FLUID MECHANICS
Fluid mechanics is the study of the behaviour of fluids (liquids and gases). These materials will look at the various aspects of fluid behaviour.
PRESSURE DUE TO DEPTH IN A LIQUID
If we consider a point below the surface of a fluid, then we can see that there must be sufficient pressure in the fluid to support the column of fluid above that point.
The weight (W) of the column of fluid is given by the equation
W = mg
Where m is the mass of fluid and g is the acceleration due to gravity.
Also, m = V x
Where V = Volume and = Density
Therefore W = V x x g
The volume of the column of fluid is given by
V = A x h
Therefore: W = A x h x x g
This weight must be supported by the pressure of the liquid, providing the force F
F = P x A
Where P = Pressure of fluid
Therefore if W = F
A x h x x g = x A
Therefore h x x g = P (cancelling A)
Or
We can now check that the units of the equation balance
N=kgxmxm
m2m3s2
Substitute 1N = 1 Kgm
s2
Kgm=Kgm2cancelling Kg=Kg
S2m2S2m2ms2ms2
We can see that it is important that we use values of, g and h in base units if we are to calculate pressure in N/m2
CONTINUITY EQUATION
.
If fluid flows along a pipe then the mass flow rate (m) at any point (1) is equal to the mass flow rate at any other point (2), assuming there are no leakages.
..
Therefore m, = m2
If the fluid has constant density then as
.
m = Q x
Where Q = Volumetric flow rate (m3/s)
= density of fluid (Kg/m3)
Q1 = Q2
And
Where A = cross sectioned area
And C = velocity of fluid flow
.
If the fluid does not have a constant density (gas) then we have m = m2
Q1 x 1 = Q2 x 2
And as Q = A x C
Bernoulli’s Equation
Bernoulli’s equation states that the total energy at one point (1) of flow in a pipe is equal to the total energy at another point (2) in the pipe providing that:
The flow is steady
The points lie on a streamline
There is no friction
Pressure Energy1 + Kinetic Energy1 + Potential Energy1
= PressureEnergy2 + Potential Energy2
P1V1 + mC12 + mgh1 = P2V2 +mC22 + mgh2
22
Where V = Volumetric flow rate
Dividing through by mass gives
P1 V1+ C12 + gh1 = P2 V2 + C22+gh2
m 2m2
As m =
V
P1 + C12 + gh1 = P2+ C22 + gh2
1222
Multiply by
P1 + C12+ 1gh1 = P2 + C22 + gh2
22
Or we can state
P + C2 + gh = Constant
2
If we consider a constant density fluid flowing horizontally (i.e. neglecting potential energy) then we have:
P1 + C12= P2 + C22
22
Also from continuity, we have
A1 C1 = A2 C2
From equation 2 we can see that as the area of flow decreases, the velocity increases.
A1 C1 = A2 C2
Also, from equation 1 we can see that the pressure decreases.
P1 + C12 = P2 + C22
22
Therefore, when fluid passes over a stationary body forcing the streamlines to get closer together, the velocity must increase (as the area decreases), and therefore the pressure decreases. This is how an airfoil works, the flow over the top surface is faster than the flow over the bottom surface, therefore there is low pressure on the top and an upward force is produced on the airfoil (lift)
The Bernoulli equation allows us to determine how energy transfers take place between elevation, velocity head and pressure head. We can examine piping systems and determine the variation of fluid properties and the balance of energy. Sometimes the changes in the energy levels are confusing and seem to contradict common sense; students can often find the outcomes of the Bernoulli equation confusing. For example, if an incompressible fluid is flowing along a horizontal pipe which expands in diameter, then, as the area increases the velocity must decrease (continuity, Q = CA). As there is no change in potential head (horizontal pipe), the loss in velocity head must result in a gain in pressure head (conservation of energy or head). Therefore, as the velocity decreases the pressure increases. This may seem counter-intuitive (surely fluid cannot flow in the direction of increasing pressure), however, we must remember that the fluid has inertia and the increase in pressure does have the effect of decelerating the fluid (F=ma), in the same way that a car moves forward when a brake (force in the opposite direction to motion) is applied.
There are many fluid problems to which Bernoulli can be applied, it can be applied to problems in which more than one flow may enter or leave the system at the same time, or to series and parallel piping systems.
Fluid/docs/Science/DC/SL