FLUID MECHANICS 2

MINI PROJECT 1

R.PANJAABEGESAN

13355

Q1:How fluids can be distinguished upon solids?

SOLID FLUID

Resists forces up to yield point

The intermolecular forces are

higher so they are rigid and also

have almost constant shape and

volume which can only be

changed if sufficient

pressure/stress is applied

Single state of matter

The bonding between atom in

solid are having more strength

Distance between atoms also less

causing the energy required to

evaporate any solid is greater than

liquid

Substances that flow easily which

causes it to be unable to resist

forces, whatever may its

magnitude be

The intermolecular forces are

weak as in fluid causing them to

take shape of the container which

the fluid(liquid) occupies, but

always the upper surface of the

liquid will always be horizontal

As for gases, the intermolecular

forces are much weaker causing

them to move in any direction so

it does not have fixed volume

A group of states which consists

of liquid, gas, vapour and plasma

Atom distance is more and

strength also less

Q2:How fluids can be classified?

A fluid can be classified as either a liquid or a gas, or even a mixture of both. A gas,

for example, is far more compressible than a liquid; a gas does not possess a free-

surface that divides it from its environment; a gas expands and occupies the entire

space of its container; most gases have no wetting characteristics, while most liquids

are wet.

In gases, the average spacing between simple molecules at normal pressures and

temperatures is of the order 10d0, whereas for liquids it is of the order d0. the

molecules are thus far apart for gases so that only weak cohesive forces act between

molecules. In liquids, each molecule is within a strong force field and the molecules

are packed as close together as the repulsive forces allow. The primary property of the

liquid and solid phases of matter is that they are condensed phases when the

molecules are within strong cohesive forces.

The primary properties of the liquid and gaseous phases are the fluidity and the ability

to change shape freely.

A fluid may first exist as a gas for some pressure and temperature, then as a liquid at a

different pressure or temperature; this, of course, reflects changes in the

intermolecular force and molecular spacing. We can understand this shift from gas to

liquid by considering a gas that is being compressed isothermally. The kinetic energy

of the molecule remains invariant

As the distance between neighbouring molecules decreases. When the specific

volume of the gas becomes so small that the spacing between molecules is only a few

times their distance, attractive forces become significant.

A fluid can be classified as either a liquid or a gas, and can be divided into seven

categorizes:

1. Fluids sensitive to changes in temperature and/or pressure.

2. Fluids that require two mechanical seals.

3. Non lubricating liquids, gases and solids.

4. Slurries, classified as solids in liquid . The solids may or may not be abrasive.

5. Liquids sensitive to agitation.

6. Liquids that react with each other to form a solid.

7. Lubricating liquids.

Some other comparison of classes of fluids

Newtonian / Non-Newtonian Fluids

Even among fluids which are accepted as fluids there can be wide differences in

behaviour under stress. Fluids obeying Newton's law where the value of m is constant

are known as Newtonian fluids. If m is constant the shear stress is linearly dependent

on velocity gradient. This is true for most common fluids.

Fluids in which the value of m is not constant are known as non-Newtonian fluids.

There are several categories of these, and they are outlined briefly below.

These categories are based on the relationship between shear stress and the velocity

gradient (rate of shear strain) in the fluid. These relationships can be seen in the graph

below for several categories

Shear stress vs. Rate of shear strain du/dy

Each of these lines can be represented by the equation

where A, B and n are constants. For Newtonian fluids A = 0, B = m and n = 1.

Below are brief description of the physical properties of the several categories:

o Plastic: Shear stress must reach a certain minimum before flow commences.

o Bingham plastic: As with the plastic above a minimum shear stress must be achieved.

With this classification n = 1. An example is sewage sludge.

o Pseudo-plastic: No minimum shear stress necessary and the viscosity decreases with

rate of shear, e.g. colloidial substances like clay, milk and cement.

o Dilatant substances; Viscosity increases with rate of shear e.g. quicksand.

o Thixotropic substances: Viscosity decreases with length of time shear force is applied

e.g. thixotropic jelly paints.

o Rheopectic substances: Viscosity increases with length of time shear force is applied

o Viscoelastic materials: Similar to Newtonian but if there is a sudden large change in

shear they behave like plastic.

