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Fluid Mechanics Chapter 11

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Expectations After this chapter, students will: know what a fluid is understand and use the physical quantities mass density and pressure calculate the change in pressure with depth in a stationary fluid distinguish between absolute and gauge pressure apply Pascal’s Principle to the operation of hydraulic devices

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Expectations After this chapter, students will: apply Archimedes’ Principle to objects immersed in fluids distinguish among several kinds of fluid flow apply the equation of continuity to enclosed fluid flows apply Bernoulli’s Equation in analyzing relevant physical situations

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Expectations After this chapter, students will: make calculations of the effect of viscosity on fluid flows

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Fluids A fluid can be defined as a material that flows. Fluids assume the shapes of their containers. More analytically: a fluid is a material whose shear modulus is negligibly small. Examples: liquids and gases.

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Mass Density Because a fluid is so continuous in its nature, when we consider its mechanics, its mass density is often more convenient and useful to us than is its mass. Mass density is the ratio of the mass of a material to its volume: SI units: kg/m 3 Greek letter “rho”

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Pressure A force-like quantity that is useful in the mechanical analysis of fluids is pressure. Pressure is the ratio of the magnitude of the force applied perpendicularly to a surface to the surface’s area: SI units: N/m 2 = Pa (the Pascal) Other popular units: lb/in 2 (“psi”), T (torr), mm of Hg, inches of Hg, bars (1.0×10 5 Pa), atmospheres

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Pressure vs. Depth in a Fluid Consider a “parcel” of water that is a part of a larger body. It is in equilibrium: Its volume is given by

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Pressure vs. Depth in a Fluid Its mass, then, is: Substitute into our equilibrium equation:

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Absolute vs. Gauge Pressure Absolute pressure is pressure by our definition: Gauge pressure is the difference between a pressure being measured and the pressure due to the atmosphere: Atmospheric pressure under standard conditions is about 1.013×10 5 Pa.

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Blaise Pascal 1623 – 1662 French mathematician

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Pascal’s Principle If a fluid is completely enclosed, and the pressure applied to any part of it changes, that change is transmitted to every part of the fluid and the walls of its container. Note that this does not mean that the pressure is the same everywhere in the fluid. We just calculated how pressure in a fluid increases with depth. Instead, the change in pressure is everywhere the same.

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Pascal’s Principle Calculate the pressures:

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Pascal’s Principle If a change in F 1 produces a change in pressure: the same change in pressure appears at A 2 :

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Pascal’s Principle

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If we take the F’s to be changes from zero: Pascal’s principle is the basis of many useful force-multiplying devices, both hydraulic (the fluid is a liquid) and pneumatic (the fluid is a gas).

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Archimedes’ Principle Born 287 BC, in Syracuse, Sicily Died 212 BC “... certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

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Archimedes’ Principle An immersed object:

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Archimedes’ Principle This says that the buoyant force, F B, exerted on the immersed object by the fluid, is equal to the weight of the fluid that has been displaced (pushed aside) by the object. If the object is not completely immersed, the displaced volume is that of the immersed portion of the object.

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Fluids in Motion Two kinds of flow: Laminar (or steady, or streamline) flows have constant, or near-constant, velocities associated with fixed points within the flow. Can be modeled as layers (“lamina”) of constant, or gradually-changing, velocity. Turbulent (or unsteady) flows have rapidly and chaotically-changing velocities.

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Fluids in Motion Two more kinds of flow: Compressible flows have variable fluid density. Gases are compressible fluids. Incompressible flows have constant fluid density. Liquids are incompressible fluids. Ideal fluid: perfectly incompressible.

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Fluids in Motion Still another two kinds of flow: In viscous flows, frictional forces act between adjacent layers of fluid, and between the fluid and its container walls. This frictional property of a fluid is called its viscosity. Non-viscous flows are free of fluid friction. Ideal fluid: zero viscosity.

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Fluids in Motion A streamline is the curve that is tangent to the fluid velocity vector from point to point in a laminar flow.

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Flow and Continuity Mass Flow Rate: mass per unit time

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Flow and Continuity Equation of continuity for any fluid flow: the mass flow rate is constant at every point in a non- branching flow (no place to get rid of fluid, or introduce new fluid).

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Flow and Continuity What if the flow is incompressible? The density is then constant: volume flow rate

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Flow and Continuity Equations of continuity (summary) For any enclosed, non-branching flow: (conservation of matter) For an incompressible, enclosed, non-branching flow:

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Bernoulli’s Equation Daniel Bernoulli 1700 – 1782 Swiss mathematician and natural philosopher Did pioneering work in elasticity and fluid mechanics

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Bernoulli’s Equation An incompressible fluid flows in a pipe: The equation of continuity tells us that v 1 > v 2 : So, the fluid must accelerate. How?

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Bernoulli’s Equation Free-body diagram of a parcel of fluid in the “accelerating” part:

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Bernoulli’s Equation A fluid changes height, at a constant cross-sectional flow area: the velocity is constant, and the pressure changes just as we would calculate it statically. h P1P1 P2P2

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Bernoulli’s Equation Now: combine changes in height with changes in cross-section.

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Bernoulli’s Equation Consider the work done on a parcel of fluid by a pressure difference across it: Notice that this work is nonconservative. volume

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Bernoulli’s Equation Apply the work-energy theorem.

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Fluid Flows and Conservation The major mathematical relationships we have derived for fluid flows are statements of conservation laws. Matter: the equations of continuity Energy: Bernoulli’s equation (any flow)(incompressible only)

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Frictional Fluid Flows Due to friction between the fluid and itself, the force required to move a layer of fluid (area A) at a constant velocity v a distance y from a stationary surface is: y

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Frictional Fluid Flows The constant of proportionality in this equation is called the coefficient of viscosity. SI units of coefficient of viscosity: Pa·s Common cgs unit: the poise (P)

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Jean Louis Marie Poiseuille 1797 – 1869 French doctor and physiologist Developed methods of measuring blood pressure

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Poiseuille’s Law Volume flow rate: pipe radius pipe length end pressure difference

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