Fluid Flow or Discharge When a fluid that fills a pipe flows through a pipe of cross-sectional area...

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Fluid Flow or Discharge Fluid Flow or Discharge When a fluid that When a fluid that fills a pipe fills a pipe flows through a flows through a pipe of cross- pipe of cross- sectional area A sectional area A with an average with an average velocity v, the velocity v, the flow or discharge flow or discharge Q is determined Q is determined by: by: v A Q

Transcript of Fluid Flow or Discharge When a fluid that fills a pipe flows through a pipe of cross-sectional area...

Fluid Flow or DischargeFluid Flow or Discharge

• When a fluid that When a fluid that fills a pipe flows fills a pipe flows through a pipe of through a pipe of cross-sectional cross-sectional area A with an area A with an average velocity v, average velocity v, the flow or the flow or discharge Q is discharge Q is determined by: determined by: vAQ

Equation of ContinuityEquation of Continuity• Suppose an Suppose an

incompressible incompressible (constant density) fluid (constant density) fluid fills a pipe and flows fills a pipe and flows through it.through it.

• If the cross-sectional If the cross-sectional area of the pipe is Aarea of the pipe is A11 at at

one point and Aone point and A22 at at another point, the flow another point, the flow through Athrough A11 must equal must equal

the flow through Athe flow through A22..

2211

2211

vAvA

vAQvAQ

Equation of ContinuityEquation of Continuity• The equation of continuity in the form The equation of continuity in the form

applies only when the density of the fluid applies only when the density of the fluid

is constant. If the density of the fluid is is constant. If the density of the fluid is not constant, the equation of continuity isnot constant, the equation of continuity is

2211 vAvA

222111 vADvAD

•mass flow rate (units kg/s):

•volume flow rate (units m3/s):

mD A v

t

VA v

t

Equation of ContinuityEquation of Continuity• The equation of continuity shows that where the The equation of continuity shows that where the

cross-sectional area is large, the fluid speed is cross-sectional area is large, the fluid speed is slow and that where the cross-sectional area is slow and that where the cross-sectional area is small, the fluid speed is large.small, the fluid speed is large.

• This explains why water shoots out of a hose This explains why water shoots out of a hose faster when you place your thumb across the faster when you place your thumb across the opening, reducing the cross-sectional area opening, reducing the cross-sectional area through which the water can come out.through which the water can come out.

ViscosityViscosity and Viscous Flow and Viscous Flow• Viscosity Viscosity of a fluid is a of a fluid is a

measure of how difficult it is measure of how difficult it is to cause the fluid to flow.to cause the fluid to flow.

• In an ideal fluid there is no In an ideal fluid there is no viscosity to hinder the fluid viscosity to hinder the fluid layers as they slide past layers as they slide past one another.one another.

• Within a pipe of uniform Within a pipe of uniform cross-sectional area, every cross-sectional area, every layer of an ideal fluid moves layer of an ideal fluid moves with the same velocity, with the same velocity,

even the layer next to even the layer next to the wall.the wall.

ViscosityViscosity and Viscous Flow and Viscous Flow• When viscosity is present, the fluid layers do When viscosity is present, the fluid layers do

not all have the same velocity. not all have the same velocity. – The fluid closest to the wall does not move at all, The fluid closest to the wall does not move at all,

while the fluid at the center of the pipe has the while the fluid at the center of the pipe has the greatest velocity.greatest velocity.

– The fluid layer next to the wall surface does not move The fluid layer next to the wall surface does not move because it is held tightly by intermolecular forces.because it is held tightly by intermolecular forces.

– The intermolecular forcesThe intermolecular forces are so strong that if aare so strong that if a solid surface moves, solid surface moves, the adjacent fluid layerthe adjacent fluid layer moves along with it andmoves along with it and remains at rest relativeremains at rest relative to the moving surface. to the moving surface.

