Fluids Dynamics Mp

Fluids DynamicsDue: 8:00pm on Thursday, September 22, 2011

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Streamlines and Fluid Flow

Learning Goal: To understand the continuity equation.Streamlines represent the path of the flow of a fluid. You can imagine that they represent a time-exposure photograph that shows the paths of small particles carried by the flowing

fluid. The figure shows streamlines for the flow of an incompressible fluid in a tapered pipe of circular cross section. The speed of the fluid as it enters the pipe on the left is . Assume that the cross-sectional areas of the pipe

are at its entrance on the left and at its exit on the right.

Part A

Find , the volume of fluid flowing into the pipe per unit of time. This quantity is also known as the volumetric flow rate.

Hint A.1

Find the volume of fluid entering the pipe

Hint not displayed

Express the volumetric flow rate in terms of any of the quantities given in the problem introduction.

ANSWER:    = Correct

Part B

Because the fluid is assumed to be incompressible and mass is conserved, at a particular moment in time, the amount of fluid that flows into the pipe must equal the amount of fluid that flows out. This fact is embodied in the continuity equation. Using the continuity equation, find the velocity of the fluid flowing out of the right end of the pipe.

Hint B.1

Find the volumetric flow rate out of the pipe

Hint not displayed

Hint B.2

Apply the continuity equation

Hint not displayed

Express your answer in terms of any of the quantities given in the problem introduction.

ANSWER:

   =Correct

Part C

If you are shown a picture of streamlines in a flowing fluid, you can conclude that the __________ of the fluid is greater where the streamlines are closer together.Enter a one-word answer.

ANSWER:

velocityCorrect

Thus the velocity of the flow increases with increasing density (number per unit area) of streamlines.

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 left-hand 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 right-hand 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

Hint not displayed

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 __________.

Hint B.1 Effects from conservation of mass

Hint not displayed

ANSWER:

decrease in speed

increase in speed

remain in equilibrium

Correct

Part C

Now assume that point 2 is at height with respect to point 1, as shown in the figure.

The ends of the flow tube have the same areas as the ends of the horizontal flow tube shown in Part A. Since the cross section of the flow tube is decreasing, Bernoulli's equation tells us that a fluid element flowing toward point 2 from point 1 moves toward a region of lower pressure. In this case, what is the pressure drop experienced by the fluid element?

Hint C.1 How to approach the problem

Hint not displayed

ANSWER:

The pressure drop is

smaller than the pressure drop occurring in a purely horizontal flow.equal to the pressure drop occurring in a purely horizontal flow.larger than the pressure drop occurring in a purely horizontal flow.

Correct

Part D

From a physical point of view, how do you explain the fact that the pressure drop at the ends of the elevated flow tube from Part C is larger than the pressure drop occurring in the similar but purely horizontal flow from Part A?

Hint D.1 Physical meaning of the pressure drop in a tube

Hint not displayed

ANSWER:

A greater amount of work is needed to balance the

increase in potential energy from the elevation change.decrease in potential energy from the elevation change.larger increase in kinetic energy.

larger decrease in kinetic energy.

Correct

In the case of purely horizontal flow, the difference in pressure between the two ends of the flow tube had to balance only the increase in kinetic energy resulting from the acceleration of the fluid. In an elevated flow tube, the difference in pressure must also balance the increase in potential energy of the fluid; therefore a higher pressure is needed for the flow to occur.

A Siphon at the Bar

Jane goes to a juice bar with her friend Neil. She is thinking of ordering her favorite drink, 7/8 orange juice and 1/8 cranberry juice, but the drink is not on the menu, so she decides to order a glass of orange juice and a glass of cranberry juice and do the mixing herself. The drinks come in two identical tall glasses; to avoid spilling while mixing the two juices, Jane shows Neil something she learned that day in class. She drinks about 1/8 of the orange juice, then takes the straw from the glass containing cranberry juice, sucks up just enough cranberry juice to fill the straw, and while covering the top of the straw with her thumb, carefully bends the straw and places the end over the orange juice glass. After she releases her thumb, the cranberry juice flows through the straw into the orange juice glass. Jane has successfully designed a siphon.

Assume that the glass containing cranberry juice has a very large diameter with respect to the diameter of the straw and that the cross-sectional area of the straw is the same at all points. Let the atmospheric pressure be and assume that the cranberry juice has negligible viscosity.

Part A

Consider the end of the straw from which the cranberry juice is flowing into the glass

containing orange juice, and let be the distance below the surface of cranberry juice at

which that end of the straw is located: . What is the initial velocity of the cranberry juice as it flows out of the straw? Let denote

the magnitude of the acceleration due to gravity.

