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M E 320 Professor John M. Cimbala Lecture 08
Today, we will :
- Begin Chapter 4 – FLUID KINEMATICS
- Discuss the material acceleration and the material derivative, and show examples
- Discuss various kinds of flow patterns and flow visualization techniques
- Begin to discuss other kinematic properties (motion and deformation of fluid particles)
III. FLUID KINEMATICS A. Descriptions of Fluid Flow – there are two ways to describe fluid flow:
- Lagrangian description
- Eulerian description
- Acceleration field and material derivative
Derivation of Material Acceleration (Section 4-1)
Recall the chain rule : If f is a function of two variables, t and some variable s which is itself also a function of t , then we take the total derivative of f with respect to t as follows: df f dt f ds f f ds dt t dt s dt t s dt
Now let’s apply this chain rule to the time derivative of the fluid particle’s velocity:
Thus, the acceleration of a fluid particle is calculated using the chain rule as follows:
Or, finally,
This is a Lagrangian description of the acceleration of a fluid particle.
Note that from the Lagrangian description (following a fluid particle, x particle is a function of time, since the particle’s location changes with time. Thus, x particle = x particle( t ). Similarly, y particle = y particle( t ) and z particle = z particle( t ).
dt / dt = dx particle/ dt = dy particle/ dt = dz particle/ dt =