There is also one more - which is not real, it does not exist - known as the ideal fluid.

This is a fluid which is assumed to have no viscosity. This is a useful concept when

theoretical solutions are being considered - it does help achieve some practically

useful solutions.

Compressible versus Incompressible Fluids

Compressible fluids are fluids whose specific volume is a function of pressure.

Compressibility is not related to a fluid's ability to change shape, as is sometimes

erroneously assumed. Conversely, an incompressible fluid is a fluid whose density if

not changed by external forces acting on the fluid.

Hydrodynamic is the study of the behaviour of incompressible fluids, whereas gas

dynamics is the study of compressible fluids. The familiar Mach number M indicates

the importance of the compressibility of gases in the dynamics of a fluid flow. The

Mach number is defined as the ratio of the velocity of the fluid to the velocity of

sound. Compressible fluids are subdivided into subsonic, transonic, supersonic and

hypersonic compressible flows, meaning speeds less than, equal to, or greater than the

speed of sound.

Q3:How fluids flow can be classified?

Laminar flow- where the stream lines are parallel so the flow is smooth

(Reynolds Number, Re <2300)

Transient Flow- The stream lines start to become disturbed so the flow becomes less

smooth (2300<Re<4000)

Turbulent flow- The stream lines cross each other regularly so the flow is rough

(Re>4000)

By knowing the pipe roughness and calculating the Re, a friction factor can be taken

from the Moody Diagram which enables the reduced velocity (Reduced by the

relationship between pipe roughness and Re) to be determined.

The Formula is :

RE = ρ * V * L / μ

Steady versus Unsteady Fluid Flows

A steady fluid flow has properties and variables that are independent of real time.

Mathematically, this can be stated as

( )=0

Equation above states that none of the dependent variables change with time at any

point in the flow. In unsteady flow, however, the fluid exhibits variations at a fixed

point in space with respect to time. Thus, we shall have to consider whether the flow

through a nozzle is steady or unsteady or if the flow past a wing is steady or unsteady.

Consider steady and unsteady flows for two different real fluids. Let one fluid be

laminar (well-behaved), and the other turbulent (random). A turbulent flow can be

viewed as steady provided that its time average velocity is constant at a specific point

in the flow. We are primarily concerned with steady fluid flow, although unsteady

motion are also treated.

One, Two, and Three-Dimensional Flows

A one-dimensional flow has spatial variations in one direction only, such a flow is

also said to be uniform at every cross section normal to the main direction of flow.

Only one independent space variable is needed to describe the variation. Usually it is

designate x to be that variable. Thus f = f(x) is one-dimensional.

Example: steady ideal fluid flow through a graduated tunnel.

Similarly, a two-dimensional flow is one in which spatial variations exist in two-

directions, or variations exist along some planar surface. Two Cartesian independent

space variables are needed to describe the variation. Thus f = f(x; y) is two-

dimensional.

Example: steady flow through a pipe.

A three dimensional flow has spatial variations everywhere in the flow field. All

turbulent fluid flows are three-dimensional. Thus f = f(x; y; z) would be three-

dimensional.

Example: steady flow rotating in a fixed wall.

Rotational versus Irrotational Flow

A flow is irrotational if it exhibits no rate of angular deformation of any fluid particle.

The converse holds for rotational flows. Irrotational flow means that the fluid masses

may deform but cannot rotate. To recognize rotation we need to consider finite

(though small) fluid masses called fluid parcels. Fluid particles, which are point

masses, have no detectable rotation. To detect rotation, a coordinate system is

attached to the fluid parcel, and if, the coordinate system rotates as the parcel moves

along a path, then we say the flow is rotational. A fluid flow that is irrotational is

defined as a potential flow.

