objectives - kntu · objectives • deals with forces applied by fluids at rest or...

Objectives

• deals with forces applied by fluids at rest or in rigid-body motion.• The fluid property responsible for those forces is pressure, which is a

normal force exerted by a fluid per unit area.

discussion of pressure(absolute & gage pressures)

pressure at a point

variation of pressure withdepth in a gravitational

field

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Objectives

pressure measurementdevices

manometer

barometer

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Objectives

hydrostatic forces appliedon submerged bodies

Plane surface

Curved surface

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Objectives

buoyant force applied byfluids

submerged bodies

floating bodies

stability of suchbodies?

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Objectives

stability of such bodies?

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Objectives

apply Newton’s 2nd law of motion to a body of fluid in motion that acts as arigid body:

Analyze the variation of pressure in fluids that undergo linear accelerationand in rotating containers.

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Pressure definition

• Pressure is defined as a normal force exerted by a fluid per unit area. Wespeak of pressure only when we deal with a gas or a liquid.

• The counterpart of pressure in solids is normal stress.

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Pressure definition

• The pressure unit pascal is too small for pressures encountered inpractice.

• Therefore, its multiples kilopascal (1 kPa = 103 Pa) and megapascal(1 MPa = 106 Pa) are commonly used.

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Pressure definition

• The actual pressure at a given position is called the absolute pressure,and it is measured relative to absolute vacuum (i.e., absolute zeropressure).

• Most pressure-measuring devices, however, are calibrated to read zero inthe atmosphere

• difference between the absolute pressure and the local atmosphericpressure: gage pressure.

• Pressures below atmospheric pressure are called vacuum pressures

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Pressure definition

• Throughout this text, the pressure P will denote absolute pressure unlessspecified otherwise. Often the letters “a” (for absolute pressure) and “g”(for gage pressure) are added to pressure units (such as psia and psig) toclarify what is meant.

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Pressure at point

• Pressure is a scalar quantity• Newton’s second law,a force balance in the x- & z-dir:

Pressure at a PointPressure at a Point

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Pressure at point

regardless of the angle

We can repeat the analysis for an element in the xz-plane and obtain asimilar result.Thus we conclude that the pressure at a point in a fluid has the samemagnitude in all directions.It can be shown in the absence of shear forces that this result is applicable

to fluids in motion as well as fluids at rest.12/118

Pressure variation

• pressure in a fluid at rest does not change in the horizontal direction.Variation of Pressure with DepthVariation of Pressure with Depth

Considering a thin horizontal layer offluid and doing a force balance in any

horizontal direction.

this is not the case in the verticaldirection in a gravity field.

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Pressure variation

• Pressure in a fluid increases with depth because more fluid rests ondeeper layers, and the effect of this “extra weight” on a deeper layer isbalanced by an increase in pressure

• force balance in the vertical z-direction:

specific weight of the fluidpressure in a fluid increases linearly with depth

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Pressure variation

• for small to moderate distances, the variation of pressure with height isnegligible for gases because of their low density.

pressure in a room filledwith air can be assumed

to be constant

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Pressure variation

• If we take point 1 to be at the free surface of a liquid open to theatmosphere, then the pressure at a depth h from the free surfacebecomes:

• Liquids are essentially incompressiblesubstances, and thus the variation ofdensity with depth is negligible.

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Pressure variation

• at great depths such as those encountered in oceans, the change in thedensity of a liquid can be significant

• For fluids whose density changes significantly with elevation:

positive z direction to be upward

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Pressure variation

• Dutch mathematician Simon Stevin (1586):

Pressure in a fluid at rest is independent of the shape or cross section of thecontainer. It changes with the vertical distance, but remains constant in

other directions.

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Pressure variation

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Pressure variation

• Pascal’s law:

“Pascal’s machine” has been thesource of many inventions thatare a part of our daily lives such

as hydraulic brakes and lifts.

The area ratio A2 /A1 is called:ideal mechanical advantage of the hydraulic lift

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Pressure variation

Using a hydraulic car jack with a piston area ratio of A2 /A1 = 10,a person can lift a 1000-kg car by applying a force of just 100 kgf

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Pressure Measurement

• A device based on the pressure head principle is called a manometer, andit is commonly used to measure small and moderate pressure differences.

• A manometer mainly consists of a glass or plastic U-tube containing oneor more fluids such as mercury, water, alcohol, or oil.

