Fluid

Mechanics

PRESSURE AND FLUIDSTATICS

Lecture slides by Aamer Raza

John Ninomiya flying a cluster of 72 helium-filled balloons over Temecula,California in April of 2003. The helium balloons displace approximately 230m3 of air, providing the necessary buoyant force. Don’t try this at home!

Objectives• Determine the variation of pressure in a fluid at

rest

• Calculate pressure using various kinds ofmanometers

• Calculate the forces exerted by a fluid at rest onplane or curved submerged surfaces.

• Analyze the stability of floating and submergedbodies.

• Analyze the rigid-body motion of fluids incontainers during linear acceleration or rotation.

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3-1 ■ PRESSUREPressure: A normal force exertedby a fluid per unit area

Somebasic

pressuregages.

68 kg 136 kg

Afeet=300cm2

0.23 kgf/cm2 0.46 kgf/cm2

P=68/300=0.23 kgf/cm2

The normal stress (or“pressure”) on the feet of a

chubby person is much greaterthan on the feet of a slim

person.

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Absolute pressure: The actual pressure at a given position. It ismeasured relative to absolute vacuum (i.e., absolute zero pressure).

Gage pressure: The difference between the absolute pressure and thelocal atmospheric pressure. Most pressure-measuring devices arecalibrated to read zero in the atmosphere, and so they indicate gagepressure.

Vacuum pressures: Pressures below atmospheric pressure.

Throughoutthis text, thepressure Pwill denoteabsolutepressure

unlessspecifiedotherwise.

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Pressure at a Point

Forces acting on a wedge-shapedfluid element in equilibrium.

Pressure is the compressiveforce per unit area but it is nota vector. Pressure at any pointin a fluid is the same in all

directions. Pressure hasmagnitude but not a specificdirection, and thus it is ascalar quantity.

Pressure is a scalar quantity,not a vector; the pressure at apoint in a fluid is the same in

7all directions.

Variation of Pressure with DepthWhen the variation of density

with elevation is known

The pressure of a fluid at restincreases with depth (as aresult of added weight).

Free-body diagram of a rectangularfluid element in equilibrium.

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Pressure in a liquid atIn a room filled with a gas, thevariation of pressure with height isnegligible.

rest increases linearlywith distance from thefree surface.

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The pressure is the same at all points on a horizontal plane in agiven fluid regardless of geometry, provided that the points areinterconnected by the same fluid.

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Pascal’s law: The pressure applied to aconfined fluid increases the pressurethroughout by the same amount.

The area ratio A2/A1 iscalled the ideal mechanicaladvantage of the hydrauliclift.

Lifting of a largeweight by a small

force by theapplication ofPascal’s law.

3-2 ■ PRESSURE MEASUREMENT DEVICESThe Barometer

• Atmospheric pressure is measured by a device called a barometer; thus,the atmospheric pressure is often referred to as the barometric pressure.

• A frequently used pressure unit is the standard atmosphere, which isdefined as the pressure produced by a column of mercury 760 mm inheight at 0°C (Hg = 13,595 kg/m3) under standard gravitationalacceleration (g = 9.807 m/s2).

The length or thecross-sectional area

of the tube has noeffect on the heightof the fluid column of

a barometer,provided that thetube diameter islarge enough to

avoid surface tension

The basic barometer.(capillary) effects.

At high altitudes, a car engine generatesless power and a person gets less oxygenbecause of the lower density of air.

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The variation of gagepressure with depth in thegradient zone of the solarpond.

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The ManometerIt is commonly used to measure small and

moderate pressure differences. A manometercontains one or more fluids such as mercury, water,alcohol, or oil.

Measuring thepressure drop across

a flow section or a flowdevice by a differential

manometer.

The basicmanometer.

In stacked-up fluid layers, thepressure change across a fluid layerof density and height h is gh.

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Other Pressure Measurement Devices• Bourdon tube: Consists of a hollow metal tube

bent like a hook whose end is closed andconnected to a dial indicator needle.

• Pressure transducers: Use various techniquesto convert the pressure effect to an electricaleffect such as a change in voltage, resistance,or capacitance.

• Pressure transducers are smaller and faster,and they can be more sensitive, reliable, andprecise than their mechanical counterparts.

• Strain-gage pressure transducers: Work byhaving a diaphragm deflect between twochambers open to the pressure inputs.

• Piezoelectric transducers: Also called solid-state pressure transducers, work on the

principle that an electric potential is generated ina crystalline substance when it is subjected tomechanical pressure.

Various types of Bourdon tubes usedto measure pressure.

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Deadweight tester: Another type of mechanical pressure gage. It is usedprimarily for calibration and can measure extremely high pressures.

A deadweight tester measures pressure directly through application of aweight that provides a force per unit area—the fundamental definition ofpressure.

