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Physics

Course Code:Lesson 7

Fluid MechanicsHow Motion Takes Place in a Fluid

Contents

Topics

• General Characteristics of Fluid Motion

• Bernoulli’s Equation

• Fluid flow can be steady or nonsteady. When the fluid velocity v at any given point is constant in time, the fluid motion is said to be steady. That is, at any given point in a steady flow the velocity of each passing fluid particle is always the same. These conditions can be achieved at low flow speeds; a gently flowing stream is an example.

• Fluid flow can be rotational or irrotational. If the element of fluid at each point has no net angular velocity about that that point, the fluid is irrotational.

• Fluid flow can be compressible or incompressible

General Characteristics of Fluid Flow

• Fluid flow can be viscous or nonviscous. Viscosity in fluid motion is the analogue of friction in the motion of solids

We shall confine our discussion of fluid dynamics to steady, irrotational, incompressible, nonviscous flow.

In steady flow, the velocity v at a given point is constant in time. If we consider the point p within the fluid, since v at p does not change in time, every particle arriving at p will pass on with the same speed in the same direction. The same is true about the point Q and R. Hence if we trace out the path of the particle, that curve will be the path of every particle arriving at p. This curve is called a streamline.


P Q

R

If we assume a steady flow and select a finite number of streamlines to form a bundle, like the streamline pattern shown below, the tubular region is called a tube of flow.


A2,V2

A1,V1

P

Q

• Let the speed be v1 for fluid particle at p and v2 for fluid particles at Q. Let A1 and A2 be the cross-sectional areas of the tubes perpendicular to the streamlines at the points p and Q respectively. At the time interval t a fluid element travels approximately the distance v t. Then the mass of fluid m1 crossing A1 in the time interval t is approximately,

Or the mass flux is approximately


tvAm 1111

tm 1

111 vA

• The mass flux at p is

• The mass flux at Q is

Where 1 and 2 are the fluid densities at p and Q respectively. Because no fluid can leave through the walls of the tube and there are no sources or sinks where fluids can be created or destroyed in the tube. The mass crossing each section of the tube per unit time must be the same. In particular the mass flux at p must equal that at Q.

vA = constant


111 vA

222 vA

222111 vAvA

If the fluid is incompressible, as we shall henceforth assume, then

or Av = constant.The product Av gives the volume flux or flow rate in


2211 vAvA

sm3

According to the continuity equation, the speed of fluid flow can vary along the paths of the fluid. The pressure can also vary; it depends on height as in the static situation and it also depends on the speed of flow.

We shall derive an important relationship called Bernoulli’s equation that relates the pressure, flow speed, and height

Bernoulli’s Equation

For flow of an ideal incompressible fluid.

The equation is an ideal tool for analysing plumbing systems, hydroelectric generating stations and the flight of aeroplanes.

The dependence of pressure on speed follows from the continuity equation. When an incompressible fluid flows along a flow tube, with varying cross section, its speed must change, and so an element of fluid must have an acceleration. If the tube is horizontal, the force that causes this acceleration has to be applied by the surrounding fluid. This means that the pressure must be different in regions of different cross section; if it were the same every where, the net force on every fluid element must be zero. When a horizontal flow tube narrows, and a fluid element speeds up, it must be moving towards a region of lower pressure in order to have a net forward force to accelerate it.


Let us compute the work done by a fluid element during a time dt. The pressure at the two ends are p1 and p2; the force on the cross section at a is p1A1 and the force at c is p2A2.

The net work dW done on the element by the surrounding fluid during this displacement is therefore

The second term has a negative sign because the force at c opposes the displacement of the fluid.

The work done would be equal to the change in the total mechanical energy (kinetic + gravitational potential energy) associated with the fluid element.


dvppdsApdsApdW 21222111

The mechanical energy between b and c does not change.

At the beginning of dt, the fluid between a and b has

volume A1ds1, mass A1ds1, and kinetic energy,

At the end of dt, the fluid between c and d has kinetic energy

The net change in kinetic energy dK during time dt is


.21 2

111 vdsA

22222

1 vdsA

21222

1 vvdvdK

For the change in gravitational potential energy, at the beginning of dt, the potential energy for the mass between a and b is

For the mass between c and d,The net change in potential energy dU during dt is

Using the energy equation dW = dK + dU we obtain


22

11

dvgydmgydvgydmgy

12 yydvgdU

)()(21

)()(21)(

1221

2221

1221

2221

yygvvpp

yydvgvvdvdvpp

This is Bernoulli's equation. It states that the work done on a unit volume of fluid by the surrounding fluid is equal to sum of the changes in kinetic and potential energies per unit volume that occur during the flow.

The first term on the right is the pressure difference associated with the change of speed of the fluid. The second term on the right is the additional pressure difference caused by the weight of the fluid and the difference in the elevation of the two ends. In a more convenient form,we can state the equation as


2222

2111 2

121 vgypvgyp

The subscripts 1 and 2 refer to any two points along the flow tube so we can also write;

Note that when the fluid is not moving, v1 = v2 = 0 and the equation reduces to the pressure relation we met before.


constvgyp 2

21

APPLICATIONS OF BERNOULLIS EQUATIONS

Bernoulli's equation is useful for describing a variety of phenomena. First we observe that if a fluid is at rest so that v = 0, P + gh = constant.

This is just the hydrostatic equation between pressure and height.

For flow in a horizontal, constant height tube,

Kv 2

21

P1,v1,

P2, v2

A1

A2

Thus pressure must be lower in a region in which a fluid is moving faster.

Consider the flowmeter below:- A device designed to measure the speed of fluids in a pipe

Applying Bernoulli's equation,

212221

222

211

21

21

21

vvpp

vpvp

121

2

2

12121

2

112

AAvpp

AAvv

Since

The pressure difference = gh, so we can solve for v1

EXAMPLE 2

Water flowing from a tank

A1P1=PA

h

A2

Consider points just outside and inside the hole and apply Bernoulli's equation

22

21

21 vpughp AfluidfluidA

22

21

21 vugh fff

22

22

221

21

vugh

vugh

ghuv 22 hence

If A1 A2

vAAu1

2 v

U2 term could be ignored

ghv 2

ASSIGNMENT

• GRP 4: WE CAN HARNESS ENERGY OF THE EARTH – GEOTHERMAL ALTERNATIVE! (15 mins)

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