Hint A.1

How to approach the problem

Hint not displayed

Hint A.2

Apply Bernoulli's principle

Hint not displayed

Express your answer in terms of and .

ANSWER:

   = Correct

The speed of fluid flowing from the outlet of a siphon tube is the same as the speed that a

body would acquire in falling from rest through a distance . This result is valid also for

fluid flowing from an opening in a container at distance below the surface of the fluid.

Part B

Given the information found in Part A, find the time it takes to Jane to transfer enough

cranberry juice into the orange juice glass to make her favorite drink if centimeters. Assume that the flow rate of the liquid is constant, and that the glasses are cylindrical with a diameter of 7.0 centimeters and are filled to height 14.0 centimeters. Take the diameter of the straw to be 0.4 centimeters.

Hint B.1

How to approach the problem

To make her favorite drink, Jane has to transfer 1/8 of the cranberry juice into the orange

juice glass. To calculate the time needed to remove 1/8 of the cranberry juice, you need

to know at what volume flow rate the cranberry juice flows into the glass containing orange juice. By simply dividing the volume you need to remove by the volume flow rate, you will find the time needed to complete the fluid transfer.

Hint B.2

Find the volume flow rate

Hint not displayed

Express your answer numerically in seconds to two significant figures.

ANSWER:    =

3.8Correct

A Water Tank That Needs Cleaning

A cylindrical open tank needs cleaning. The tank is filled with water to a height

meter, so you decide to empty it by letting the water flow steadily from an opening at the

side of the tank, located near the bottom. The cross-sectional area of the tank is

square meters, while that of the opening is square meters.

Part A

How much time does it take to empty half the tank? (Note: A useful antiderivative is

.)

Hint A.1

How to approach the problem

Hint not displayed

Hint A.2

Find the discharge rate

Hint not displayed

Hint A.3

Find the outflow speed as a function of the fluid speed at the surface

Hint not displayed

Hint A.4

Rewrite the expression for the outflow speed as a function of the cross-sectional areas of the tank and the opening

Hint not displayed

Hint A.5

Find the rate of change of the level of water in the tank

Hint not displayed

Hint A.6

How to solve a separable first-order ODE

Hint not displayed

Hint A.7

The limits of integration

Hint not displayed

Express your answer numerically in seconds. Take the free-fall acceleration due to gravity

to be meters per second per second.

ANSWER:    =

51.9Correct

Venturi Meter with Two Tubes

A pair of vertical, open-ended glass tubes inserted into a horizontal pipe are often used together to measure flow velocity in the pipe, a configuration called a Venturi meter.

Consider such an arrangement with a horizontal pipe carrying fluid of density . The fluid

rises to heights and in the two open-ended tubes (see figure). The cross-sectional area of

the pipe is at the position of tube 1, and at the position of tube 2.

Part A

Find , the gauge pressure at the bottom of tube 1. (Gauge pressure is the pressure in excess of outside atmospheric pressure.)

Hint A.1

How to approach the problem

Hint not displayed

Hint A.2

Simplified Bernoulli's equation

Hint not displayed

Express your answer in terms of quantities given in the problem introduction and , the magnitude of the acceleration due to gravity.

ANSWER:    = Correc

t

The fluid is pushed up tube 1 by the pressure of the fluid at the base of the tube, and not by its kinetic energy, since there is no streamline around the sharp edge of the tube. Thus energy is not conserved (there is turbulence at the edge of the tube) at the entrance of the tube. Since Bernoulli's law is essentially a statement of energy conservation, it must be applied separately to the fluid in the tube and the fluid flowing in the main pipe. However, the pressure in the fluid is the same just inside and just outside the tube.

Part B

Find , the speed of the fluid in the left end of the main pipe.

Hint How to approach the problem

B.1

Hint not displayed

Hint B.2 Find in terms of

Hint not displayed

Hint B.3

Find in terms of given quantities

Hint not displayed

Express your answer in terms of , , , and either and or , which is equal to .

ANSWER:

   =

Correct

Note that this result depends on the difference between the heights of the fluid in the tubes, a quantity that is more easily measured than the heights themselves.

Water Flowing from a Tank

Water flows steadily from an open tank as shown in the figure.

The elevation of point 1 is 10.0 meters, and the elevation of points 2 and 3 is 2.00 meters. The cross-sectional area at point 2 is 0.0480 square meters; at point 3, where the water is discharged, it is 0.0160 square meters. The cross-sectional area of the tank is very large compared with the cross-sectional area of the pipe.

