Fluids in MotionFluids in Motion

The lower falls at Yellowstone National Park: the water at the top of the falls passes through a narrow slot, causing the velocity to increase at that point. In this chapter, we will study the physics of fluids in motion.

Fluid Motion

Paul E. Tippens

Objectives: After completing Objectives: After completing this module, you should be this module, you should be able to:able to:• Define the Define the rate of flowrate of flow for a fluid and for a fluid and

solve problems using velocity and solve problems using velocity and cross-section. cross-section.

• Write and apply Write and apply Bernoulli’s equationBernoulli’s equation for the general case and apply for (a) a for the general case and apply for (a) a fluid at rest, (b) a fluid at constant fluid at rest, (b) a fluid at constant pressure, and (c) flow through a pressure, and (c) flow through a horizontal pipe.horizontal pipe.

Fluids in MotionFluids in Motion

All fluids are assumed in this treatment to

exhibit streamline flow.

• Streamline flow is the motion of a fluid in which every particle in the fluid follows the same path past a particular point as that followed by previous particles.

• Streamline flow is the motion of a fluid in which every particle in the fluid follows the same path past a particular point as that followed by previous particles.

Assumptions for Fluid Assumptions for Fluid Flow:Flow:

Streamline flow Turbulent flow

• All fluids move with streamline flow.

• The fluids are incompressible.

• There is no internal friction.

• All fluids move with streamline flow.

• The fluids are incompressible.

• There is no internal friction.

Rate of FlowRate of FlowThe The rate of rate of flowflow R R is defined as the volume is defined as the volume VV of a fluid of a fluid that passes a certain cross-section that passes a certain cross-section AA per unit of time per unit of time tt..

The volume The volume VV of fluid is given by of fluid is given by the product of area the product of area A A and and vtvt:: V AvtV Avt

AvtR vA

t Rate of flow = velocity x area

vt

Volume = A(vt)

A

Constant Rate of FlowConstant Rate of FlowFor an incompressible, frictionless fluid, the For an incompressible, frictionless fluid, the velocity increases when the cross-section velocity increases when the cross-section decreases:decreases:

1 1 2 2R v A v A

A1

A2

R = A1v1 = A2v2

v1

v2

v2

2 21 1 2 2v d v d2 2

1 1 2 2v d v d

Example 1:Example 1: Water flows through a Water flows through a rubber hose rubber hose 2 cm2 cm in diameter at a in diameter at a velocity of velocity of 4 m/s4 m/s. What must be the . What must be the diameter of the nozzle in order that the diameter of the nozzle in order that the water emerge at water emerge at 16 m/s16 m/s??The area is proportional to the square of diameter, so:

2 21 1 2 2v d v d

d2 = 0.894 cmd2 = 0.894 cmd2 = v1d12/v2

Example 1 (Cont.):Example 1 (Cont.): Water flows Water flows through a rubber hose through a rubber hose 2 cm2 cm in in diameter at a velocity of diameter at a velocity of 4 m/s4 m/s. What is . What is the the rate of flowrate of flow in m in m33/min?/min?

2 21

1 1

(4 m/s) (0.02 m)

4 4

dR v

R1 = 0.00126 m3/s

1 1 2 2R v A v A

21

1 1 1; 4

dR v A A

R1 = 0.0754 m3/minR1 = 0.0754 m3/min

Problem Strategy for Rate of Problem Strategy for Rate of Flow:Flow:

• Read, draw, and label given information.Read, draw, and label given information.

• The rate of flow R is volume per unit time.The rate of flow R is volume per unit time.

• When cross-section changes, R is constant.When cross-section changes, R is constant.

• Read, draw, and label given information.Read, draw, and label given information.

• The rate of flow R is volume per unit time.The rate of flow R is volume per unit time.

• When cross-section changes, R is constant.When cross-section changes, R is constant.

1 1 2 2R v A v A

• Be sure to use consistent units for area and velocity.

• Be sure to use consistent units for area and velocity.

Problem Strategy (Continued):Problem Strategy (Continued):

• Since the area Since the area AA of a pipe is proportional to of a pipe is proportional to its diameter its diameter dd, a more useful equation is:, a more useful equation is:

• Since the area Since the area AA of a pipe is proportional to of a pipe is proportional to its diameter its diameter dd, a more useful equation is:, a more useful equation is:

• The units of area, velocity, or diameter chosen for one section of pipe must be consistent with those used for any other section of pipe.

• The units of area, velocity, or diameter chosen for one section of pipe must be consistent with those used for any other section of pipe.

2 21 1 2 2v d v d

Work in Moving Work in Moving a Volume of a Volume of FluidFluid

P1

A1

P1

A1

P2

A2

A2

P2

h

Volume V

Note differences in pressure P and area

A

Fluid is raised to a height h.

22 2 2 2

2

; F

P F P AA

11 1 1 1

1

; F

P F P AA

F1

, F2

Work on a Fluid (Cont.)Work on a Fluid (Cont.)

Net work done on fluid is sum of work done by input force Fi less the work done by resisting force F2,

as shown in figure.

