01 statics

ADVANCED FLUID MECHANICS

Fluid Statics

Advanced Fluid Mechanics. Fluid statics

LaCàN www-lacan.upc.edu 2

Introduction

• Why do boats stay afloat? • How can submarines stay submerged? • What forces are applied on a dam? • How does a hydraulic press work? • … These questions (and many others) can be answered by analyzing the behavior of a fluid at rest.

Falkirk Wheel, Scotland http://en.wikipedia.org/wiki/Falkirk_Wheel

http://www.youtube.com/watch?v=n61KUGDWz2A

Mercury manometer http://en.wikipedia.org/wiki/Blood_pressure "

El Atazar Dam, Spain http://www.pbs.org/wgbh/buildingbig/

dam/basics.html



Forces on a fluid

Forces applied on a fluid can be of two different types:

1.  Surface forces The stress (σ) is the force per unit area, that can be separated in •  Thermodynamic pressure (p)

-  Depends on the position of the fluid molecules -  Normal to the surface

•  Viscous stress -  Due to the motion of the fluid molecules -  Depends on the rate of distortion -  Usually, proportional to the fluid viscosity µ

2.  Body forces Act on the whole volume: gravity, electromagnetic, Coriolis, centrifugal …

viscous stress is zero for a static analysis



Pressure in a static fluid

When a force is applied on the surface of a fluid, a uniform pressure is created in any point of the liquid. This pressure originates forces that are always perpendicular to the walls of the fluid container. A static fluid is an isotropic structure, with no preferred directions. Therefore, the stress must have the direction of the normal vector and have the same magnitude for any direction.

Pascal’s Law!

fluids can only sustain compression

See also http://www.mauilab.com/guests/hydrostatics/



Proof of Pascal’s law

To prove that the magnitude of σ is independent of the normal direction, we consider the fluid element of the picture and impose a force balance in the x direction.

• If the element shrinks to zero the two forces are applied at the same point P.

• The same applies if we choose y or z direction.



Hydraulic force transmission



Hydrostatic equilibrium

•  The pressure force applied on a surface can be obtained by integrating the pressure per unit area:

Thus, is the pressure force per unit volume. •  The body force per unit volume is . Hydrostatic equilibrium In order for a fluid to remain motionless, the applied forces must be balanced:

Gauss’ theorem


The same equilibrium equation can be obtained by balancing forces on an infinite domain.


Body forces are: The forces due to hydrostatic pressure are

in the horizontal direction and

in the vertical direction. Imposing equilibrium for an arbitrary volume, one gets

Hydrostatic equilibrium (2)

x

y b



Pressure in a gravitational field (1) Hydrostatic equilibrium for a body in a gravitational field: with Integrate along a line c connecting two points in the fluid • The first integral is easily computed:

• For the second one, we need to take into account that the gravity acceleration can be expressed as



Pressure in a gravitational field (2)

Then, if we consider constant density: Substituting in the equilibrium equation yields: or, using Cartesian coordinates with the z-axis opposite to the direction of :

Hydrostatic equilibrium for a fluid with constant density under a gravitational force"



Pressure in a gravitational field (3)

A force f is called conservative if: • Its curl is zero or • The work done along a closed path is zero or • It can be written as the gradient of a potential F: For any conservative force, the equilibrium equation can be integrated along a line



Pressure measurement (1)

Mercury barometer A glass tube of 1m length is filled with mercury. Its open end is placed in a pool of mercury. The mercury column falls, leaving a space which is almost a perfect vacuum.

ρHg = 1.306x104 kg/m3 g = 9.8066 m/s2

patm = 1.0133x105 Pa h = 0.760 m

p1 = 0 (vacuum)



Pressure measurement (2)

Manometer The same principle can be used to measure the pressure in closed containers Most pressure measuring devices calculate the pressure difference between the fluid container and the atmosphere. Definition: • Gauge pressure: absolute pressure – atmospheric pressure > 0 • Vacuum pressure: atmospheric pressure – absolute pressure > 0


Example

Multiple-liquid manometer Water flows through pipes A and B. Lubricating oil is in the upper portion of the inverted U. Mercury is in the bottom of the manometer bends. Determine the pressure difference, pA – pB in units of kPa. ρwater = 1000 kg/m3

SG = ρ / ρwater SGoil = 0.88 SGHg = 13.6




Pressure forces on solid surfaces

In many applications, it is important to compute the force exerted by a fluid at rest on a surface. We can calculate the force F and the moment T by integrating the differential force and moment applied on a differential surface dΓ: The pressure forces can be replaced by a single force acting at the center of pressure xcp: In structures completely or partially surrounded by the atmosphere, the absolute pressure p is replaced by the gauge pressure p-patm



We can use the gauge pressure p-patm because a uniform pressure produces no net force or moment on the structure:

+"



Example: gravity dam (1)

First of all, the pressure distribution is obtained by applying the hydrostatic equilibrium condition: The total force applied to the surface is: where the normal vector is

We want to compute the pressure force of the water on the dam.



Example: gravity dam (2)

The moment is Finally, we can compute the center of pressure:

ez"


Example: inclined plane submerged surface

The inclined surface shown, hinged along edge A, is 5m wide. Determine the resultant force FR of the water and the air on the inclined surface.



Example: curved submerged surface

The gate shown is hinged at O and has a constant width w=5m. The equation of the surface is x=y2/a, where a=4m. The depth of water to the right of the gate is D=4m. Find the magnitude of the force Fa applied as shown, required to maintain the gate in equilibrium if the weight of the gate is neglected.




Stratified fluids

Density is not uniform but varies with altitude (atmosphere, deep oceans).

For these fluids, the hydrostatic equilibrium equation

is no longer valid, because pressure is not constant.

Pressure at any point can be computed using the differential form of the equilibrium equation:

Pressure changes only on the z direction and gravity acts in the same direction. Thus, the equation can be written

and integrating between a point z0 with known pressure p0 and any other point z



Pressure forces on bodies immersed in fluids

A body immersed in a fluid withstands a pressure force (buoyant force) Fb. This force is equal in magnitude but opposite in direction to the force of gravity acting on the displaced fluid. For a completely immersed body, the buoyant force is

where V is the volume of the body and ρ the density of the fluid. This buoyant force Fb can be considered to act on the center of buoyancy because the moment of the gravity force is

Archimedes’ Principle!



Pressure forces on floating bodies

If a body is floating in the interface of two fluids, the same principle applies but we have to consider the force acting on the displaced volume of each fluid. However, when a body is floating in water, we can neglect the force acting by air because ρair<<ρwater.



Stable equilibrium

Forces applied on an immersed body are: •  Buoyant force Fb applied on B (center of buoyancy)

•  Gravitational force ρgV applied on G (center of mass) As B and G are vertically aligned, the body is in static equilibrium.

If the body is rotated a small angle ε, there will be a restoring couple ρgVε[BG] whenever G lies below B. On the contrary, if G lies above B, the couple will tend to increase the angle of rotation and the body will turn upside down.

01 statics

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