The Momentum Equation
As in the two-dimensional, planar case, the vector-momentum equation (Eq. 3.66) written for axisymmetric flow may be reduced to the Bernoulli Equation for the special case of incompressible, inviscid flow. To understand this, we expand Eq. 3.66 in cylindrical coordinates. Then, we simplify this vector equation by assuming axial symmetry. Next, we apply this vector equation along a streamline by taking the dot product of each term with an incremental streamline length ds = dxex + drer . Then, we appeal to the fact that a streamline is defined by:
and substitute this relationship as appropriate. The final result is:
Assuming incompressible flow, this may be integrated to give:
2 pV +p = constant,
which simply is Bernoulli’s Equation, as seen in Chapter 4 in Cartesian coordinates. The student should follow the procedure outlined to verify Eq. 7.12.
This result, Eq. 7.12, could have been deduced directly by realizing that the Bernoulli Equation represents the momentum equation for any steady, inviscid, incompressible flow. Because the Bernoulli Equation is an algebraic equation containing two scalar quantities (i. e., velocity magnitude and pressure magnitude at a point) and because scalar quantities are independent of any coordinate system, then it is valid for any coordinate system.
Eq. 7.12 indicates that if the magnitude of the velocity at any point in an axisym- metric flow field can be found, then the static pressure at that point may be found directly. As in two-dimensional, planar flow, it is most convenient to introduce scalar functions to find the velocity components and, hence, the required velocity magnitude.