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2.016 Hydrodynamics Reading #3 2.016 Hydrodynamics Prof. A.H. Techet Introduction to basic principles of fluid mechanics I. Flow Descriptions 1. Lagrangian (following the particle): In rigid body mechanics the motion of a body is described in terms of the body’s position in time. This body can be translating and possibly rotating, but not deforming. This description, following a particle in time, is a Lagrangian description, with velocity vector JK ± ± = ± Vu i + v j + w z . (3.1) Using the Lagrangian approach, we can describe a particle located at point JJK x = (, xy , z ) for some time t = t o , such that the particle velocity is o o o o G G x V = t , (3.2) and particle acceleration is JK K V a = . (3.3) t G We can use Newton’s Law of motion ( F = ma G ) on the body to determine the acceleration and thus, the velocity and position. However, in fluid mechanics, it is difficult to track a single fluid particle. But in the lab we can observe many particles passing by one single location. 2. Eulerian (observing at one location): In the lab, we can easily observe many particles passing a single location, and we can make measurements such as drag on a stationary model as fluid flows past. Thus it is useful to use the Eulerian description, or control volume approach, and describe the flow at every fixed point in space ( xyz ) as a function of time, t . ,, version 3.0 updated 8/30/2005 -1- © 2005 A. Techet

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2.016 Hydrodynamics Reading #3 version 3.0 updated 8/30/2005 -2- © 2005 A. Techet z x w u Figure 1: An Eulerian description gives a velocity vector at every point in x,y,z as a function of time. In an Eulerian velocity field, velocity is a function of the position vector and time, (,) Vxt JK K . For example: ± 22 (,) 6 3 1 0 t xi z y j x y tz =++ ±± 3. Reynolds Transport Theorem (the link between the two views): In order to apply Newton’s Laws of motion to a control volume, we need to be able to link the control volume view to the motion of fluid particles. To do this, we use the Reynolds Transport Theorem, which you’ll derive in graduate fluids classes, like 2.25. Suffice it to say that the theorem exists. For this class, we’ll use control volumes to describe fluid motion. 4. Description of Motion: Streamlines: Line everywhere tangent to velocity (Eulerian) (No velocity exists perpendicular to the streamline!) Streaklines : instantaneous loci of all fluid particles that pass through a given point x o .
2.016 Hydrodynamics Reading #3 Particle Pathlines: Trajectory of fluid particles (“more” lagrangian) In steady flow stream, streak, and pathlines are identical!! (Steady flow has no time dependence.) II. Governing Laws The governing laws of fluid motion can be derived using a control volume approach.

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