• To keep the size of the manometer to a manageable level, heavy fluidssuch as mercury are used if large pressure differences are anticipated.

Manometer: Single fluidManometer: Single fluid

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• measure the pressure in the tank

Since the gravitational effects ofgases are negligible, the pressureanywhere in the tank is assumed

to be constant

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• cross-sectional area of the tube has no effect on the differential height h

• the diameter of the tube should be large enough (more than a fewmillimeters) to ensure that the surface tension effect and thus thecapillary rise is negligible.

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• Many engineering problems and some manometers involve multipleimmiscible fluids of different densities stacked on top of each other.

Manometer: multi-fluidManometer: multi-fluid

Pascal’s law: 2 points at the same elevation in a continuous fluid at rest areat the same pressure.

There is jump over a different fluid

+ + + =27/118


• Manometers are particularly well-suited to measure pressure dropsacross a horizontal flow section between two specified points due to thepresence of a device such as a valve or heat exchanger or any resistance toflow.

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BarometerBarometer

Atmospheric pressure is measured by a device called a barometer;atmospheric pressure = barometric pressure.

The Italian Evangelista Torricelli (1608–1647) was the first to conclusivelyprove that the atmospheric pressure can be measured by inverting a

mercury-filled tube into a mercury container that is open to theatmosphere

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g is the local gravitational accel.

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• length and the cross-sectional area of the tube have no effect on theheight of the fluid column of a barometer

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• A frequently used pressure unit is the standard atmosphere, (weatherforecasters)

• If water instead of mercury were used to measure the standardatmospheric pressure, a water column of about 10.3 m would be needed.

Nose bleeding is a common experience at high altitudes since the differencebetween the blood pressure and the atmospheric pressure is larger in this

case, and the delicate walls of veins in the nose are often unable towithstand This extra stress.

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= 15238/118

Hydrostatic Forces On Submerged Surfaces

• A plate exposed to a liquid, such as a gate valve in a dam, the wall of aliquid storage tank, or the hull of a ship at rest, is subjected to fluidpressure distributed over its surface.

• Hydrostatic forces form a system of parallel forces,• we need to determine the magnitude of the force;• and its point of application,

Hydrostatic Forces On Submerged Plane SurfacesHydrostatic Forces On Submerged Plane Surfaces

center of pressure

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When analyzing hydrostatic forceson submerged surfaces:

the atmospheric pressure can besubtracted for simplicity when it

acts on both sides of the structure.

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• Hydrostatic force on an inclined plane surface completely submerged ina liquid

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• absolute pressure at any point on the plate

• resultant hydrostatic force:

• y-coord of the centroid (or center) of the surface:

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where PC is the pressure at the centroid of the surface,which is equivalent to the average pressure on the surface,

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• The pressure P0 is usually atmospheric pressure, which can be ignored inmost cases since it acts on both sides of the plate.

• If not:

• Then, assume the presence of an additional liquid layer of thickness hequivon top of the liquid with absolute vacuum above.

Pressure magnitude: OkayPressure point of application: ???

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• The line of action of the resultant hydrostatic force, in general, does notpass through the centroid of the surface. it lies underneath where thepressure is higher.

• Two parallel force systems are equivalent if they have the samemagnitude and the same moment about any point.

• equating the moment of the resultant force to the moment of thedistributed pressure force about the x-axis

line of action of the resultant forceline of action of the resultant force

Center of Pressure

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• The Ixx are widely available for common shapes in engineeringhandbooks, but they are usually given about the axes passing through thecentroid of the area.

• Fortunately, the second moments of area about two parallel axes arerelated to each other by the parallel axis theorem,

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• For P0=0:

For areas that possess symmetry about the y-axis, the center ofpressure lies on the y-axis directly below the centroid.

