Basic Fluid Mechanics

Summary of introductory concepts

By Engr Sarfaraz Khan Turk

Lecture no 1 to 10

Introduction What is Fluid Mechanics? Fluid mechanics deals with the study of all fluids

under static and dynamic situations. Fluid mechanics is a branch of continuous mechanics which deals with a relationship between forces, motions, and statically conditions in a continuous material. This study area deals with many and diversified problems such as surface tension, fluid statics, flow in enclose bodies, or flow round bodies (solid or otherwise), flow stability, etc.

HistoryFaces of Fluid Mechanics

Archimedes(C. 287-212

BC)

Newton(1642-1727)

Leibniz(1646-1716)

Euler(1707-1783)

Navier(1785-1836)

Stokes(1819-1903)

Reynolds(1842-1912)

Prandtl(1875-1953)

Bernoulli(1667-1748)

Taylor(1886-1975)

Introduction contd In fact, almost any action a person is doing

involves some kind of a fluid mechanics problem. Furthermore, the boundary between the solid mechanics and fluid mechanics is some kind of gray shed and not a sharp distinction for the complex relationships between the different branches which only part of it should be drawn in the same time.). The fluid mechanics study involve many fields that have no clear boundaries between them.

Introduction contd

Fluids omnipresentWeather & climateVehicles: automobiles, trains, ships, and

planes, etc.EnvironmentPhysiology and medicineSports & recreationMany other examples!

Introduction

Field of Fluid Mechanics can be divided into 3 branches:

Fluid Statics: mechanics of fluids at rest Kinematics: deals with velocities and

streamlines w/o considering forces or energy Fluid Dynamics: deals with the relations

between velocities and accelerations and forces exerted by or upon fluids in motion

Streamlines

A streamline is a line that is tangential to the instantaneous velocity direction (velocity is a vector that has a direction and a magnitude)

Instantaneous streamlines in flow around a cylinder

Intro…con’t Mechanics of fluids is extremely important in many

areas of engineering and science. Examples are:

Biomechanics & Bio-fluid mechanics. Blood flow through arteries Flow of cerebral fluid

Meteorology and Ocean Engineering. Movements of air currents and water currents

Chemical Engineering Design of chemical processing equipment

Intro…con’t

Mechanical EngineeringDesign of pumps, turbines, air-conditioning

equipment, pollution-control equipment, etc. Civil Engineering

Transport of river sedimentsPollution of air and waterDesign of piping systemsFlood control systems

Intro…con’t

Fluids essential to life Human body 65% water Earth’s surface is 2/3 water Atmosphere extends 17km above the earth’s surface

History shaped by fluid mechanics Geomorphology Human migration and civilization Modern scientific and mathematical theories and methods Warfare

Affects every part of our lives

Dimensions and Units

Before going into details of fluid mechanics, we stress importance of units

In U.S, two primary sets of units are used: 1. SI (Systeme International) units2. English units

Unit Table

Quantity SI Unit English Unit

Length (L) Meter (m) Foot (ft)

Mass (m) Kilogram (kg) Slug (slug) = lb*sec2/ft

Time (T) Second (s) Second (sec)

Temperature ( ) Celcius (oC) Farenheit (oF)

Force Newton (N)=kg*m/s2

Pound (lb)

Dimensions and Units con’t

1 Newton – Force required to accelerate a 1 kg of mass to 1 m/s2

1 slug – is the mass that accelerates at 1 ft/s2 when acted upon by a force of 1 lb

To remember units of a Newton use F=ma (Newton’s 2nd Law) [F] = [m][a]= kg*m/s2 = N

More on Dimensions

To remember units of a slug also use F=ma => m = F / a

[m] = [F] / [a] = lb / (ft / sec2) = lb*sec2 / ft

1 lb is the force of gravity acting on (or weight of ) a platinum standard whose mass is 0.45359243 kg

Weight and Newton’s Law of Gravitation

Weight Gravitational attraction force between two bodies

Newton’s Law of Gravitation F = G m1m2/ r2

G - universal constant of gravitation m1, m2 - mass of body 1 and body 2, respectively r - distance between centers of the two masses F - force of attraction

Weight m2 - mass of an object on earth’s surface

m1 - mass of earth r - distance between center of two masses r1 - radius of earth

r2 - radius of mass on earth’s surface

r2 << r1, therefore r = r1+r2 ~ r1

Thus, F = m2 * (G * m1 / r2)