It is constructed with an internal chamber filled with a fluid (usually oil),along with a tight-fitting piston, cylinder, and plunger.

Weights are applied to the top of the piston, which exerts a force on the oilin the chamber. The total force F acting on the oil at the piston-oil interfaceis the sum of the weight of the piston plus the applied weights.

A deadweight tester isable to measureextremely high

pressures (up to 10,000psi in some

applications). 25

3-3 ■ INTRODUCTION TO FLUID STATICSFluid statics: Deals with problems associated with fluids at rest.

The fluid can be either gaseous or liquid.

Hydrostatics: When thye fluid is a liquid.

Aerostatics: When the fluid is a gas.In fluid statics, there is no relative motion between adjacent fluidlayers, and thus there are no shear (tangential) stresses in the fluidtrying to deform it.

The only stress we deal with in fluid statics is the normal stress, whichis the pressure, and the variation of pressure is due only to theweight of the fluid.

The topic of fluid statics has significance only in gravity fields.

The design of many engineering systems such as water dams andliquid storage tanks requires the determination of the forces actingon the surfaces using fluid statics.

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3-4 ■ HYDROSTATICFORCES ON SUBMERGEDPLANE SURFACES

A plate, such as a gate valve in a dam,the wall of a liquid storage tank, or the

hull of a ship at rest, is subjected tofluid pressure distributed over itssurface when exposed to a liquid.

On a plane surface, the hydrostaticforces form a system of parallel forces,and we often need to determine themagnitude of the force and its point ofapplication, which is called the centerof pressure.

HooverDam.

When analyzing hydrostatic forces onsubmerged surfaces, the atmospheric

pressure can be subtracted for simplicitywhen it acts on both sides of the structure.

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Hydrostatic forceon an inclinedplane surfacecompletelysubmerged in aliquid.

The pressure at thecentroid of a surface is

equivalent to the averagepressure on the surface. 28

The resultant force acting on aplane surface is equal to theproduct of the pressure at thecentroid of the surface and the

surface area, and its line ofaction passes through the

center of pressure.

second moment of area(area moment of inertia)about the x-axis.

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The centroid and the centroidal moments ofinertia for some common geometries.

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Pressure acts normal to the surface, andthe hydrostatic forces acting on a flatplate of any shape form a volume whosebase is the plate area and whose lengthis the linearly varying pressure.

This virtual pressure prism has aninteresting physical interpretation: itsvolume is equal to the magnitude of theresultant hydrostatic force acting on theplate since FR = PdA, and the line ofaction of this force passes through thecentroid of this homogeneous prism.

The projection of the centroid on the plateis the pressure center.Therefore, with the concept of pressureprism, the problem of describing the

resultant hydrostatic force on a planesurface reduces to finding the volume

and the two coordinates of the centroid ofthis pressure prism.

The hydrostatic forces acting on aplane surface form a pressure prismwhose base (left face) is the surfaceand whose length is the pressure.

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Special Case:Submerged

Rectangular PlateHydrostatic force actingon the top surface of a

submerged tiltedrectangular plate.

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Hydrostatic forceacting on the topsurface of a

submerged verticalrectangular plate.

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Hydrostatic force actingon the top surface of asubmerged horizontal

rectangular plate.

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3-5 ■ HYDROSTATIC FORCES ONSUBMERGED CURVED SURFACES

Determination of the hydrostatic force acting on a submerged curved surface.

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When a curved surface is above the liquid,the weight of the liquid and the verticalcomponent of the hydrostatic force act inthe opposite directions.

The hydrostatic force acting on acircular surface always passes

through the center of the circle sincethe pressure forces are normal to

the surface and they all pass 38through the center.

in a multilayered fluid of different densities can be determined byconsidering different parts of surfaces in different fluids as differentsurfaces, finding the force on each part, and then adding them usingvector addition. For a plane surface, it can be expressed as

The hydrostatic force on asurface submerged in amultilayered fluid can be

determined by considering partsof the surface in different fluids

as different surfaces.

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Schematic for Example 3-9 and the free-body diagram ofthe liquid underneath the cylinder.

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3-6 ■ BUOYANCY AND STABILITYBuoyant force: The upward force a fluid exerts on a body immersed in it.The buoyant force is caused by the increase of pressure with depth in a fluid.

The buoyant force acting onthe plate is equal to theweight of the liquiddisplaced by the plate.

For a fluid with constantdensity, the buoyant force is

independent of the distance ofthe body from the free surface.

It is also independent of thedensity of the solid body.

A flat plate of uniform thickness h submergedin a liquid parallel to the free surface.

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The buoyant forces acting on asolid body submerged in a fluid andon a fluid body of the same shapeat the same depth are identical.The buoyant force FB acts upwardthrough the centroid C of thedisplaced volume and is equal inmagnitude to the weight W of thedisplaced fluid, but is opposite indirection. For a solid of uniformdensity, its weight Ws also actsthrough the centroid, but itsmagnitude is not necessarily equalto that of the fluid it displaces.(Here Ws > W and thus Ws > FB;this solid body would sink.)