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Special Case: Submerged Rectangular PlateSpecial Case: Submerged Rectangular Plate

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:

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• When the upper edge of the plate is at the free surface and thus s=0

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• Vertical rectangle;• When P0 is ignored since it acts on both sides of the plate• and whose top edge is horizontal and at the free surface:

• Pressure center:

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pressure distribution on a submergedhorizontal surface is uniform

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Hydrostatic Forces On Submerged Curved SurfacesHydrostatic Forces On Submerged Curved Surfaces

Integration of thepressure forces

that changedirection along

the curved surface

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Easiest way to determine theresultant hydrostatic force: FR

Easiest way to determine theresultant hydrostatic force: FR

???Determine horizontalcomponent: FH

???Determine horizontalcomponent: FH

???Determine verticalcomponent: FV

???Determine verticalcomponent: FV

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Vertical surface of the liquid blockconsidered is simply the projection of

the curved surface on a verticalplane,

Horizontal surface is the projectionof the curved surface on a horizontal

plane

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• in static equilibrium, the force balances:

add magnitudes if both act in thesame direction and subtract ifthey act in opposite directions

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• exact location of the line of action of the resultant force?

• The exact location of the line of action of the resultant force (e.g., itsdistance from one of the end points of the curved surface) can bedetermined by taking a moment about an appropriate point.

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The hydrostatic force acting on a circularsurface always passes through the centerof the circle since the pressure forces arenormal to the surface and they all pass

through the center.

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• hydrostatic forces acting on a plane or curved surface submerged in amultilayered 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:

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• pressure at the centroid of the portion of the surface in fluid i and Ai isthe area of the plate in that fluid. The line of action of this equivalentforce can be determined from the requirement that the moment of theequivalent force about any point is equal to the sum of the moments ofthe individual forces about the same point.

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Buoyancy

• The force that tends to lift the body is called the buoyant force: FB

• The buoyant force is causedby the increase of pressure in afluid with depth

Buoyancy & StabilityBuoyancy & Stability

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Buoyancy

• the buoyant force acting on the plate is equal to the weight of the liquiddisplaced by the plate.

• The weight and the buoyant force must have the same line of action tohave a zero moment: Archimedes’ principle

Buoyant force is independent of the distance of the body from the freesurface.

It is also independent of the density of the solid body.

The buoyant force acting on a body immersed in a fluid is equal to theweight of the fluid displaced by the body, and it acts upward through the

centroid of the displaced volume.73/118

Buoyancy

For floating bodies:the weight of the entire

body must be equal to thebuoyant force,

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Buoyancy

• a body immersed in a fluid– (1) remains at rest at any point in the fluid when its density is equal to

the density of the fluid,– (2) sinks to the bottom when its density is greater than the density of

the fluid, and– (3) rises to the surface of the fluid and floats when the density of the

body is less than the density of the fluid

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Buoyancy

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Buoyancy

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Buoyancy

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Buoyancy

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Buoyancy

• The buoyant force is proportional to the density of the fluid, and thus wemight think that the buoyant force exerted by gases such as air isnegligible.

• This is certainly the case in general,but there are significant exceptions.

• For example, the volume of a person is about 0.1 m3, and taking thedensity of air to be 1.2 kg/m3, the buoyant force exerted by air on theperson is

• The weight of an 80-kg person is 80x9.81=788 N.• Therefore, ignoring the buoyancy in this case results in an error in

weight of just 0.15 percent, which is negligible. 80/118

Buoyancy & Stability

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Buoyancy & Stability

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Stability

Stability of Immersed and Floating BodiesStability of Immersed and Floating Bodies

• An important application of the buoyancy concept is the assessment ofthe stability of immersed and floating bodies with no externalattachments.

design of ships and submarines

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Stability

• For an immersed or floating body in static equilibrium, the weight andthe buoyant force acting on the body balance each other, and such bodiesare inherently stable in the vertical direction.

• If an immersed neutrally buoyant body is raised or lowered to a differentdepth, the body will remain in equilibrium at that location.

• If a floating body is raised or lowered somewhat by a vertical force, thebody will return to its original position as soon as the external effect isremoved.

• Therefore, a floating body possesses vertical stability, while an immersedneutrally buoyant body is neutrally stable since it does not return to itsoriginal position after a disturbance.

vertical & rotational stability: Verticalvertical & rotational stability: Vertical

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Stability: Immersed body

• The rotational stability of an immersed body depends on the relativelocations of the center of gravity G of the body and the center ofbuoyancy B, which is the centroid of the displaced volume.

• An immersed body is stable if the body is bottom-heavy and thus point Gis directly below point B.

• A rotational disturbance of the body in such cases produces a restoringmoment to return the body to its original stable position.

rotational stability of an immersed bodyrotational stability of an immersed body

Hot-air or helium balloons (which can be viewed as being immersed in air)are also stable since the cage that carries the load is at the bottom.

