14. Fluid Mechanics

WELCOME TO GROUP WELCOME TO GROUP 1414

BYBY::

1.Lailatul Fitriyani1.Lailatul Fitriyani

2.Dina Rachmawati2.Dina Rachmawati

3.Indah Dwiphayanti3.Indah Dwiphayanti

FLUID MECHANICSFLUID MECHANICS

Chapter outline:Chapter outline:1.1.PressurePressure2.2.Variation of pressure with depthVariation of pressure with depth3.3.Pressure measurementPressure measurement4.4.Buoyant forces and Archimedes’s Buoyant forces and Archimedes’s prinsipleprinsiple5.5.Fluid DinamicsFluid Dinamics6.6.Bernoulli’s EquationBernoulli’s Equation7.7.Other Aplications of Fluid DynamicsOther Aplications of Fluid Dynamics

PRESSUREPRESSURE Pressure is force apllied to unit area of Pressure is force apllied to unit area of

surface.Measured in pascal( SI unit ) or in dynes ( cgs surface.Measured in pascal( SI unit ) or in dynes ( cgs unit ).1 Pa = 1 N/munit ).1 Pa = 1 N/m².².

The fluid pressure can be The fluid pressure can be measured directly if the measured directly if the

spring is spring is calibrated in advance. calibrated in advance. If If F F is the is the magnitude of the force magnitude of the force exerted exerted on theon thepiston and piston and A A is the surface area of the piston, then the is the surface area of the piston, then the pressure pressure P P of the fluid at theof the fluid at thelevel to which the device has been submerged is defined level to which the device has been submerged is defined as the ratio as the ratio FF//AA::

P=F/A

QUICK QUIZQUICK QUIZ

ANSWERANSWER

(a). Because the basketball player’s weight is (a). Because the basketball player’s weight is distributed over the larger surface area of the distributed over the larger surface area of the shoe, the pressure (shoe, the pressure (FF//AA) that he applies is ) that he applies is relatively small. The woman’s lesser weight relatively small. The woman’s lesser weight is distributed over the very small is distributed over the very small crosssectional area of the spiked heel, so the crosssectional area of the spiked heel, so the pressure is high.pressure is high.

VARIATION OF PRESSURE WITH VARIATION OF PRESSURE WITH DEPTHDEPTH

Atmospheric pressure decreases with Atmospheric pressure decreases with increasing altitude; for this reason, increasing altitude; for this reason, aircraft flying at high altitudes must have aircraft flying at high altitudes must have pressurized cabins. pressurized cabins. Density Density of a of a substance is defined as its mass per unit substance is defined as its mass per unit volume. Under standard conditions (at volume. Under standard conditions (at 0°C and at atmospheric pressure) the 0°C and at atmospheric pressure) the densities of gases are about 1/1 000 the densities of gases are about 1/1 000 the densities of solids and liquids.densities of solids and liquids.

PASCAL’S LAWPASCAL’S LAW

““A change in the pressure applied to a fluid is A change in the pressure applied to a fluid is transmitted undiminished to every point of the transmitted undiminished to every point of the fluid and to the walls of the container”. fluid and to the walls of the container”.

Therefore,Therefore,the force the force FF2 is greater than the force 2 is greater than the force FF1 by a 1 by a factor factor AA2/2/AA1. By designing a hydraulic1. By designing a hydraulicpress with appropriate areas press with appropriate areas AA1 and 1 and AA2, a 2, a large output force can be applied bylarge output force can be applied bymeans of a small input force.The aplicant of means of a small input force.The aplicant of the pascal’s law is hydraulic brake,car the pascal’s law is hydraulic brake,car lift,hydraulic jacks,fork lift,etc.lift,hydraulic jacks,fork lift,etc.

2

2

1

1

21

A

F

A

F

PP

QUICK QUIZQUICK QUIZ

The pressure at the bottom of a filled glass of The pressure at the bottom of a filled glass of water ( water (ρρ==1 000 kg/m1 000 kg/m³³) is ) is PP. The water is . The water is poured out and the glass is filled with ethyl poured out and the glass is filled with ethyl alcohol(alcohol(ρρ== 806 kg/m 806 kg/m³³). The pressure at the ). The pressure at the bottom of the glass is (a) smaller than bottom of the glass is (a) smaller than PP(b) (b) equal to equal to P P (c) larger than (c) larger than P P (d) indeterminate(d) indeterminate

ANSWERANSWER

(a). Because both fluids have the same (a). Because both fluids have the same depth, the one with the smaller density depth, the one with the smaller density (alcohol) will exert the smaller pressure.(alcohol) will exert the smaller pressure.

