An Introduction to Fluid Mechanics and Transport Phenomena by G. Hauke

By G. Hauke

This e-book provides the rules of fluid mechanics and delivery phenomena in a concise means. it really is appropriate as an advent to the topic because it comprises many examples, proposed difficulties and a bankruptcy for self-evaluation.

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The streakline is the geometric place occupied by fluid particles that have passed by the same point at previous times. Fig. 8. The plume of a chimney is a streakline. Streamlines indicate the velocity direction. They can be visualized by implanting little flags inside the fluid and observing their orientation. The streamlines can be obtained by drawing lines tangent to the flags. They are a rather mathematical object. The trajectory is the path followed by a fluid particle. For example, the braking marks on a road indicate the position that a tire has been occupying while the wheel was being dragged.

If we hang a thermometer from a bridge and register the water temperature versus time at various locations on the bridge, we would be using the Eulerian description. The thermometer would be measuring the temperature of different fluid particles as they pass through a fixed point. However, if we took a boat that was so light that it moved at the flow speed, then a thermometer glued to it would be registering the temperature of the (approximately) same fluid particle. In this case we would be using the Lagrangian description because we would be describing the temperature of the same fluid particle with time.

For the flow field of the above example, determine the trajectory of the fluid particle that passes through the point (x0 , y0 ), at t = 0. Solution. 4 Streamlines, Trajectories and Streaklines 25 which implies C1 = x0 /e C2 = y0 /e Finally, the trajectory is given in parametric form through the combination of 2 x = e(t+1) −1 x0 y y0 = e(t−1) 2 −1 This is a valid curve in two dimensions. Sometimes it is possible to eliminate t and write the same curve in explicit form, that is, as y(x). Getting t from the first equation, t ln xx0 + 1 − 1 = and substituting in the second one, y y0 √ ln =e x x0 +1−2 2 −1 which is the equation of the trajectory in explicit form.

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