Equation of continuity and Bernoulli’s Principle (Ch. 10)

1. Equation of continuity and Bernoulli’s Principle(Ch. 10) Owen von KugelgenHead-Royce School

2. Moving Fluids • Continuity Principle A1v1 = A2v2 • Bernoulli’s PrincipleP1 + Dgh1 + (1/2)Dv12 =P2 + Dgh2 + (1/2)Dv22(really just conservation of energy!)

3. Continuity Principle A1v1 = A2v2 A1 A2 v2 v1 When the diameter of a pipe decreases, the speed of the water increases(imagine a garden hose)

4. Continuity Principle A1v1 = A2v2 A1 A2 ∆x2 ∆x1 Vol1/t = Vol2/t A1 ∆x1 / t = A2 ∆x2 / t A1v1 = A2v2

5. PE = mghPE/Vol = Dgh Pressure = F/AP = (F*d)/Vol = W/VolP = Energy/Vol KE = (1/2)mv2KE/Vol = (1/2)Dv2 Can we apply energy concepts to fluids?

6. Bernoulli’sPrinciple

7. Bernoulli’s Principle P2 h P1 P2 = P1 + Dgh∆Pressure = ∆PE/V = Dg∆h

8. Bernoulli’s Principle v2 P1 P2 v1 P1 + Dgh1+ (1/2)Dv12 = P2 + Dgh2 + (1/2)Dv22P1 = P2 + (1/2)Dv22 - (1/2)Dv12 P1 = P2 + (1/2)D[v22 - v12] Due to the continuity principle: v2 > v1 so P1 > P2

9. Bernoulli’s Principle Conceptual meaning: Higher fluid speed produces lower pressure This helps wings lift balls curve atomizers and carburetors Do their job

12. Wing lift The air across the top of a conventional airfoil experiences constricted flow lines and increased air speed relative to the wing. This causes a decrease in pressure on the top according to theBernoulli equation and provides a lift force

13. Curve Ball