PRESSURE MEASUREMENTPRESSURE MEASUREMENT

We can measure the pressure with barometer. This tool We can measure the pressure with barometer. This tool consist of a glass tube which filled mercury, where end consist of a glass tube which filled mercury, where end up of close-tube and end down of open-tube entered into up of close-tube and end down of open-tube entered into baller of mercury. The closed end of the tube is nearly a baller of mercury. The closed end of the tube is nearly a vacuum. The different hight between vacuum. The different hight between surface up and down from surface up and down from mercury is effect of air pressure. mercury is effect of air pressure. If the air pressure increase, If the air pressure increase, part of mercury in barrel will be enter part of mercury in barrel will be enter to tube. And if the air pressure decrease, to tube. And if the air pressure decrease, part of mercury in barrel will be out and part of mercury in barrel will be out and push the water surface in tube. push the water surface in tube.

Buoyant Forces and Buoyant Forces and Archimedes’s PrincipleArchimedes’s Principle

Buoyant force is upward force exerted by a Buoyant force is upward force exerted by a fluid on any immersed object. We can determine fluid on any immersed object. We can determine the magnitude of a buoyant force by applying the magnitude of a buoyant force by applying some logic. Imagine a beach ball&sized parcel some logic. Imagine a beach ball&sized parcel of water beneath the water surface, as in Figureof water beneath the water surface, as in Figure

Because this parcel is in equilibrium, there must Because this parcel is in equilibrium, there must be an upward force that balances the downward be an upward force that balances the downward gravitational force on the parcel. This upward gravitational force on the parcel. This upward force is the buoyant force, and its magnitude is force is the buoyant force, and its magnitude is equal to the weight of the water in the parcel.equal to the weight of the water in the parcel.

Now imagine replacing the beach ball&sized Now imagine replacing the beach ball&sized

parcel of water with a beach ball of the same parcel of water with a beach ball of the same size. The resultant force applied by the fluid size. The resultant force applied by the fluid surrounding the beach ball is the surrounding the beach ball is the same,regardless of whether it is applied to a same,regardless of whether it is applied to a beach ball or to a parcel of water. Consequently, beach ball or to a parcel of water. Consequently, we can claim that we can claim that the magnitude of the the magnitude of the buoyant force always equals the weight of buoyant force always equals the weight of the fluid displaced by the objectthe fluid displaced by the object. This . This statement is known as statement is known as Archimedes’s principleArchimedes’s principle..

Quick QuizQuick Quiz

AnswerAnswer

(c). The ice cube displaces a volume of (c). The ice cube displaces a volume of water that has a weight equal to that of the water that has a weight equal to that of the ice cube. When the ice cube melts, it ice cube. When the ice cube melts, it becomes a parcel of water with the same becomes a parcel of water with the same weight and exactly the volume that was weight and exactly the volume that was displaced by the ice cube before.displaced by the ice cube before.

Fluid DynamicsFluid Dynamics

The flow is said to be steady, or laminar, if each particle The flow is said to be steady, or laminar, if each particle of the fluid follows a smooth path, such that the paths of of the fluid follows a smooth path, such that the paths of different particles never cross each other. In steady flow, different particles never cross each other. In steady flow, the velocity of fluid particles passing any point remains the velocity of fluid particles passing any point remains constant in time. constant in time.

The term viscosity is commonly used in the description The term viscosity is commonly used in the description

of fluid flow to characterize the degree of internal friction of fluid flow to characterize the degree of internal friction in the fluid. This internal friction, or in the fluid. This internal friction, or viscous forceviscous force, is , is associated with the resistance that two adjacent layers of associated with the resistance that two adjacent layers of fluid have to moving relative to each other. Viscosity fluid have to moving relative to each other. Viscosity causes part of the kinetic energy of a fluid to be causes part of the kinetic energy of a fluid to be converted to internal energy.converted to internal energy.

Because the motion of real fluids is very Because the motion of real fluids is very complex and not fully understood, we make complex and not fully understood, we make some simplifying assumptions in our approach. some simplifying assumptions in our approach. In our model of ideal fluid flow,we make the In our model of ideal fluid flow,we make the following four assumptions:following four assumptions:

1.1. The fluid is nonviscous. The fluid is nonviscous.

2.2. The flow is steady. The flow is steady.

3.3. The fluid is incompressible. The fluid is incompressible.

4.4. The flow is irrotationalThe flow is irrotational. .

Quick QuizQuick Quiz

AnswerAnswer

(b). The liquid moves at the highest speed in (b). The liquid moves at the highest speed in the straw with the smaller cross sectional the straw with the smaller cross sectional area.area.

Bernoulli’s EquationBernoulli’s Equation

As a fluid moves through a region where its As a fluid moves through a region where its speed and/or elevation above the Earth’s speed and/or elevation above the Earth’s surface changes, the pressure in the fluid surface changes, the pressure in the fluid varies with these changes. Consider the flow varies with these changes. Consider the flow of a segment of an ideal fluid through a of a segment of an ideal fluid through a nonuniform pipe in a time interval nonuniform pipe in a time interval ∆t.∆t.

A fluid in laminar flow through a A fluid in laminar flow through a constricted pipe. The volume of the constricted pipe. The volume of the shaded portion on the left is equal to the shaded portion on the left is equal to the volume of the shaded portion on the right.volume of the shaded portion on the right.

