Problem 5.51
[Difficulty: 2]
Given:
Sediment concentration fates in a river after a rainfall are:
t
c
100
ppm
hr
x
c
50
ppm
mi
Stream speed is 0.5 mph, where a boat is used to survey the concentration.
The boat speed is 2.5 mph.
Find:
Physically the observed rates of change differ because the
observer is convected through the flow. The convective
change may add to or subtract from the local rate of change.
Solution:
We will apply the concept of substantial derivative
t
c
z
c
w
y
c
v
x
c
u
Dt
Dc
Governing
Equation:
(Substantial Derivative)
Assumptions:
(1) Onedimensional motion (velocity not a function of y or z)
(2) Steady flow (velocity is not a function of t)
t
c
x
c
u
Dt
Dc
Based on the above assumptions, the substantial derivative reduces to:
To obtain the rates of change from the boat, we set
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 Fall '07
 Lear
 Fluid Mechanics, Derivative

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