Question 8.5: Note that Bernoulli and mass balance provide two equations:p...

Because we do not know \nu as a function of (x, y, z) or (r, \theta , z), we cannot determine if Bernoulli holds across the streamline (i.e., if \triangledown \times \nu =0, then Bernoulli may hold for any two points). Hence, let us recall the original Euler equations for a steady ideal flow:

from Eqs. (8.58) and (8.64). In particular, from the n-direction equation with g\sim0,

-\frac{\partial P}{\partial s}-\rho g\frac{\partial z}{\partial n} =\rho \left(\frac{\partial \nu _{s}}{\partial t}+\nu _{s}\frac{\partial \nu _{s}}{\partial s} \right).                                (8.58)

-\frac{\partial P}{\partial n}-\rho g\frac{\partial z}{\partial n}=\rho \left(-\frac{\nu ^{2}_{s} }{R} \right).                              (8.64)

FIGURE 8.9 A simple internal flow that converges from a larger to a smaller diameter tube. Assume that the pressure gauges are connected flush to the wall of the tubing and that they are filled with an incompressible fluid. where R is the radius of curvature of the streamline. Noting that R\rightarrow \infty for the locally parallel horizontal streamlines within the regions associated with gauges A and B, then at each gauge, \partial p/\partial n=0, which states that p does not vary in the normal direction when the streamlines are locally parallel. Hence, the pressure measured by these gauges, at the wall, equals the pressures at 1 and 2, and Bernoulli and mass balance can determine \nu _{1} and \nu _{2} in terms of measured p_{1}, p_{2}, A_{1} and A_{2}.