Net work done on fluid is sum of work done by input force Fi less the work done by resisting force F2,

as shown in figure.

Net Work = P1V - P2V = (P1 - P2) V

F1 = P1A1

F2 = P2A2

v1

v2

A1

A2

h2

h1 s1

s2

Conservation of EnergyConservation of EnergyKinetic Energy

K:2 22 1½ ½K mv mv

Potential Energy U:

2 1U mgh mgh

Net Work = K + U

2 21 2 2 1 2 2( ) (½ ½ ) ( )P P V mv mv mgh mgh

also Net Work = (P1 - P2)V

F1 = P1A1

F2 = P2A2

v1

v2

A1

A2

h2

h1 s1

s2

Conservation of EnergyConservation of Energy2 2

1 2 2 1 2 2( ) (½ ½ ) ( )P P V mv mv mgh mgh

Divide by V, recall that density m/V, then simplify:

2 21 1 1 2 2 2½ ½P gh v P gh v

Bernoulli’s Theorem:2

1 1 1½P gh v Const

v1

v2

h1

h2

Bernoulli’s Theorem (Horizontal Pipe):Bernoulli’s Theorem (Horizontal Pipe):2 2

1 1 1 2 2 2½ ½P gh v P gh v

h1 = h2

v1 v2

Horizontal Pipe (h1 = h2)

2 21 2 2 1½ ½P P v v

h

Now, since the difference in pressure P = gh,

2 22 1½ ½P gh v v Horizontal

Pipe

Example 3:Example 3: Water flowing at Water flowing at 4 m/s4 m/s passes passes through a Venturi tube as shown. If through a Venturi tube as shown. If h h = 12 = 12 cmcm, what is the velocity of the water in the , what is the velocity of the water in the constriction?constriction?

v1 = 4 m/s

v2h

h = 6 cm

2 22 1½ ½P gh v v

Bernoulli’s Equation (h1 = h2)

2gh = v22 - v1

2Cancel , then clear fractions:

2 2 22 12 2(9.8 m/s )(0.12 m) (4 m/s)v gh v

v2 = 4.28 m/s v2 = 4.28 m/s Note that density is not a factor.

Bernoulli’s Theorem for Fluids at Bernoulli’s Theorem for Fluids at Rest.Rest.

For many situations, the fluid remains at rest so that v1

and v2 are zero. In such cases we have:

2 21 1 1 2 2 2½ ½P gh v P gh v

P1 - P2 = gh2 - gh1P = g(h2 -

h1)

P = g(h2 - h1)

h = 1000

kg/m3

This is the same relation seen earlier for finding the pressure P at a given depth h = (h2 - h1) in a fluid.

Torricelli’s TheoremTorricelli’s Theorem

2v gh

h1

h2h

When there is no change of pressure, P1 = P2.

2 21 1 1 2 2 2½ ½P gh v P gh v

Consider right figure. If surface v2 and P1= P2 and v1 = v we have:

Torricelli’s theorem:

2v gh

v2

Interesting Example of Interesting Example of Torricelli’s Theorem:Torricelli’s Theorem:

v

vv

Torricelli’s theorem:

2v gh

• Discharge velocity increases with depth.

• Holes equidistant above and below midpoint will have same horizontal range.

• Maximum range is in the middle.

Example 4:Example 4: A dam springs a A dam springs a leak at a point leak at a point 20 m20 m below the below the surface. What is the emergent surface. What is the emergent velocity?velocity?

2v ghhTorricelli’s theorem:

2v gh

Given: h = 20 m g = 9.8 m/s2

22(9.8 m/s )(20 m)v

v = 19.8 m/s2v = 19.8 m/s2

Strategies for Bernoulli’s Equation:Strategies for Bernoulli’s Equation:

• Read, draw, and label a rough sketch with givens.

• The height h of a fluid is from a common reference point to the center of mass of the fluid.

• In Bernoulli’s equation, the density is mass density and the appropriate units are kg/m3.

• Write Bernoulli’s equation for the problem and simplify by eliminating those factors that do not change.

• Read, draw, and label a rough sketch with givens.

• The height h of a fluid is from a common reference point to the center of mass of the fluid.

• In Bernoulli’s equation, the density is mass density and the appropriate units are kg/m3.

• Write Bernoulli’s equation for the problem and simplify by eliminating those factors that do not change.

2 21 1 1 2 2 2½ ½P gh v P gh v

Strategies (Continued)Strategies (Continued)2 2

1 1 1 2 2 2½ ½P gh v P gh v

• For a stationary fluid, v1 = v2 and we have:

P = g(h2 - h1)

P = g(h2 - h1)

• For a horizontal pipe, h1 = h2 and we obtain:

h = 1000

kg/m3

2 21 2 2 1½ ½P P v v

• For no change in pressure, P1 = P2 and we have:

Strategies (Continued)Strategies (Continued)2 2

1 1 1 2 2 2½ ½P gh v P gh v

2v gh

Torricelli’s Theorem