Physics 101: Lecture 25, Pg 1 Physics 101: Lecture 25 Fluids in Motion: Bernoulli’s Equation l Today’s lecture will cover Textbook Sections è Fluids in motion: Continuity & Bernoulli’s equation Note: Everything we do assumes fluid is non-viscous and incompressible.

Physics 101: Lecture 25, Pg 2 Physics 101: Lecture 24 Archimedes Principle (summary) l Buoyant Force (F B ) è F B =weight of fluid displaced è F B =  fluid V displ g è W = Mg =  object V object g l If object floats…. è F B =W è Therefore  fluid g V displ. =  object g V object è Therefore V displ. /V object =  object /  fluid

Physics 101: Lecture 25, Pg 3 Suppose you float a large ice-cube in a glass of water, and that after you place the ice in the glass the level of the water is at the very brim. When the ice melts, the level of the water in the glass will: 1. Go up, causing the water to spill out of the glass. 2. Go down. 3. Stay the same. CORRECT Concept Question F B =  W g V displaced W =  ice g V ice V displaced = V ice under water = V ice  ice /  W

Physics 101: Lecture 25, Pg 4 Fluids in Motion l Consider an ideal fluid (incompressible and nonviscous) that flows steadily. l Steady Flow: Every fluid particle passing trough the same point in the stream has the same velocity. Streamlines are used to visualize the trajectory of fluid particles in motion. The velocity vector of the fluid particle is tangent to the streamline. The fluid velocity can vary from point to point along a streamline but at a given point the velocity is constant in time.

Physics 101: Lecture 25, Pg 5 Equation of Continuity l Mass is conserved as the fluid flows. If a certain mass of fluid enters a pipe at one end at a certain rate, the same mass exits at the same rate at the other end of the tube (if nothing gets lost in between through holes, for instance). Mass flow rate at position 1 = Mass flow rate at position 2   1 A 1 v 1 =  2 A 2 v 2  A v = constant along a tube that has a single entry and a single exit point for fluid flow.

Physics 101: Lecture 25, Pg 6 Concept Question A stream of water gets narrower as it falls from a faucet (try it & see). This phenomenon can be explained using the equation of continuity A1A1 A2A2 V1V1 V2V2 The water's velocity is increasing as it flows down, so in order to compensate for the increase in velocity, the area must be decreased because the density*area*speed must be conserved

Physics 101: Lecture 25, Pg 7 Bernoulli’s Equation l Work-Energy Theorem : W nc = change of total mechanical energy applied to fluid flow : Difference in pressure => net force is not zero => fluid accelerates Pressure is due to collisional forces which is a nonconservative force: W nc = (P 2 -P 1 ) V Consider a fluid moving from height h 1 to h 2. Its total mechanical energy is given by the sum of kinetic and potential energy. Thus, W nc = E tot,1 –E tot,2 = ½ m v 1 2 +m g h 1 –( ½ m v 2 2 +m g h 2 )

Physics 101: Lecture 25, Pg 8 Fluid Flow (summary) Fluid Flow (summary) Mass flow rate:  Av (kg/s) Continuity:  1 A 1 v 1 =  2 A 2 v 2 i.e., mass flow rate the same everywhere e.g., flow of river For fluid flow without friction (nonviscous): Bernoulli: P / 2  v  gh 1 = P / 2  v  gh 2 A 1  1 A 2  2 v1v1 v2v2