HW15.2
Due: 11:10pm on Monday, October 4, 2010
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Understanding Bernoulli's Equation
Bernoulli's equation is a simple relation that can give useful insight into the balance among fluid pressure,
flow speed, and elevation. It applies exclusively to ideal fluids with steady flow, that is, fluids with a constant
density and no internal friction forces, whose flow patterns do not change with time. Despite its limitations,
however, Bernoulli's equation is an essential tool in understanding the behavior of fluids in many practical
applications, from plumbing systems to the flight of airplanes.
For a fluid element of density
that flows along a streamline, Bernoulli's equation states that
,
where
is the pressure,
is the flow speed,
is the height,
is the acceleration due to gravity, and
subscripts 1 and 2 refer to any two points along the streamline. The physical interpretation of Bernoulli's
equation becomes clearer if we rearrange the terms of the equation as follows:
.
The term
on the lefthand side represents the total work done on a unit volume of fluid by the
pressure forces of the surrounding fluid to move that volume of fluid from point 1 to point 2. The two terms on
the righthand side represent, respectively, the change in potential energy,
, and the change in
kinetic energy,
, of the unit volume during its flow from point 1 to point 2. In other words,
Bernoulli's equation states that the work done on a unit volume of fluid by the surrounding fluid is equal to the
sum of the change in potential and kinetic energy per unit volume that occurs during the flow. This is nothing
more than the statement of conservation of mechanical energy for an ideal fluid flowing along a streamline.
Part A
Consider the portion of a flow tube shown in the figure. Point 1 and point 2 are at the same height. An
ideal fluid enters the flow tube at point 1 and moves
steadily toward point 2. If the cross section of the
flow tube at point 1 is greater than that at point 2,
what can you say about the pressure at point 2?
Hint A.1
How to approach the problem
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Hint A.2
Apply Bernoulli's equation
Hint not displayed
Hint A.3
Determine
with respect to
Hint not displayed
ANSWER:
The pressure at point 2 is
lower than the pressure at point 1.
equal to the pressure at point 1.
higher than the pressure at point 1.
Correct
Thus, by combining the continuity equation and Bernoulli's equation, one can characterize the flow of
an ideal fluid.When the cross section of the flow tube decreases, the flow speed increases, and
therefore the pressure decreases. In other words, if
, then
and
.
Part B
As you found out in the previous part, Bernoulli's equation tells us that a fluid element that flows through a
flow tube with decreasing cross section moves toward a region of lower pressure. Physically, the pressure
drop experienced by the fluid element between points 1 and 2 acts on the fluid element as a net force that
causes the fluid to __________.
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 Fall '07
 Rainey
 Physics, Fluid Dynamics, Force, Shear Stress, Strength of materials

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