Q4:How fluids motion can be governed?

Navier-Stokes Equations

The motion of a non-turbulent, Newtonian fluid is governed by the Navier-

Stokes equations. The equation can be used to model turbulent flow, where the

fluid parameters are interpreted as time-averaged values.

The Bernoulli Equation

The Bernoulli Equation - A statement of the conservation of energy in a form

useful for solving problems involving fluids. For a non-viscous,

incompressible fluid in steady flow, the sum of pressure, potential and kinetic

energies per unit volume is constant at any point.

Conservation laws

The conservation laws states that particular measurable properties of an

isolated physical system does not change as the system evolves.

Conservation of energy (including mass)

Fluid Mechanics and Conservation of Mass - The law of conservation of mass

states that mass can neither be created or destroyed.

The Continuity Equation - The Continuity Equation is a statement that mass is

conserved.

Darcy-Weisbach Equation

Pressure Loss and Head Loss due to Friction in Ducts and Tubes - Major loss -

head loss or pressure loss - due to friction in pipes and ducts.

Head loss can be calculated with

where

hf is the head loss due to friction;

L is the length of the pipe;

D is the hydraulic diameter of the pipe (for a pipe of circular section, this

equals the internal diameter of the pipe);

V is the average velocity of the fluid flow, equal to the volumetric flow

rate per unit cross-sectional wetted area;

g is the local acceleration due to gravity;

f is a dimensionless coefficient called the Darcy friction factor. It can be found

from a Moody diagram or more precisely by solving Colebrook equation.

where the pressure loss due to friction Δp is a function of:

the ratio of the length to diameter of the pipe, L/D;

the density of the fluid, ρ;

the mean velocity of the flow, V, as defined above;

a (dimensionless) coefficient of laminar, or turbulent flow, f.

Euler Equations

In fluid dynamics, the Euler equations govern the motion of a compressible,

inviscid fluid. They correspond to the Navier-Stokes equations with zero

viscosity, although they are usually written in the form shown here because

this emphasizes the fact that they directly represent conservation of mass,

momentum, and energy.

Laplace's Equation

The Laplace Equations describes the behavior of gravitational, electric, and

fluid potentials.

Ideal Gas Law

The Ideal Gas Law - For a perfect or ideal gas the change in density is directly

related to the change in temperature and pressure as expressed in the Ideal Gas

Law.

Properties of Gas Mixtures - Special care must be taken for gas mixtures when

using the ideal gas law, calculating the mass, the individual gas constant or the

density.

The Individual and Universal Gas Constant - The Individual and Universal

Gas Constant is common in fluid mechanics and thermodynamics.

Mechanical Energy Equation

The Mechanical Energy Equation - The mechanical energy equation in:-

Terms of Energy per Unit Mass

pin / ρ + vin2 / 2 + g hin + wshaft = pout / ρ + vout

2 / 2 + g hout + wloss

where

p = static pressure

ρ = density

v = flow velocity

g = acceleration of gravity

h = elevation height

wshaft = net shaft energy inn per unit mass for a pump, fan or similar

wloss = loss due to friction

Terms of Energy per Unit Volume

pin + ρ vin2 / 2 + γ hin + ρ wshaft = pout + ρ vout

2 / 2 + γ hout + wloss

where

γ = ρ g = specific weight

Terms of Energy per Unit Weight involves Heads.

pin / γ + vin2 / 2 g + hin + hshaft = pout / γ + vout

2 / 2 g + hout + hloss

where

γ = ρ g = specific weight

hshaft = wshaft / g = net shaft energy head inn per unit mass for a pump, fan or similar

hloss = wloss / g = loss head due to friction

Pressure

Static Pressure and Pressure Head in a Fluid - Pressure and pressure head in a

static fluid.

Static Pressure

p2 - p1 = - γ (z2 - z1)

where

p2 = pressure at level 2

p1 = pressure at level 1

z2 = level 2

z1 = level 1

Pressure Head

h = (p2 - p1) / γ