Consider the same two concentric cylinders of Prob. 4-44. This time, however, the inner cylinder is rotating. but the outer cylinder is stationary. In the limit, as the outer cylinder is very large compared to the inner cylinder (imagine the inner cylinder spinning very fast while its radius gets very small), what kind of flow does this approximate? Explain. After a long time has passed, generate an expression for the tangential velocity profile, namely u θ as a function of (at most) ω i , R j , R o , ρ , and μ .. Hint: Your answer may contain an (unknown) constant, which can be obtained by specifying a boundary condition at the inner cylinder surface.
Consider the same two concentric cylinders of Prob. 4-44. This time, however, the inner cylinder is rotating. but the outer cylinder is stationary. In the limit, as the outer cylinder is very large compared to the inner cylinder (imagine the inner cylinder spinning very fast while its radius gets very small), what kind of flow does this approximate? Explain. After a long time has passed, generate an expression for the tangential velocity profile, namely u θ as a function of (at most) ω i , R j , R o , ρ , and μ .. Hint: Your answer may contain an (unknown) constant, which can be obtained by specifying a boundary condition at the inner cylinder surface.
Solution Summary: The author explains the expression for tangential velocity: V=wR_i2r.
Consider the same two concentric cylinders of Prob. 4-44. This time, however, the inner cylinder is rotating. but the outer cylinder is stationary. In the limit, as the outer cylinder is very large compared to the inner cylinder (imagine the inner cylinder spinning very fast while its radius gets very small), what kind of flow does this approximate? Explain. After a long time has passed, generate an expression for the tangential velocity profile, namely
u
θ
as a function of (at most)
ω
i
,
R
j
,
R
o
,
ρ
, and
μ
.. Hint: Your answer may contain an (unknown) constant, which can be obtained by specifying a boundary condition at the inner cylinder surface.
The velocity field for a line vortex in the r?-plane is given byur = 0 u? = K / rwhere K is the line vortex strength. For the case with K = 1.5 m/s2, plot a contour plot of velocity magnitude (speed). Specifically, draw curves of constant speed V = 0.5, 1.0, 1.5, 2.0, and 2.5 m/s. Be sure to label these speeds on your plot.
consider the 2 dimensional velocity field V= -Ayi +Axj
where in this flow field does the speed equal to A? Where does the speed equal to 2A?
Consider the steady, two-dimensional, incompressible velocity field, namely, V-›= (u, ?) = (ax + b) i-›+ (−ay + cx) j-›. Calculate the pressure as a function of x and y.
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