ViscosityViscosity and Viscous Flow and Viscous Flow• This is why a layer of dust lies on the This is why a layer of dust lies on the

surface of fan blades even at high speeds. surface of fan blades even at high speeds. The layer of air in contact with the fan The layer of air in contact with the fan blade has no velocity relative to the fan blade has no velocity relative to the fan blade and does not blow off the dust.blade and does not blow off the dust.

• Force F needed to move a layer of viscous Force F needed to move a layer of viscous fluid with constant velocity:fluid with constant velocity:– = viscosity= viscosity– A = areaA = area– v = velocityv = velocity– d = distance from the immobile surfaced = distance from the immobile surface

dvAη

F

ViscosityViscosity and Viscous Flow and Viscous Flow

• Viscosity of liquids and gases depend Viscosity of liquids and gases depend on temperature. on temperature. – Usually, the viscosities of liquids Usually, the viscosities of liquids

decrease as the temperature increases. decrease as the temperature increases. – The viscosities of gases increase as the The viscosities of gases increase as the

temperature increases.temperature increases.

• Viscous fluids have a high viscosity, Viscous fluids have a high viscosity, such as tar and molasses.such as tar and molasses.

Forces Exerted By a FluidForces Exerted By a Fluid

• If a fluid were If a fluid were subjected to a subjected to a tangential force F, the tangential force F, the layers of the fluid layers of the fluid would slide past one would slide past one another without another without friction.friction.

• This means that a This means that a fluid can sustain only fluid can sustain only a perpendicular force, a perpendicular force, and conversely, can and conversely, can exert only a force exert only a force perpendicular to the perpendicular to the surface.surface.

•If a fluid were If a fluid were subjected to a subjected to a tangential force F, the tangential force F, the layers of the fluid would layers of the fluid would slide past one another slide past one another without friction.without friction.

•This means that a fluid This means that a fluid can sustain only a can sustain only a perpendicular force, and perpendicular force, and conversely, can exert conversely, can exert only a force only a force perpendicular to the perpendicular to the surface.surface.

Forces Exerted By a FluidForces Exerted By a Fluid• Suppose that a Suppose that a

nonelastic fluid isnonelastic fluid is between 2 plates. between 2 plates. If the velocityIf the velocity v of v of the upper plate is the upper plate is not toonot too large, the large, the fluid shears in the fluid shears in the wayway indicated. The indicated. The viscosity viscosity is is related to the force related to the force F required to F required to produce the produce the velocity v by: velocity v by:

dηAv

F

Forces Exerted By a FluidForces Exerted By a Fluid

• A = area of either plateA = area of either plate

• d = distance between d = distance between platesplates

• units for units for : N·s/m: N·s/m22 or or kg/m·s or lb· s/ftkg/m·s or lb· s/ft22

• 1 poiseuille (Pl) = 1 poiseuille (Pl) = 1 N·s/m1 N·s/m22 = 1 kg/m·s = 1 kg/m·s

• 1 poise (P) = 0.1 1 poise (P) = 0.1 kg/m·skg/m·s

Poiseuille’s LawPoiseuille’s Law

• The fluid flow through a cylindrical The fluid flow through a cylindrical pipe of length l and cross-sectional pipe of length l and cross-sectional radius r is given by:radius r is given by:

• PP11 - P - P22 is the pressure difference is the pressure difference

between the two ends of the pipe.between the two ends of the pipe.

lη8

)PP(rπQ 21

4

Work Done by a PistonWork Done by a Piston• Work done by a piston in forcing a Work done by a piston in forcing a

volume V of fluid into a cylinder volume V of fluid into a cylinder against an opposing pressure P is against an opposing pressure P is given by:given by: W = P W = P··VV

Bernoulli’s Principle Bernoulli’s Principle • If a fluid is incompressible - a change in pressure If a fluid is incompressible - a change in pressure

does not cause a change in volume - the volume does not cause a change in volume - the volume of fluid entering per second must equal the of fluid entering per second must equal the volume leaving per second.volume leaving per second.

