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Chapter%203%20Kinematics

# Chapter%203%20Kinematics - III Kinematics of Fluid Motion...

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III. Kinematics of Fluid Motion Fluid is continuum. Geometric point for a fluid: x = x i e i . Two ways to describe the flow kinematics: i. the Lagrangian description ii. the Eulerian description. 3.1 Lagrangian Descriptions Identify a material point by a name, say a vector ξ , -- usually use the initial geometric position at time t=t 0 , so that ξ and time t are the independent variables ξ = ( ξ 1 , ξ 2 , ξ 3 ) = x at t = t 0 . (3.1) Dependent variable = the position of each material point (fluid element) at later times x = x ( ξ , t). (3.2) Velocity & acceleration of a material point are: u ( ξ , t) = | ξ fixed , (3.3) & a ( ξ , t) = | ξ fixed = | ξ fixed . (3.4) Lagrangian approach is a natural way of describing the motion of fluid particle because Newton's 2 nd law can be easily formulated for a particle. However, Lagrangian approach is not quite convenient to use. Not interested in the fate of individual fluid element.

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Interested in fluid flow behavior at some fixed points or regions in the flow field as time goes on. Eulerian description is more frequently used. 3.2 Eulerian Description Independent variables: the geometrical point x and t Velocity and acceleration of an fluid element passing x at time t are: u = u ( x , t), (3.5) & a = a ( x , t). (3.6) Identity of each fluid particle is irrelevant. Velocity, acceleration, pressure, density ... are field variables, not functions of individual fluid particles. 3.3 Relation between Lagrangian & Eulerian Description An analogy: Lagrangian approach of describing the fluid motion ~ policemen monitoring individual automobiles (fluid particles) by following them and measuring their speed. If the information on the traffic condition in some areas at certain time is needed, especially in some congested areas, "Lagrangian" approach is inefficient because many policemen are needed to follow individual automobiles. Eulerian approach
It is better to monitor the traffic flow at fixed x on some specific streets or intersections. Pay no attention to where the car comes from and where it is going. Only need collective behavior of the automobiles (speed & concentration) passing those locations. Velocity Evaluation Consider a fluid particle starting at x = ξ at time t0. It occupies x at instant t. After a short time interval t, it moves to a new position: x ( ξ , t+ t). The fluid velocity at ( x , t) of this same fluid particle at instant t can be obtained as u ( x , t) = lim 0 t ∆ → = | ξ fixed . (3.7) If x ( ξ , t) is already known, u ( x , t) is easy to obtain.

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