Answered: Consider a two-dimensional (planar),…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Transcribed Image Text:Consider a two-dimensional (planar), steady and incompressible flow in the en- trance region of a duct leading to the fully developed region, as shown. Velocity profile at the entrance is U, which is uniform. At length L the flow becomes fully developed. Fluid viscosity is μ and its density of p. Pressure is uniform in the vertical (y) direction. Consider unity width normal to the page. Vo h L เง I x u Find the fully-developed velocity profile u(y) at x ≥ L. Using continuity equa- tion, express the velocity profile in terms of Uo, y and h only. Find the pressure gradient dp/dr and wall shear stress in the fully developed region (x > L) in terms of μ, U₁ and h. Calculate the total friction force acting on the walls in the entrance region (0 < x <L). (Note that the flow is not fully-developed in this region and thus, the solution in part (a) is not valid.) Assume that Blasius boundary layer with constant free-stream velocity U₁ de- velops over the walls in the entrance region. Estimate the streamwise velocity U(x) and pressure p(x) outside the boundary layer. (Note that outside the boundary layer the velocity is uniform in y and the flow is irrotational.)

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