Chapter04

Chapter04 - Kinematics Kinematics z Kinematics deals with...

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Unformatted text preview: Kinematics Kinematics z Kinematics deals with the way we describe spatial coordinates and velocities z Fluid mechanics differs from classical physics – Classical Physics: Lagrangian description – Fluid Mechanics: Eulerian description z Will develop the Reynolds Transport Theorem – Makes derivation of conservation principles simple – Lays the foundation for the control-volume approach that we will use throughout this course Lagrangian Description Lagrangian Description-- 1 1 z In terms of rectangular Cartesian coordinates our notation is as follows z In classical physics we use the Lagrangian description and write z The position, r , of a fluid particle at time t is a function of t and its initial position r o Lagrangian Description Lagrangian Description-- 2 2 z The velocity is given by z Subscript r o means differentiate with r o held constant…corresponds to following the fluid particle as it moves z With a denoting acceleration and F denoting force, Newton’s second law of motion is Lagrangian Description Lagrangian Description-- 3 3 z The Lagrangian description is very convenient for describing motion of individual particles z But, it’s inconvenient in Fluid Mechanics – Would have to track an extremely-large number of fluid particles – Viscous stresses are very-complicated functions z Primary applications – Rarefied (very-low density) gasdynamics – Free-surface (H 2 O-Air interface) problems – Combustion problems Eulerian Description Eulerian Description-- 1 1 z The vast majority of fluid mechanics applications use the Eulerian description z We focus on a point in space and observe particles passing by…velocity is now a function of time and space z Acceleration is still ∂ u / ∂ t computed with r o held constant since we must follow the same fluid particle Eulerian Description Eulerian Description-- 2 2 Imagine that you observe traffic from the roadside – Granny B is driving at a constant speed of 60 mph – 4 sec later, a decelerating Leadfoot D goes by doing 90 mph – You conclude that the traffic is accelerating because Granny B has zero acceleration Leadfoot D is decelerating Eulerian Description Eulerian Description-- 3 3 z ∂ u / ∂ t computed with current position, r , held constant gives spurious results because we’re looking at different “fluid particles” z Acceleration must be computed following the same fluid particle…we must implicitly regard r as a function of t z Take the following limit Eulerian Description Eulerian Description-- 4 4 z Expanding in Taylor series… z For small ∆ t , we can say z Then, noting that ∆ r = u ∆ t z Hence, Eulerian Description Eulerian Description-- 5 5 z Taking the limit ∆ t → … z Denote the acceleration by a = d u /dt so that z We call this the Eulerian, material, substantial (and even the Lagrangian) derivative z Whatever we call it, the differential operator is the rate of change following a fluid particle Eulerian Description Eulerian Description...
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This note was uploaded on 10/12/2009 for the course AME 309 at USC.

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Chapter04 - Kinematics Kinematics z Kinematics deals with...

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