Volumetric flow rate:Volumetric flow rate: 1 = entering 1 = entering 2 = leaving2 = leaving Let v represent the speed with which a liquid moves Let v represent the speed with which a liquid moves in a cylindrical pipe, so that during the time t the in a cylindrical pipe, so that during the time t the liquid moves a distance equal to v·t (where v is liquid moves a distance equal to v·t (where v is velocity.velocity.

t

V

tV 21

Bernoulli’s PrincipleBernoulli’s Principle

• The volume of liquid passing a cross-sectional The volume of liquid passing a cross-sectional

area A is given byarea A is given by

• Volumetric flow rate Q:Volumetric flow rate Q:

• Because the liquid is incompressible, the Because the liquid is incompressible, the volumetric flow rate is the same entering and volumetric flow rate is the same entering and leaving the system. leaving the system.

• Volumetric flow rate (system): Volumetric flow rate (system):

tvAV

vAtV

Q

2211 vAvA

Bernoulli’s PrincipleBernoulli’s Principle• The quantity The quantity has the has the

samesame value at every point in an incompressible value at every point in an incompressible fluid moving in streamline (non-turbulent) flow.fluid moving in streamline (non-turbulent) flow.

• Bernoulli’s equation:Bernoulli’s equation:

PgDhvD5.0 2

222

2112

1 PgDhvD5.0PgDhvD5.0

Bernoulli’s PrincipleBernoulli’s Principle• If the fluid is not moving, then both speeds are zero. The If the fluid is not moving, then both speeds are zero. The

fluid is static. If the height at the top of the column is hfluid is static. If the height at the top of the column is h11 is is defined as zero, and hdefined as zero, and h22 is the depth, then Bernoulli’s is the depth, then Bernoulli’s equation reduces to the equation for pressure as a function equation reduces to the equation for pressure as a function of depth:of depth:

• If the fluid is flowing through a horizontal pipe with a If the fluid is flowing through a horizontal pipe with a constriction, as shown in the figure on the next slide, there constriction, as shown in the figure on the next slide, there is no change in height and the gravitational potential is no change in height and the gravitational potential energy does not change. Bernoulli’s equation reduces to:energy does not change. Bernoulli’s equation reduces to:

221 hgDPP

222

211 vD5.0PvD5.0P

2 21 1 1 2 2 2

1 12 2

P h D g D v P h D g D v

Potential energy per unit volume

Kinetic energy per unit volume

Work per unit volume done by the fluid

Points 1 and 2 must be on the same streamline

Bernoulli’s PrincipleBernoulli’s Principle• The flow rate Q in the tube has The flow rate Q in the tube has

to be constant, therefore, the to be constant, therefore, the fluid has to move faster through fluid has to move faster through the constriction to maintain the the constriction to maintain the constant flow rate Q.constant flow rate Q.

• The velocity at point a is The velocity at point a is greater than at either the meter greater than at either the meter entrance or the meter exit.entrance or the meter exit.

• The pressure in a fluid is related The pressure in a fluid is related to the speed of flow, therefore to the speed of flow, therefore the pressure in the fluid is less the pressure in the fluid is less at point a and greater at the at point a and greater at the meter entrance, as illustrated meter entrance, as illustrated by the liquid levels in the U-by the liquid levels in the U-tube manometer.tube manometer.

• The pressure difference is equal The pressure difference is equal to:to: gDhP

Bernoulli’s PrincipleBernoulli’s Principle• Bernoulli's principle describes the Bernoulli's principle describes the

relationship between pressure and relationship between pressure and velocity in a fluid and describes the velocity in a fluid and describes the conservation of energy as it applies to conservation of energy as it applies to fluids.fluids.