The net work done on the segment by these forces The net work done on the segment by these forces in the time interval in the time interval ∆t is:∆t is:

VPPW

VPVPW

WWW

VPxAPW

VPxAPW

)( 21

21

21

22222

11111

Part of this work is goes into changing the kinetic energy of Part of this work is goes into changing the kinetic energy of fluid, and part the goes into changing the gravitational fluid, and part the goes into changing the gravitational potential energy of the segment-Earth system. The change potential energy of the segment-Earth system. The change in the kinetic energy of the segment of fluid isin the kinetic energy of the segment of fluid is

Considering the gravitational potential energy of the Considering the gravitational potential energy of the segment–Earth system, once again there is no change segment–Earth system, once again there is no change during the time interval for the unshaded portion of the fluid.during the time interval for the unshaded portion of the fluid.

21

22 2

1

2

1mvmvK

12 mgymgyU

The total work done on the system by the fluid The total work done on the system by the fluid outside the segment is equal to the change in outside the segment is equal to the change in mechanical energy of the system:mechanical energy of the system:

If we divide each term by the portion volume If we divide each term by the portion volume V V and and recall that recall that ρρ== mm//VV, this expression, this expressionreduces toreduces to

122

12

221 2

1

2

1)( mgymgymvmvVPP

UKW

122

12

221 2

1

2

1gygyvvPP

This is This is Bernoulli’s equationBernoulli’s equation as applied to an ideal as applied to an ideal fluid. It is often expressed asfluid. It is often expressed as

When the fluid is at rest, and Equation 14.8 When the fluid is at rest, and Equation 14.8 becomesbecomes

tconsgyvP tan2

1 2

021 vv

ghyygPP )( 1221

While Equation 14.9 was derived for an While Equation 14.9 was derived for an incompressible fluid, the general behaviorincompressible fluid, the general behavior

of pressure with speed is true even for gases—as of pressure with speed is true even for gases—as the speed increases, the pressurethe speed increases, the pressure

decreases. This decreases. This Bernoulli effect Bernoulli effect explains the explains the experience with the truck on the highway atexperience with the truck on the highway at

the opening of this section.the opening of this section.

QUICK QUIZQUICK QUIZ

You observe two helium balloons floating next to You observe two helium balloons floating next to each other at the ends of strings secured to a each other at the ends of strings secured to a table. The facing surfaces of the balloons aretable. The facing surfaces of the balloons are

separated by 1–2 cm. You blow through the separated by 1–2 cm. You blow through the small space between the balloons. What small space between the balloons. What happens to the balloons? (a) They move toward happens to the balloons? (a) They move toward each other. (b) They move away from each each other. (b) They move away from each other. (c) They are unaffected.other. (c) They are unaffected.

AnswerAnswer

(a). The high-speed air between the balloons (a). The high-speed air between the balloons results in low pressure in this region. The higher results in low pressure in this region. The higher pressure on the outer surfaces of the balloons pressure on the outer surfaces of the balloons pushes them toward each other.pushes them toward each other.

Other Applications of Fluid Other Applications of Fluid DynamicsDynamics

Consider the streamlines that flow Consider the streamlines that flow around an airplane wing. Let us assume around an airplane wing. Let us assume that the air stream approaches the wing that the air stream approaches the wing horizontally from the right with a velocity horizontally from the right with a velocity v1. The tilt of the wing causes the air v1. The tilt of the wing causes the air stream to be deflected downward with a stream to be deflected downward with a velocity v2. velocity v2.

According to Newton’s third law, the air stream According to Newton’s third law, the air stream exerts a force F on the wing that is equal in exerts a force F on the wing that is equal in magnitude and opposite in direction. This force magnitude and opposite in direction. This force has a vertical component called the lift (or has a vertical component called the lift (or aerodynamic lift) and a horizontal component aerodynamic lift) and a horizontal component called drag.called drag.

In general, an object moving through a fluid experiences In general, an object moving through a fluid experiences lift as the result of any effect that causes the fluid to lift as the result of any effect that causes the fluid to change its direction as it flows past the object. Some change its direction as it flows past the object. Some factors that influence lift are the shape of the object, its factors that influence lift are the shape of the object, its orientation with respect to the fluid flow, any spinning orientation with respect to the fluid flow, any spinning motion it might have, and the texture of its surface. For motion it might have, and the texture of its surface. For example, a golf ball struck with a club is given a rapid example, a golf ball struck with a club is given a rapid backspin due to the slant of the club.backspin due to the slant of the club.

A number of devices operate by means of the A number of devices operate by means of the pressure differentials that result from differences pressure differentials that result from differences in a fluid’s speed. For example, a stream of air in a fluid’s speed. For example, a stream of air passing over one end of an open tube, the other passing over one end of an open tube, the other end of which is immersed in a liquid, reduces the end of which is immersed in a liquid, reduces the pressure above the tubepressure above the tube