Homework6

# Homework6 - MAE 101A Introductory Fluid Mechanics Homework...

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MAE 101A: Introductory Fluid Mechanics Homework 6 Due Friday March 12, 5:00 PM Problem 1 The y component of velocity in a steady incompressible ﬂow ﬁeld in the xy plane is v = 2 xy ( x 2 + y 2 ) 2 Show that the simplest expression for the x component of velocity is u = 1 x 2 + y 2 - 2 y 2 ( x 2 + y 2 ) 2 Problem 2 The y component of velocity in a steady, incompressible ﬂow ﬁeld in the xy plane is v = Axy ( y 2 - x 2 ), where A = 2 m - 3 · s - 1 and x and y are measured in meters. Find the simplest x component of velocity for this ﬂow ﬁeld. Problem 3 Consider the incompressible ﬂow of a ﬂuid through a nozzle as shown in ﬁgure. The area of the nozzle is given by A = A 0 (1 - bx ) and the inlet velocity varies according to U = U 0 (1 - e - λt ) where A 0 = 0 . 5 m 2 , L = 5 m , b = 0 . 1 m - 1 , λ = 0 . 2 s - 1 , and U 0 = 5 m/s . Find and plot the acceleration on the centerline, with time as a parameter. Figure 1: Problem 3 Problem 4 Consider the incompressible, inviscid ﬂow of air between two parallel disks of radius R = 75 mm , as shown in the ﬁgure. Air enters through a pipe of radius R i = 25 mm and exits radially, reaching R = 75 mm at a velocity V = 15 m/s . (a) Apply continuity equation and simplify the equation. 1

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(b) From (a) show that ~ V = V ( R/r ) ~ e r , for R i < r < R . (c) Calculate the radial acceleration of a particle at r = R i and at r = R . Figure 2: Problem 4 Problem 5 The inviscid incompressible ﬂow between two parallel disks. The upper disk is rotating at an angular velocity ω as shown in the ﬁgure, while the lower disk is stationary. It
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Homework6 - MAE 101A Introductory Fluid Mechanics Homework...

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