• Bernoulli’s principle also explains why Bernoulli’s principle also explains why a roof blows off of a house in violent a roof blows off of a house in violent winds.winds. – Wind creates a low pressure region above Wind creates a low pressure region above

the peak of the roof, creating a pressure the peak of the roof, creating a pressure difference inside and outside the house difference inside and outside the house which results in the loss of the roof.which results in the loss of the roof.

Bernoulli's Equation and LiftBernoulli's Equation and Lift • The shape of a wing The shape of a wing

forces air to travel faster forces air to travel faster over the curved upper over the curved upper surface than it does over surface than it does over the flatter lower surface. the flatter lower surface.

• According to Bernoulli’s According to Bernoulli’s equation, the pressure equation, the pressure above the wing is lower above the wing is lower (faster moving air), while (faster moving air), while the pressure below the the pressure below the wing is higher (slower wing is higher (slower moving air).moving air).

• The wing is lifted upward The wing is lifted upward due to the higher due to the higher pressure on the bottom pressure on the bottom of the wing.of the wing.

Bernoulli's Equation and LiftBernoulli's Equation and Lift

• Air flows over the top of an airplane wing of area A with Air flows over the top of an airplane wing of area A with speed vt, and past the underside of the wing (also of area speed vt, and past the underside of the wing (also of area A) with speed vu.A) with speed vu.

• the magnitude Fthe magnitude FLL of the upward lift force on the wing will of the upward lift force on the wing will be: be:

• Ski jumpers use this same principle to help themselves stay Ski jumpers use this same principle to help themselves stay in the air longer during jumps.in the air longer during jumps.

• A boomerang with a curved surface will turn in the A boomerang with a curved surface will turn in the direction of the curved face due to pressure differences direction of the curved face due to pressure differences created by the different air velocities over the two surfacescreated by the different air velocities over the two surfaces..

2u

2tL vvAD5.0F

Control surfaces on a plane

• By extending the slats, the wing area A can be increased to generate more lift at low speeds for take off and landing.

Curveball Pitch

Torricelli’s TheoremTorricelli’s Theorem

• If an opening exists in a If an opening exists in a tank containing a liquid tank containing a liquid at a distance h below the at a distance h below the top of the liquid, then top of the liquid, then the velocity v of outflow the velocity v of outflow from the opening is:from the opening is:

provided the liquid obeys provided the liquid obeys Bernoulli’s equation and Bernoulli’s equation and the top of the liquid may the top of the liquid may be regarded as be regarded as motionless (v = 0 m/s).motionless (v = 0 m/s).

2v g h

Tidal WavesTidal Waves • Tidal waves are the dissipation of energy in a viscous fluid Tidal waves are the dissipation of energy in a viscous fluid

over an inclined plane; tidal waves have nothing to do with over an inclined plane; tidal waves have nothing to do with tides.tides.

• The energy source is usually an under-sea earthquake (it The energy source is usually an under-sea earthquake (it could also be an under-sea explosion or a meteor strike); the could also be an under-sea explosion or a meteor strike); the viscous fluid is the ocean; the inclined plane is the ocean viscous fluid is the ocean; the inclined plane is the ocean floor sloping upward toward land.floor sloping upward toward land.

• Earthquake:Earthquake: – When the Earth moves up and down it also moves the ocean When the Earth moves up and down it also moves the ocean

water up and down. This generates a huge wave traveling water up and down. This generates a huge wave traveling outward in a series of concentric rings. outward in a series of concentric rings.

– In deep water, most of the tidal wave (tsunami) remains hidden In deep water, most of the tidal wave (tsunami) remains hidden beneath the surface. But as the tidal wave moves toward more beneath the surface. But as the tidal wave moves toward more shallow water, its enormous energy is forced to the surface. shallow water, its enormous energy is forced to the surface.

– In the open ocean, tidal waves are hundreds of miles wide and In the open ocean, tidal waves are hundreds of miles wide and travel at jetliner speeds. Near land they slow down to freeway travel at jetliner speeds. Near land they slow down to freeway speeds. speeds.

