PHY183-Lecture44

# PHY183-Lecture44 - Review Pascal's Principle Pascal's...

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April 10, 2006 1 Engineers 1 Engineers 1 Spring Semester 2006 Lecture 44 April 10, 2006 2 Review - Pascal Review - Pascal ’s Principle s Principle ! Pascal’s Principle for incompressible fluids “When there is a change in pressure at any point in a confined fluid, there is an equal change in pressure at every point in the fluid” ! Hydraulic advantage F out = F in A out A in April 10, 2006 3 Ideal Fluid Motion Ideal Fluid Motion ! The motion of real-life fluids is complicated ! Here we make some simplifying assumptions ! Assume an “ideal fluid”, exhibiting Laminar flow • The velocity of the fluid at a given point in space does not change with time Incompressible flow • The density of the fluid does not change as the fluid flows Non-viscous flow • The fluid flows freely, no friction or losses Irrotational flow • No part of the fluid rotates about its center of mass; no turbulence April 10, 2006 4 Examples of Flow Examples of Flow Laminar Flow Laminar to Turbulent Flow Turbulent Flow

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5 Continuity Equation Continuity Equation ! Consider the motion of an ideal fluid flowing with speed v in a container with cross sectional area A ! The volume ! V that moves past a point in a time t is given by ! The volume per unit time is then ! V = A ! x = Av ! t ! V ! t = Av April 10, 2006 6 Continuity Continuity Eqn Eqn (2) (2) ! Now consider a fluid flowing through a container that changes cross sectional area ! The fluid is initially flowing with speed v 1 in a container with area A and then enters a section of the container where the fluid flows with speed 2 and cross sectional area ! The volume of fluid entering this section of the container must equal the volume of fluid leaving the container April 10, 2006 7 Continuity Continuity Eqn Eqn (3) (3) ! The volume per unit time flowing in the first part of the container is ! The volume per unit time flowing in the second part of the container is ! Using the fact that the volume per unit time passing any point must be the same we get ! This equation is called the continuity equation ! V ! t = A 1 v 1 ! V ! t = A 2 v 2 A 1 v 1 = A 2 v 2 April 10, 2006 8 Bernoulli Bernoulli ’s Equation s Equation ! We can express a constant volume flow rate R (=volume per time) ! We can also express a constant mass flow rate m (=mass per time) ! Now let’s consider the case where an incompressible fluid (constant density " ) is flowing at a steady rate through the container below R p 2 - pressure v 2 - speed y 2 - height
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PHY183-Lecture44 - Review Pascal's Principle Pascal's...

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