Fluid Mechanics for Mechanical Engineers/Fluid Statics

From Wikiversity
Jump to navigation Jump to search

>back to Chapters

Definitions[edit]

Fluid at rest(left) and in rigid body motion (right)

Fluid statics is the study of fluids which are either at rest or in rigid body motion with respect to a fixed frame of reference. Rigid body motion means that there is no relative velocity between the fluid particles.

In a fluid at rest, there is no shear stress, i. e. fluid does not deform, but fluid sustains normal stresses.

We can apply Newton's second law of motion to evaluate the reaction of the particle to the applied forces.

Forces created by pressure on the surfaces of a differential fluid volume in a static fluid

Force balance in direction:

We can also say,

Force created by pressure is :

is the vector having the surface area as magnitude and surface normal as direction.

Thus,

Force caused by the pressures opposite to the surface normal.

For a differential fluid element:

Remember Taylor Series expansion:

P is the pressure in the center of the fluid element, therefore the pressure on the surface in direction of is .

Thus,


Thus,

or,


or,


or,


for

Pressure changes only in direction.

Pressure variation in an incompressible and static fluid[edit]

Pressure variation with depth of water column


is constant since and are constants.



If we take at the surface, then:



h is measured from the surface.

Any two points at the same elevation in a continuous length of the same liquid are at the same pressure. Pressure increases as one goes down in a liquid column. Remember:

similar depth in same fluid experience similar hydrostatic pressure

For incompressible flow,

Consider 3 immiscible fluids in a container and find out a relation for the pressure at the bottom of the fluid shown in the schematics besides.

Hydrostatic pressure profile for fluids with different densities and height

Transmission of Pressure[edit]

Concept of transmission of pressure is very important for hydraulic and pneumatics system. Neglecting elevation changes the following relation can be written:

which could be stated in the famous Pascal's law like below:

A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid

Pascal's law

Communicating containers[edit]

Lets consider two closed containers(which means the free surface pressure could be different than atmospheric pressure) both contain same fluid are connected via a connector valve. When the valve is open, the heights of the fluid columns can give an indication about the pressure in both chamber.

For closed container

(of course,when we calculate the small 'h', it should be measured at the height of connecting valve for both column distinctively.)

for

So from the picture above, we can understand that the pressure in the right column is higher than the left column.

For open Containers,

If both fluid columns are at the same level


so, the depth of the fluid from free surface in both column will be the same. This nice principle was used for Water-based Barometer [1] a.k.a 'Storm Barometer' or 'Goethe Barometer'. Try to see if you understand the device.


variation of height of fluid column due to difference in pressure from free surface
Goethe Barometer

Pressure Measurement Equipments[edit]

Barometer[edit]

From the equations which we derived before , it is also possible to measure the pressure exerted by almost 100 km[2] thick earth atmosphere which is above us. Since the constituents and the density varies over the height of the atmosphere , we will consider a fluid column which have free surface with no atmospheric pressure but connected with a fluid which experience atmospheric pressure like communicating container. Let consider first (from previous section),

for atm = kPa

water height will be:


where as height of mercury will be around So now taking this barometer to desired location and observing the mercury column height, the atmospheric pressure could be measured.

Barometer

U-tube manometer[edit]

The volume rate of flow, Q, through a pipe can be determined by means of a flow nozzle located in the pipe as illustrated in the figure. The nozzle creates a pressure drop, , along the pipe which is related to the flow through the equation where K is a constant depending on the pipe and nozzle size. The pressure drop is frequently measured with a differential U-tube manometer of the type illustrated. (a) Determine an equation for in terms of the specific weight of the flowing fluid, , the specific weight of the gage fluid, , and the various heights indicated. (b) For , , and , what is the value of the pressure drop, ? Solution

(a) Although the fluid in the pipe is moving, the fluids in the columns of the manometer are at rest so that the pressure variation in the manometer tubes is hydrostatic. If we start at point A and move vertically upward to level (1), the pressure will decrease by and will be equal to the pressure at (2) and (3). We can move from (3) to (4) where the pressure has been further reduced by . The pressures at levels (4) and (5) are equal, and as we move from (5) to B the pressure will increase by .

Thus, in equation form

or

It is to be noted that the only column height of importance is the differential reading, . The diferential manometer could be placed 0.5 m or 5.0 m above the pipe or and the value of would remain the same. Relatively large values for the differential reading can be obtained for small pressure differences, ,if the difference between and is small.


(b) The specific value of the pressure drop for the data given is

Application of U tube manometer to measure pressure difference

Inclined-Tube Manometer[edit]

To measure small pressure changes, a monometer of the type shown in the figure, is frequently used. One leg of the manometer is inclined at an angle , and the differential reading is measured along the inclined tube. The difference in pressure can be expressed as


or

where the pressure difference between points (1) and (2) is due to the vertical distance between the points, which can be expressed as . Thus, for relatively small angles the differential reading along the inclined tube can be made large even for small pressure differences. The inclined-tube manometer is often used to measure small differences in gas pressures so that if pipes A and B contain a gas then


or

where the contributions of the gas columns and have been neglected. The equation above shows that the differential reading (for a given pressure difference) of the inclined-tube manometer can be increased over that obtained with a conventional U-tube manometer by the factor . Recall that as .


inlclined-Tube manometer to measure small difference

Buoyancy[edit]

Derivation of buoyancy on a body immersed in a fluid

Static pressure rises in fluids as the location of interest gets closer to the earth due to gravitational force. Owing to this change of pressure at different elevations, a lifting force, namely buoyancy, occurs.

Buoyancy can be best explained when a body is immersed in a liquid. Consider a differential column volume of the immersed body. At the bottom surface, a higher hydrostatic force is applied than that on the top surface because of higher hydrostatic pressure on the bottom surface. Hence, the net differential force in direction is

Hence, the total force becomes:

This is the buoyancy force and it exists on all bodies immersed in a fluid which is under the effect of gravitation.

Hydrostatic Forces on Submerged Surfaces[edit]

When an object is submerged into liquid, forces due to hydrostatic pressure act on the surface of the body. These forces are distributed on the surface of the object and their magnitude and direction change with the local depth and the surface normal, respectively. When designing technical applications, it is essential to know:

  • The magnitude of the resultant force on the surface (integrated force)
  • The direction of the resultant force
  • Line of action of the resultant force

so that the structure can be designed to sustain the hydrostatic surface forces. Examples technical applications are: under water tunnels, buildings, gates, submarines, etc..

Magnitude of Resultant Force[edit]

Consider the submerged flat surface. The magnitude of the resultant hydrostatic force on the liquid side can be calculated by integrating distributed hydrostatic force over the surface:

since

where is the first moment of the area about . Hence

where is the depth of the centroid of the submerged surface and is the pressure at the centroid of the surface. Though the resultant force proportional to the pressure at the centroid, the line of action does not pass through the centroid.

Line of action of the Resultant Force[edit]

The location where the resultant force acts can be found by using the first moments of the forces about and .

The first moment about is:

and the first moment about is:

Using

where

Direction of the Resultant Force[edit]

The direction of the resultant force is opposite to the surface normal. If a surface is composed of several subsurfaces having different surface normal, either those surfaces should be treated individually or the vector sum of the surface forces on each surface will give the direction of the resultant force.

Hydrostatic Forces on Curved Submerged Surfaces[edit]

Reference[edit]