– What makes a tidal wave so destructive is the speed and What makes a tidal wave so destructive is the speed and tremendous volume of water delivered onto a coastal or island tremendous volume of water delivered onto a coastal or island environment as the tidal wave is forced by the inclining ocean environment as the tidal wave is forced by the inclining ocean floor onto the land.floor onto the land.

Bernoulli’s Principle and SyringesBernoulli’s Principle and Syringes• The force applied to the plunger is equal to the The force applied to the plunger is equal to the

pressure times the area of the plunger.pressure times the area of the plunger.• Viscous flow will occur within the barrel of the Viscous flow will occur within the barrel of the

syringe and only a little pressure difference is syringe and only a little pressure difference is needed to move the fluid through the barrel to point needed to move the fluid through the barrel to point 2, where the fluid will enter the2, where the fluid will enter the

narrow needle.narrow needle.• The pressure appliedThe pressure applied to the plunger is nearlyto the plunger is nearly equal to the pressureequal to the pressure

PP22 at point 2. at point 2.• The pressure at point 1, PThe pressure at point 1, P11,, is also known as the gaugeis also known as the gauge pressure.pressure.

Bernoulli’s Principle and SyringesBernoulli’s Principle and Syringes

• Apply Bernoulli’s principle,Apply Bernoulli’s principle,

and if the needle is held horizontally,and if the needle is held horizontally,

• Poiseuille’s Law may also be needed Poiseuille’s Law may also be needed to solve this type of problem.to solve this type of problem.

222

211 vD5.0PvD5.0P

222

2112

1 PgDhvD5.0PgDhvD5.0

421rπ

Qlη8PP

Bernoulli’s Principle and SiphonsBernoulli’s Principle and Siphons• A A siphonsiphon is an inverted U-shaped pipe or tube that can is an inverted U-shaped pipe or tube that can

transfer water from a higher container to a lower transfer water from a higher container to a lower container by lifting the water upward from the higher container by lifting the water upward from the higher container and then lowering it into the lower container. container and then lowering it into the lower container. The water is simply seeking its level, just as it would if The water is simply seeking its level, just as it would if you connected the two containers with a pipe at their you connected the two containers with a pipe at their bottoms. In that case, the water in the higher container bottoms. In that case, the water in the higher container would flow out of it and into the lower container, would flow out of it and into the lower container, propelled by the higher water pressure at the bottom propelled by the higher water pressure at the bottom of the higher container. In the case of a of the higher container. In the case of a siphonsiphon, it's , it's still the higher water pressure in the higher container still the higher water pressure in the higher container that causes the water to flow toward the lower that causes the water to flow toward the lower container, but in the container, but in the siphonsiphon the water must the water must temporarily flow above the water level in the higher temporarily flow above the water level in the higher container on its way to the lower container.container on its way to the lower container.

Bernoulli’s Principle and SiphonsBernoulli’s Principle and Siphons• Two means of initiating theTwo means of initiating the

liquid flow (assume the liquid is liquid flow (assume the liquid is water):water):– You can make a siphon using a You can make a siphon using a

rubberrubber

hose and gravity is the key to getting hose and gravity is the key to getting it to work. A siphon needs to have the it to work. A siphon needs to have the "dry" end of the hose lower than the "dry" end of the hose lower than the end that is stuck in the water (the end that is stuck in the water (the "wet" end). You can get the siphon "wet" end). You can get the siphon started by first filling the hose with started by first filling the hose with water. Once the hose is full, use your water. Once the hose is full, use your thumb to plug the end of the hose thumb to plug the end of the hose that will be removed from the water. that will be removed from the water. Place the “dry” end into the second Place the “dry” end into the second container and then remove your container and then remove your thumb. Gravity does all the work from thumb. Gravity does all the work from there... there...

