We approximate the flow of air into a vacuum cleaners floor attachment by the stream

Chapter 9, Problem 9-126

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We approximate the flow of air into a vacuum cleaners floor attachment by the stream function c 5 2V # 2pL arctan sin 2u cos 2u 1 b2/r2 in the center plane (the xyplane) in cylindrical coordinates, where L is the length of the attachment, b is the height of the attachment above the floor, and V # is the volume flow rate of air being sucked into the hose. Shown in Fig. P9124 is a threedimensional view with the floor in the xz-plane; we model a two-dimensional slice of the flow in the xy-plane through the centerline of the attachment. Note that we have (arbitrarily) set c 5 0 along the positive x-axis (u 5 0). (a) What are the primary dimensions of the given stream function? (b) Nondimensionalize the stream function by defining c* 5 (2pL/V # )c and r* 5 r/b. (c) Solve your nondimensionalized equation for r* as a function of c* and u. Use this equation to plot several nondimensional streamlines of the flow. For consistency, plot in the range 22 , x* , 2 and 0 , y* , 4, where x* 5 x/b and y* 5 y/b. (Hint: c* must be negative to yield the proper flow direction.)