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.
he velocity at apoint in aflued for one-dimensional
Plow wmay be aiven in The Eutkerian coordinater by
U=Ax+ Bt, Show That X
Coordinates Canbe obtained from The Eulerian system.
The intial position
by Xo and The intial time to zo man be assumeal ·
1.
x = foxo, yo) in The Lagrange
of The fluid parficle is designated
4. Problem 4-42: The velocity field for solid-body rotation in the re-plane (Fig. P4-42) is
given by
u, = 0
ue = wr
Where w is the magnitude of the angular velocity (@ points in the z-direction). For the
case with w = 1.5 s, 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.
FIGURE P4-42
What can be explained from the contour plot velocity magnitude around the cylinder? relates to the theory in mechanics of fluids.
This figure obtains from the streamlined formula and it is extracted in cartesian coordinates.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.