We approximate the flow of air into a vacuum cleaner’s

Chapter 9, Problem 122P

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Problem 122P

We approximate the flow of air into a vacuum cleaner’s floor attachment by the stream function  arctan in the center plane (the xy-plane) in cylindrical coordinates, where L is the length of the attachment, b is the height of the attachment above the floor, and  is the volume flow rate of air being sucked into the hose. Shown in Fig. P9-126 is a three-dimensional 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 ψ = 0 along the positive x-axis (θ = 0). ( a) What are the primary dimensions of the given stream function? ( b) Nondimensionalize the stream function by defining ψ* = (2πL/ )ψ and r* = r/ b.( c) Solve your nondimensionalized equation for r* as a function of ψ*and θ. Use this equation to plot several nondimensional streamlines of the flow. For consistency, plot in the range -2