How does it work? Think of the water in terms of distinct How does it work? Think of the water in terms of distinct “packets". Since the dry end of the hose is lower than the wet “packets". Since the dry end of the hose is lower than the wet end, there are more water "packets" towards the dry end. As such, end, there are more water "packets" towards the dry end. As such, the column of water being pulled downward by gravity is heavier the column of water being pulled downward by gravity is heavier than the column of water at the wet end of the tube. Gravity pulls than the column of water at the wet end of the tube. Gravity pulls on one “packet" of water on the dry end of the tube causing it to on one “packet" of water on the dry end of the tube causing it to move down the tube. As it moves, it creates a small vacuum move down the tube. As it moves, it creates a small vacuum behind itself. This vacuum pulls the next “packet” forward behind itself. This vacuum pulls the next “packet” forward (downward) as well. This suction is strong enough to pull other (downward) as well. This suction is strong enough to pull other “packets” up the tube (against gravity) at the wet end. Once a “packets” up the tube (against gravity) at the wet end. Once a given “packet” passes the highest point in the tube, gravity pulls it given “packet” passes the highest point in the tube, gravity pulls it downward and thedownward and the

process continues. The siphon will workprocess continues. The siphon will work as long as the vertical (up and down)as long as the vertical (up and down) column of water outside the container is column of water outside the container is larger than the vertical column inside thelarger than the vertical column inside the container. If the two ends of the hose arecontainer. If the two ends of the hose are exactly the same height (the columns areexactly the same height (the columns are equal), the pull of gravity will be the sameequal), the pull of gravity will be the same on each side and the flow of water will stop.on each side and the flow of water will stop. If you then lower the free end, the flow ofIf you then lower the free end, the flow of water will begin once again. water will begin once again.

Bernoulli’s Principle and SiphonsBernoulli’s Principle and Siphons– Sucking on the lower end of the tube Sucking on the lower end of the tube

causes a partial vacuum (a region of causes a partial vacuum (a region of space with a pressure that's less space with a pressure that's less than atmospheric pressure) at the than atmospheric pressure) at the top of the top of the siphonsiphon. The partial . The partial vacuum results in a difference in vacuum results in a difference in pressure between the bottom of the pressure between the bottom of the tube and the top of the tube. With tube and the top of the tube. With greater fluid pressure at the top than greater fluid pressure at the top than the bottom, the water is pushed up the bottom, the water is pushed up into the tube and over to the lower into the tube and over to the lower container. The same kind of partial container. The same kind of partial vacuum exists in a drinking straw vacuum exists in a drinking straw when you suck on it and is what when you suck on it and is what allows atmospheric pressure to push allows atmospheric pressure to push the beverage up toward your mouth.the beverage up toward your mouth.

Bernoulli’s Principle and SiphonsBernoulli’s Principle and Siphons

Bernoulli’s Principle and SiphonsBernoulli’s Principle and Siphons• To get water up to the top of the tube, need To get water up to the top of the tube, need

to reduce pressure in the tube so that the to reduce pressure in the tube so that the atmospheric pressure will do the work to push atmospheric pressure will do the work to push the fluid up to the top of the tube where the fluid up to the top of the tube where gravity will do the work to pull the fluid out gravity will do the work to pull the fluid out the end of the hose.the end of the hose.

• Apply Bernoulli’s equation Apply Bernoulli’s equation

between points A and B to between points A and B to

determine the maximum heightdetermine the maximum height

hh11 to which water can be to which water can be

lifted above the water surface.lifted above the water surface.

