1. CIEG 305 DERIVATION OF BERNOULLI”S EQN FROM LINEAR MOMENTUM ALONG A STREAMLINE P+dP A A+dA V,ρ P ds dz θ V+dV ρ+dρ dW≈ρgdVol Gory Details 1 2

2. Take a narrow stream tube at some angle (shown on previous slide Require cross-sections to be small to that we take P and V essentially constant Neglect friction and assume incompressible. Now conserve mass using RTT for control volume defined by stream tube

3. OR REFERENCING EVERYTHING TO STATION 1

4. FROM MASS CONSERVATION AND ASSUMPTION THAT VALUES ARE MORE OR LESS EQUAL Product rule This is the differential form of mass conservation along streamline

5. LINEAR MOMENTUM EQUATION IN STREAMWISE DIRECTION The last part comes from ugly chain rule, product rule and substitution of mass conservation

6. WHAT ARE THE FORCES?? GRAVITY PRESSURE

7. SUBSTITUTE INTO THE LINEAR MOMENTUM EQUATION By mass continuity

8. Divide by ρA and rearrange THIS IS BERNOULLI’S EQUATION FOR UNSTEADY FRICTIONLESS FLOW ALONG A STREAMLINE INTEGRATE ALONG STREAMLINE

9. If we consider only steady state and incompressible flow THIS IS BERNOULLI’S EQUATION FOR STEADY, FRICTIONLESS, INCOMPRESSIBLE FLOW ALONG A STREAMLINE