Archimedes’ principle: The buoyant force actingon a body immersed in a fluid is equal to the weightof the fluid displaced by the body, and it acts upward

through the centroid of the displaced volume. 44

For floating bodies, the weight of the entire body must be equal to thebuoyant force, which is the weight of the fluid whose volume is equal to thevolume of the submerged portion of the floating body:

A solid body droppedinto a fluid will sink,

float, or remain at restat any point in thefluid, depending on itsaverage densityrelative to the density

of the fluid. 45

The altitude of a hot airballoon is controlled by thetemperature differencebetween the air inside andoutside the balloon, sincewarm air is less dense thancold air. When the balloonis neither rising nor falling,the upward buoyant forceexactly balances thedownward weight.

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Stability of Immersed and Floating Bodies

Stability is easilyunderstood by

analyzing a ballFor floating bodies such asships, stability is an importantconsideration for safety.

on the floor.50

A floating body possesses vertical stability, while an immersedneutrally buoyant body is neutrally stable since it does notreturn to its original position after a disturbance.

An immersed neutrally buoyant body is (a) stable if thecenter of gravity G is directly below the center of buoyancyB of the body, (b) neutrally stable if G and B arecoincident, and (c) unstable if G is directly above B.

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A ball in a troughWhen the center of gravity G of an immersedneutrally buoyant body is not vertically aligned

with the center of buoyancy B of the body, it is notin an equilibrium state and would rotate to its

stable state, even without any disturbance.

between two hills isstable for smalldisturbances, butunstable for largedisturbances.

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A floating body is stable if the body is bottom-heavy and thus the center ofgravity G is below the centroid B of the body, or if the metacenter M is abovepoint G. However, the body is unstable if point M is below point G.

Metacentric height GM: The distance between the center of gravityG and the metacenter M—the intersection point of the lines of actionof the buoyant force through the body before and after rotation.

The length of the metacentric height GM above G is a measure of thestability: the larger it is, the more stable is the floating body. 53

3-7 ■ FLUIDS IN RIGID-BODY MOTIONPressure at a given point has the

same magnitude in all directions, andthus it is a scalar function.In this section we obtain relations forthe variation of pressure in fluidsmoving like a solid body with orwithout acceleration in the absence ofany shear stresses (i.e., no motionbetween fluid layers relative to eachother).

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Special Case 1: Fluids at RestFor fluids at rest or moving on a straight path at constant velocity, allcomponents of acceleration are zero, and the relations reduce to

The pressure remains constant in any horizontal direction (P isindependent of x and y) and varies only in the vertical direction asa result of gravity [and thus P = P(z)]. These relations areapplicable for both compressible and incompressible fluids.

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Special Case 2: Free Fall of a Fluid BodyA freely falling body accelerates under the influence of gravity. When the airresistance is negligible, the acceleration of the body equals the gravitationalacceleration, and acceleration in any horizontal direction is zero. Therefore,ax = ay = 0 and az = -g.

In a frame of reference moving withthe fluid, it behaves like it is in anenvironment with zero gravity. Also,the gage pressure in a drop of liquidin free fall is zero throughout.

The effect of acceleration on thepressure of a liquid during free

fall and upward acceleration. 5

Acceleration on a Straight Path

Rigid-body motion of a liquid in alinearly accelerating tank.

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Lines of constant pressure(which are the projections of thesurfaces of constant pressure onthe xz-plane) in a linearlyaccelerating liquid. Also shown isthe vertical rise.

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Rotation in a Cylindrical ContainerConsider a vertical cylindrical container partiallyfilled with a liquid. The container is now rotatedabout its axis at a constant angular velocity of .After initial transients, the liquid will move as arigid body together with the container. There isno deformation, and thus there can be no shearstress, and every fluid particle in the containermoves with the same angular velocity.

Rigid-body motion of aliquid in a rotating verticalcylindrical container.

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The 6-meter spinning liquid-mercury mirror of the LargeZenith Telescope located near

Vancouver, British Columbia. Surfaces of constantpressure in a rotatingliquid.

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Note that at a fixed radius, the pressure varies hydrostatically in thevertical direction, as in a fluid at rest.

For a fixed vertical distance z, the pressure varies with the square of theradial distance r, increasing from the centerline toward the outer edge.In any horizontal plane, the pressure difference between the center andedge of the container of radius R is

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Summary• Pressure

• Pressure Measurement Devices• Introduction to Fluid Statics

• Hydrostatic Forces on Submerged PlaneSurfaces

• Hydrostatic Forces on Submerged CurvedSurfaces

• Buoyancy and Stability

• Fluids in Rigid-Body Motion

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