is parallel to so When we integrate (2) along a streamline the term on the right hand side returns 0. Since the streamline is arbitrary we have along a streamline. 