Part A

Assuming that Bernoulli's equation applies, compute the discharge rate .

Hint A.1

How to approach the problem

Hint not displayed

Hint A.2

The volume flow rate

Hint not displayed

Hint A.3

Find the fluid speed at the end of the pipe

Hint not displayed

Express your answer in cubic meters per second.

ANSWER:

   =0.200Correct

Part B

What is the gauge pressure at point 2?

Hint B.1

Definition of gauge pressure

Hint not displayed

Hint B.2

How to approach the problem

Hint not displayed

Hint B.3

Apply Bernoulli's principle

Hint not displayed

Hint B.4

Find the fluid speed at point 2

Hint not displayed

Hint B.5

Density of Water

Hint not displayed

Express your answer in pascals.

ANSWER:

6.98×104

Correct  

± Playing with a Water Hose

Two children, Ferdinand and Isabella, are playing with a water hose on a sunny summer day. Isabella is holding the hose in her hand 1.0 meters above the ground and is trying to spray Ferdinand, who is standing 10.0 meters away.

Part A

Will Isabella be able to spray Ferdinand if the water is flowing out of the hose at a constant speed of 3.5 meters per second? Assume that the hose is pointed parallel to the ground and take the magnitude of the acceleration due to gravity to be 9.81 meters per second, per second.

Hint A.1

General approach: considerations on particle motion

Hint not displayed

Hint A.2

Projectile motion

Hint not displayed

ANSWER:

Yes

No

Correct

Part B

To increase the range of the water, Isabella places her thumb on the hose hole and partially

covers it. Assuming that the flow remains steady, what fraction of the cross-sectional area of the hose hole does she have to cover to be able to spray her friend?

Assume that the cross section of the hose opening is circular with a radius of 1.5 centimeters.

Hint B.1

General approach: considerations on fluid mechanics

Hint not displayed

Hint B.2

Find the outflow speed needed

Hint not displayed

Hint B.3

Find the cross-sectional area needed

Hint not displayed

Express your answer as a percentage to the nearest integer.

ANSWER:    =

84Correct

 %

Problem 14.91

Two very large open tanks A and F (the figure

) contain the same liquid. A horizontal pipe BCD, having a constriction at C and open to the air at D, leads out of the bottom of tank A, and a vertical pipe E opens into the constriction at C and dips into the liquid in tank F. Assume streamline flow and no viscosity.

Part A

If the cross-sectional area at C is one-half the area at D and if D is a distance below the

level of the liquid in A, to what height will liquid rise in pipe E?Express your answer using one significant figure.

ANSWER:

   =3Correct

Test Your Understanding 14.4: Fluid Flow

An ideal incompressible fluid flows through a horizontal tube of radius 2.00 . At one point along the tube's length there is a constriction where the radius is only 1.00 .

Part A

Compared to the volume flow rate in the 2.00- -radius portion of the tube, the volume flow rate in the 1.00- -radius portion of the tube is

ANSWER:

2 times as great

the same

as great4 times as great

as greatnot enough information is given to decide

Correct

The volume flow rate of an incompressible fluid flowing in a tube is the same at points in the tube, no matter what the tube's radius. The fluid speed is greater where the radius is smaller and vice versa, but the volume flow rate is unaffected by changes in the radius.

Test Your Understanding 14.5: Bernoulli's Equation

Consider two points along a certain streamline in a pattern of incompressible fluid flow. At

point #1 the fluid is moving at 10.0 , whereas at point #2 the fluid is moving at 20.0 . Both points are at the same height.

Part A

Compared to the pressure at point #1, the pressure at point #2 is

ANSWER:

4 times as great

as great2 times as great

as greatthe same

not enough information given to decide

Correct

Bernoulli's equation states that

The height is the same at both points, so and the third term on the left-hand side cancels the third term on the right-hand side:

The fluid speed at point #2 ( ) is twice as great as the fluid speed at point #1 ( ), so the

term is four times greater than the term . Hence Bernoulli's equation becomes

or

This result shows that the pressure at point #2 ( ) is less than the pressure at point #1 ( ). To determine the value of , however, we would need to know the value of and the value of the fluid density ( ). Since we are not given this information, we cannot determine the ratio of the pressure at point #2 to the pressure at point #1 - which is what the question asks us to do.

Score Summary:Your score on this assignment is 102.9%.You received 97.73 out of a possible total of 95 points.

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