2 21

2

2

1 2

1 2

1

1 1

2 20 set top surface of water as 0 ; 0

0 (want this to be as small as possible)

2 (d h ) to be explained soon

12 (d h )

21

2 (d h )2

atm t B B

t

B

B C

atm

atm

atm

P h D g D v P h D g D v

h v

P

v v g

P h D g D g

P h D g D g

P h

2(d h )D g D g

Bernoulli’s Principle and SiphonsBernoulli’s Principle and Siphons• To determine the speed of the liquid flow at To determine the speed of the liquid flow at

the bottom of the siphon, start with Bernoulli’s the bottom of the siphon, start with Bernoulli’s equation:equation:

• Atmospheric pressure is found at Atmospheric pressure is found at

the top of the liquid and at the the top of the liquid and at the

bottom of the siphon, therefore, bottom of the siphon, therefore, PPtt and P and Pbb are equal and cancel are equal and cancel out.out.

• Consider the velocity vConsider the velocity vtt at the top at the top

of the liquid to be 0 m/s.of the liquid to be 0 m/s.

bb2

btt2

t PgDhvD5.0PgDhvD5.0

Bernoulli’s Principle and SiphonsBernoulli’s Principle and Siphons• Consider the lower end of the Consider the lower end of the

siphon to be the point at which siphon to be the point at which

the height is 0 m. From the figure,the height is 0 m. From the figure,

the distance from the bottom of thethe distance from the bottom of the

to the siphon to the upper level ofto the siphon to the upper level of

the liquid is d + hthe liquid is d + h22..

• The density cancels out:The density cancels out:

• Solve for vSolve for vbb::

2b2 vD5.0gD)hd(

2b2 v5.0g)hd(

ghd2v 2b

Equation of Continuity ExampleEquation of Continuity Example

• What is the flow rate of water in a pipe What is the flow rate of water in a pipe whose diameter is 10 cm when the water whose diameter is 10 cm when the water is moving with a velocity of 0.322 m/s?is moving with a velocity of 0.322 m/s?

sm

0.002529Q

sm

0.322m0.007854vAQ

m0.007854m0.05πrπA

m0.05rm;0.1cm10d

3

2

222

Equation of Continuity ExampleEquation of Continuity Example

• If the diameter If the diameter of the pipe to of the pipe to the right is the right is reduced to 4 reduced to 4 cm, what is the cm, what is the velocity of the velocity of the fluid in the fluid in the right-hand side right-hand side of the pipe?of the pipe?

sm

.v

m0.02sm

0.322m0.05v

vrvr

vrπvrπ

vAvA;QQ

R

2

2

r

vrR

R2

RL2

L

R2

RL2

L

RRLLrightleft

2R

L2

L

01252

Bernoulli’s ExampleBernoulli’s Example• The pressure PThe pressure P11= 53913.24 N/m= 53913.24 N/m22, whereas the velocity of the , whereas the velocity of the

water vwater v11 = 0.322 m/s. The diameter of the pipe at location 1 is = 0.322 m/s. The diameter of the pipe at location 1 is 10 cm and it is at ground level. If the diameter of the pipe at 10 cm and it is at ground level. If the diameter of the pipe at location 2 is 4 cm, and the pipe is 5 m above the ground, find location 2 is 4 cm, and the pipe is 5 m above the ground, find the pressure Pthe pressure P22 of the water at position 2. of the water at position 2.– From the previous example, we know that the velocity of the water From the previous example, we know that the velocity of the water

at location 2 is 2.015 m/s.at location 2 is 2.015 m/s.

Bernoulli’s ExampleBernoulli’s Example

22

22222

2

3

23

2

322

22

2112

222

211

31

222

2111

m

N.P

m

N.

m

N

m

N.

m

N.5P

sm

2.015m

kg10000.5

s

m9.8

m

kg1000m5

sm

0.322m

kg10000.5

m

N2940P

vD0.5gDhvD0.5PP

vD0.5gDhPvD0.5P

m

kg1000Dlevel);(groundm0h

vD0.5gDhPvD0.5gDhP

2

2

2

972934

112520304900084251243913

Helpful Online LinksHelpful Online Links

•Hyperphysics Fluids

•Work-Energy Applet (to determine the power needed in the pump for the water-jet to pass over the wall)

•Gallery of Fluid Mechanics