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• if the floating body is bottom-heavy and thus the center of gravity G isdirectly below the center of buoyancy B, the body is always stable.

Stable: A rotational disturbance of the body in such cases produces arestoring moment to return the body to its original stable position.

EquilibriumExists

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An immersed body whose center of gravity G is directly above point B isunstable, and any disturbance will cause this body to turn upside down.

A body for which G and B coincide is neutrally stable.

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• it cannot be at rest and would rotate toward its stable state even withoutany disturbance.

What about a case where the center of gravity is not vertically aligned withthe center of buoyancy

Not in a stateof equilibrium

restoring moment to align point G vertically with point B88/118

Stability: Floating body

• The rotational stability criteria are similar for floating bodies.

• Again, if the floating body is bottom-heavy and thus the center of gravityG is directly below the center of buoyancy B, the body is always stable.

• Unlike immersed bodies, a floating body may still be stable when G isdirectly above B.

rotational stability of a floating bodyrotational stability of a floating body

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• this is because the centroid of the displaced volume shifts to the side to apoint B’ during a rotational disturbance while the center of gravity G ofthe body remains unchanged.

• If point B’ is sufficiently far, these two forces create a restoring momentand return the body to the original position.

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• distance between the center of gravity G and the metacenter M (theintersection point of the lines of action of the buoyant force through thebody before and after rotation)

• A floating body is stable if point M is above point G, and thus GM ispositive, and unstable if point M is below point G, and thus GM isnegative.

A measure of stability for floating bodies is the metacentric height GMA measure of stability for floating bodies is the metacentric height GM

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• The metacenter may be considered to be a fixed point for small rollingangles up to about 20°.

• In the latter case, the weight and the buoyant force acting on the tiltedbody generate an overturning moment instead of a restoring moment,causing the body to capsize.

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

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

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• reduction of metacentric height and increases the possibility of capsizing

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Fluids in Rigid-body Motion

• In this section we obtain relations for the variation of pressure in fluidsmoving like a solid body with or without acceleration in the absence ofany shear stresses.

Rotation in a Cylindrical ContainerRotation in a Cylindrical Container

Straight PathStraight Path

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Consider a differential rectangular fluid element

pressure at the center ofthe element: P

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• Newton’s second law of motion for this element

• Forces: - Body forces: Gravity; - Surface forces: Pressure forces

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• Similarly, the net surface forces in the x- and y-directions are:

• Where:97/118


• The total force acting on the element is:

• Substituting into Newton’s 2nd law of motion:

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• Special Case: Fluids at Rest

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Fluids in Rigid-body Motion: Straight path

• Consider a container partially filled with a liquid. The container ismoving on a straight path with a constant acceleration.

Acceleration on a Straight PathAcceleration on a Straight Pathrigid bodyrigid body

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• Pressure is independent of y;

• Taking point 1 to be the origin (x=0, z=0) where the pressure is P0 andpoint 2 to be any point in the fluid (no subscript),

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• vertical rise (or drop) of the free surface:

– ℎ =• Isobars:

• Thus we conclude that the isobars (including the free surface) in anincompressible fluid with constant acceleration in linear motion areparallel surfaces whose slope in the xz-plane is:

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Fluids in Rigid-body Motion: Cylindrical container

Rotation in a Cylindrical ContainerRotation in a Cylindrical Container

We know from experience that when aglass filled with water is rotated about itsaxis, the fluid is forced outward as a resultof the so-called centrifugal force, and thefree surface of the liquid becomesconcave. This is known as the forcedvortex motion.

rigid bodyrigid body

every fluid particle in the containermoves with the same angular velocity

every fluid particle in the containermoves with the same angular velocity

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which is the equation of a parabola. Thus we conclude that the surfaces ofconstant pressure, including the free surface, are paraboloids of revolution. 109/118


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• For the free surface at r=0:

• Where hc is the distance of the free surface from the bottom of thecontainer along the axis of rotation.

• Free surface equation:

Zs is the distance of the free surface from the bottom of the container at radius r

volume of the paraboloid formed by the free surface=original volume of the fluid in the container

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• volume of the paraboloid formed by the free surface:

• original volume of the fluid in the container• Where h0 is the original height of the fluid in the container with no

rotation.

• Free surface equation:

• Maximum height difference:

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• Taking point 1 t o be the origin (r=0, z=0):

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