Chapter 3_sec.3.1_3.8

Chapter 3_sec.3.1_3.8 - Sec. 3.1-3.8 1 Sections 3.1-3.8...

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Unformatted text preview: Sec. 3.1-3.8 1 Sections 3.1-3.8 Fluid Mechanics EM 3313 Sec. 3.1-3.8 2 Whats the Point? Co n t e x t We looked at fluid statics in Chapter 2. Now lets look at fluid dynamics. Specifically, lets look at the Bernoulli equation application of Newtons second law to flow along, and normal to, a streamline. Motivation We can predict and/or analyze various flows using the Bernoulli equation. Learning objectives Demonstrate skill in applying simple equations of fluid dynamics, such as the Bernoulli equation and the I-D continuity equation. Demonstrate familiarity with the equations of fluid motion, their properties and usefulness in design and analysis. Sec. 3.1-3.8 3 3.1 Newtons 2 nd Law Newtons 2 nd Law F = m a This follows a particle Aside: Two ways to describe the flow Lagrangian description (doesnt require continuum) We describe the temporal variation for a specific fluid particle , i.e., we follow a fluid particle The function p i ( t ) tells how the pressure varies in time for the i th fluid particle Note that we also have x i ( t ), y i ( t ), and z i ( t ) that tell us the position of the i th fluid particle as a function of time Eulerian description (requires continuum) We describe the temporal variation at a point in space The function p ( x,y,z,t ) tells us how the pressure varies in time at the point ( x,y,z ) Newtons 2 nd Law is a Lagrangian concept To apply, we need to determine forces acting on a fluid particle and the acceleration of a fluid particle Sec. 3.1-3.8 4 3.1 Newtons 2 nd Law (cont) Forces Pressure force We know from hydrostatics that we need a pressure gradient to have a net pressure force Gravity force Neglect viscous forces Assume that viscosity is zero Inviscid flow The resulting equations are called the Euler equations If viscous terms are included we get the Navier-Stokes equations Sec. 3.1-3.8 5 3.1 Newtons 2 nd Law (cont) Definitions Velocity vector and speed Streamline: Space curves that are everywhere tangent to the local velocity vector at a given instance (Eulerian) Pathline: Path followed by a specific fluid particle (Lagrangian) Pathline and streamline are identical in steady flow w u dz dx w v dz dy u v dx dy streamline streamline streamline = = = | , | , | ( ) ( ) ( ) ( ) ( ) ( ) t z y x t z y x V t z y x w t z y x v t z y x u t z y x , , , , , , , , , ,...
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Chapter 3_sec.3.1_3.8 - Sec. 3.1-3.8 1 Sections 3.1-3.8...

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