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9.2 Find Dv/Dt in polar coordinates by taking the derivative of the velocity. (Hint: v = vr (r, θ, t) er + vθ (r, θ, t)eθ . Remember that the unit vectors have derivatives.)
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9.3 Using the Navier–Stokes equations and the continuity equation, obtain an expression for the velocity profile between two flat, parallel plates.
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9.4 The atmospheric density may be approximated by the relation ρ = ρ0 exp(−y/β), where β = 22,000 ft. Determine the rate at which the density changes with respect to body falling at v fps. If v = 20,000 fps at 100,000 ft, evaluate the rate of density change.
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9.5 Write equations (9-17) in component form for Cartesian coordinates....
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9.6 In polar coordinates, the continuity equation is...Show that
a. if vθ = 0, then vr = F(θ)/r
b. if vr =0, then vθ =f (r)
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9.7 Obtain the equations for a one-dimensional steady, viscous, compressible flow in the x direction from the Navier–Stokes equations. (These equations, together with an equation of state and the energy equation, may be solved for the case of weak shock waves.)
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9.8 Using the Navier–Stokes equations, find the differential equation for a radial flow in which vz = vθ = 0, and vr = f (r).
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9.9 Determine the velocity profile in a fluid situated between two coaxial rotating cylinders. Let the inner cylinder have radius R1, and angular velocity Ω1; let the outer cylinder have radius R2 and angular velocity Ω2.
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9.10 Beginning with the appropriate form of the Navier–Stokes equations, develop an equation in the appropriate coordinate system to describe the velocity of a fluid that is flowing in the annular space as shown in the figure. The fluid is Newtonian, and is flowing in steady, incompressible, fully developed, laminar flow through an infinitely long vertical round pipe annulus of inner radius RI and outer radius RO. The inner cylinder (shown in the figure as a gray solid) is solid, and the fluid flows between the inner and outer walls as shown in the figure. The center cylinder moves downward in the same direction as the fluid with a velocity v0. The outside wall of the annulus is stationary. In developing your equation, please state the reason for eliminating any terms in the original equation....
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9.11 Two immiscible fluids are flowing down between two flat, infinitely long flat parallel plates as shown in the figure below. The plate on the left is moving down with a velocity of vA, and the plate on the right is moving up with a velocity vB. The dotted line is the interface between the two fluids. The fluids maintain constant widths as they flow downward and L1 and L2 are not equal. The flow is incompressible, parallel, fully developed, and laminar. You may ignore surface tension, and assume that the fluids are not open to the atmosphere and that the interface between the fluids is vertical at all times. Derive equations for the velocity profiles of both fluids when the flow is at steady state....
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9.12 Consider steady, continuous, incompressible, fully developed laminar flow of a Newtonian fluid in an infinitely long round pipe of diameterDinclined at an angle a. The fluid is not open to the atmosphere and flows down the pipe due to an applied pressure gradient and from gravity. Derive an expression for the shear stress using the appropriate form of the Navier–Stokes equations....
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