17.16.* In Example 15.1 we considered the laminar flow of a Newtonian fluid between two parallel plates and showed that well downstream from the entrance the velocity distribution was parabolic. At the entrance to such a pair of plates the flow will initially have a uniform velocity, Vo independent of x and y. Boundary layers will grow from the walls, eventually meeting in the center, as sketched in Fig. 17.10.Show that if we make the simplest possible assumption, that the growing boundary layers do not interact with each other and that the fluid between the boundary layers has a constant velocity, then the distance downstream required for the boundary layers to grow together (the "entrance length") would be given by Le / h = 0.01 R. These assumptions are gross simplifications; the worst one says that the fluid in the center does not speed up. By material balance we may show that it must reach a velocity of twice the entrance velocity when the layers meet. More complicated analyses that take this into account [3, p. 178] lead to an approximate formula for parallel plates of Le / h≈ 0.04 R. To see the magnitude of this entrance length, calculate it for air flowing at 5 ft/s between plates 1.0 in apart. (Here is the Reynolds number based on distance between plates, not on distance from the leading edge.) Vo Edges of boundary layer 2V0 at centerline Parabolic velocity profile FIGURE 17.10 The entrance length for flow between two parallel plates.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

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Transcribed Image Text:In Example 15.1 we considered the laminar flow of a Newtonian fluid between two parallel plates and showed that well downstream from the entrance the velocity distribution was parabolic. At the entrance to such a pair of plates the flow will initially have a uniform velocity, Vo independent of x and y. Boundary layers will grow from the walls, eventually meeting in the center, as sketched in Fig. 17.10.Show that if we make the simplest possible assumption, that the growing boundary layers do not interact with each other and that the fluid between the boundary layers has a constant velocity, then the distance downstream required for the boundary layers to grow together (the "entrance length") would be given by Leh 0.01 R. These assumptions are gross simplifications; the worst one says that the fluid in the center does not speed up. By material balance we may show that it must reach a velocity of twice the entrance velocity when the layers meet. More complicated analyses that take this into account [3, p. 178] lead to an approximate formula for parallel plates of Le / h ≈ 0.04 R. To see the magnitude of this entrance length, calculate it for air flowing at 5 ft/s between plates 1.0 in apart. (Here is the Reynolds number based on distance between plates, not on distance from the leading edge.) 17.16.* Vo Edges of boundary layer = 2V0 at centerline Parabolic velocity profile FIGURE 17.10 The entrance length for flow between two parallel plates.

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