Problem Description

Sign Convention for the
Gravitational Acceleration

(a) For steady flow, Euler's equation in vector form is given by:

     

If one assumes the z-axis is positive upward, then the gravitational acceleration g can be written as (see figure):

Now substitute the vector identity

to write Euler's equation as

In order to obtain Bernoulli's equation, one needs to integrate the above equation along a streamline. Hence, take the dot product of Euler's equation with a differential length ds along the streamline as follows:

Recognize that the differential length ds can be expressed in terms of the Cartesian coordinates as

     ds = dx i + dy j + dz k

Each term on the left hand side of Euler's equation can be evaluated as follows:

Since streamlines are constructed such that they are tangent to the velocity field, the differential length ds is parallel to the velocity V. The vector is then perpendicular to V, which gives

Euler's equation thus becomes

which can be readily integrated to yield

For constant density, the above equation reduces to the famous Bernoulli's equation

(b) Apply Bernoulli's equation between the inlet and outlet of the nozzle (i.e., along the streamline between points A and B),