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2.1 For the two-dimensional stress field in Fig. P2.1, let σ xx =150 kPa σ yy =100 kPa σ xy =25 kPa Find the shear and normal stresses on plane AA cutting through at 30°. Solution: Make cut “AA” so that it just hits the bottom right corner of the element. This gives the freebody shown at right. Now
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5.1 For axial flow through a circular tube, the Reynolds number for transition to turbulence is approximately 2300 [see Eq. (6.2)], based upon the diameter and average velocity. If d = 5 cm and the fluid is kerosene at 20°C, find the volume flow rate in m3/h which causes transition.
Solution: For
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Solution: For water, take ρ = 998 kg/m3 and ? = 0.001 kg/m-s. The surface wave speed is co = gy = (9.81 m/s2)(0.1 m) = 0.99m/s
The average velocity is determined from the given flow rate and area:
V = Q A = (80,000 cm3 /s) (30 cm)(10 cm) = 267 cm s = 2.67 m s Froude number: Fr = V co = 2
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Solution: Make a plot of density ρ versus altitude z in the atmosphere, from Table A.6:
1.2255 kg/m3
ρ
0 z 30,000 m
This writer’s approximation: The curve is approximately an exponential, ρ ? ρ o exp(-b z), with b approximately equal to 0.00011 pe
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Solution: The area element for this axisymmetric flow is dA = 2 π r dr. From Eq. (3.7),
Q= u dA= C 0 R ∫ (R2 ?r2)2 π rdr=
π 2CR4 Ans.∫
2
P3.4 A fire hose has a 12.5-cm inside diameter and is flowing at 2.27 m3/min. The flow exits through a nozzle contraction at a d
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P7.1 An ideal gas, at 20°C and 1 atm, flows at 12 m/s past a thin flat plate. At a position 60 cm downstream of the leading edge, the boundary layer thickness is 5 mm. Which of the 13 gases in Table A.4 is this likely to be?
Solution: We are looking for the kinematic viscosity. For a gas at l
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P6.1 An engineer claims that flow of SAE 30W oil, at 20°C, through a 5-cm-diameter smooth pipe at 1 million N/h, is laminar. Do you agree? A million newtons is a lot, so this sounds like an awfully high flow rate. Solution: For SAE 30W oil at 20°C (Table A.3), take ρ = 891 kg/m3 and ? = 0.29 kg/m
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fluid, but there is always a limiting very-high viscosity for which performance will deteriorate. Can you explain the probable reason? Solution: High-viscosity fluids take a long time to enter and fill the inlet cavity of a PDP. Thus a PDP pumping high-viscosity liquid should be run slowly to ensure
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P4.1 An idealized velocity field is given by the formula
Is this flow field steady or unsteady? Is it two- or three-dimensional? At the point (x, y, z) = (?1, +1, 0), compute (a) the acceleration vector and (b) any unit vector normal to the acceleration. Solution: (a) The flow is unsteady beca
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8.1 Prove that the streamlines ψ (r, θ ) in polar coordinates, from Eq. (8.10), are orthogonal to the potential lines φ (r, θ ).
Solution: The streamline slope is represented by
Since the ψ ? slope = ?1/( φ ? slope), the two sets of lines are orthogonal. Ans.
8.2 The steady plane
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