FLUID MECHANICS  CASE STUDY



Introduction

Problem Description


Kenneth is planning to enter the doctoral
program, and he needs to pass the qualifying exam to be admitted. One
question in the recent qualifying exam is from fundamental
fluid mechanics. The first part is to derive Bernoulli's equation
from Euler's equation. The second part is to use the derived Bernoulli's
equation to predict the pressure drop across a nozzle, as shown in the
figure. The nozzle is used to accelerate and double the air velocity. 





Questions



(a) Derive Bernoulli's equation from Euler's
equation. (b) What is the pressure difference between the inlet and outlet
of the nozzle? 





Approach



 Begin the derivation process with Euler's equation in vector
form.
 Use the following vector identity:
 Assume the change in the elevation is negligible when determining
the pressure drop, i.e., a horizontal nozzle.
 The density of air is 1.23 kg/m^{3}.