Weight Weight (W) of object (with mass m2) on surface of earth

(with mass m1) is defined as

W = m2g ; g =(Gm1/r2) gravitational acceleration

g = 9.31 m/s2 in SI units g = 32.2 ft/sec2 in English units

See back of front cover of textbook for conversion tables between SI and English units

Properties of Fluids - Preliminaries

Consider a force, , acting on a 2D region of area A sitting on x-y plane

Cartesian components:

F

x

y

z

F

F F i F j F kx y z ( ) ( ) ( )

A

Cartesian components

i

F z

F x

j

k

- Unit vector in x-direction

- Unit vector in y-direction

- Unit vector in z-direction

- Magnitude of in x-direction (tangent to surface)F

F y - Magnitude of in y-direction (tangent to surface)

- Magnitude of in z-direction (normal to surface)

FF

F

Ashear stressx ( )

- For simplicity, let F y 0

• Shear stress and pressure

pF

Anorm al stress pressurez ( ( ))

• Shear stress and pressure at a point

F

Ax

Alim 0

pF

Az

A

lim 0

[ ]

[ ]( )

F

A

N

mPa Pasca l in SI un its 2

• Units of stress (shear stress and pressure)

[ ]

[ ]( )

F

A

lb

inpsi pounds per square inch in Eng lish un its 2

[ ]

[ ]( )

F

A

lb

ftpounds per square foo t Eng lish un its 2

Properties of Fluids Con’t Fluids are either liquids or gases Liquid: A state of matter in which the molecules

are relatively free to change their positions with respect to each other but restricted by cohesive forces so as to maintain a relatively fixed volume

Gas: a state of matter in which the molecules are practically unrestricted by cohesive forces. A gas has neither definite shape nor volume.

More on properties of fluids

Fluids considered in this course move under the action of a shear stress, no matter how small that shear stress may be (unlike solids)

Continuum view of Fluids

Convenient to assume fluids are continuously distributed throughout the region of interest. That is, the fluid is treated as a continuum

This continuum model allows us to not have to deal with molecular interactions directly. We will account for such interactions indirectly via viscosity

A good way to determine if the continuum model is acceptable is to compare a characteristic length of the flow region with the mean free path of molecules,

If , continuum model is valid

( )L

L

Mean free path ( ) – Average distance a molecule travels before it collides with another molecule.

Density and specific weight

Density (mass per unit volume): m

V

[ ][ ]

[ ]( )

m

V

kg

min SI un its3Units of density:

Specific weight (weight per unit volume):

[ ] [ ][ ] ( ) gkg

m

m

s

N

min SI un its3 2 3

Units of specific weight:

g

Viscosity ( ) Viscosity can be thought as the internal stickiness of a fluid Representative of internal friction in fluids Internal friction forces in flowing fluids result from cohesion

and momentum interchange between molecules. Viscosity of a fluid depends on temperature:

In liquids, viscosity decreases with increasing temperature (i.e. cohesion decreases with increasing temperature)

In gases, viscosity increases with increasing temperature (i.e. molecular interchange between layers increases with temperature setting up strong internal shear)

More on Viscosity

Viscosity is important, for example, in determining amount of fluids that can be

transported in a pipeline during a specific period of time

determining energy losses associated with transport of fluids in ducts, channels and pipes

No slip condition

Because of viscosity, at boundaries (walls) particles of fluid adhere to the walls, and so the fluid velocity is zero relative to the wall

Viscosity and associated shear stress may be explained via the following: flow between no-slip parallel plates.

Flow between no-slip parallel plates -each plate has area A

Moving plate

Fixed plate

F U,

Y

x

z

y

F F i

U U i

Force induces velocity on top plate. At top plate flow velocity is F

U

U

At bottom plate velocity is 0

The velocity induced by moving top plate can be sketched as follows:

y

u y( )

Y

U

u yU

Yy( )

The velocity induced by top plate is expressed as follows:

u y( ) 0 0

u y Y U( )

For a large class of fluids, empirically, FAU

Y

More specifically, FAU

Y ; is coeffic ien t o f vis itycos

Shear stress induced by is F F

A

U

Y

From previous slide, note that du

dy

U

Y

Thus, shear stress is du

dy

In general we may use previous expression to find shear stress at a pointinside a moving fluid. Note that if fluid is at rest this stress is zero because

du

dy 0

Newton’s equation of viscosity

du

dy

- viscosity (coeff. of viscosity)

Fixed no-slip plate

u y velocity pro file( ) ( )

Shear stress due to viscosity at a point:

fluid surface

e.g.: wind-driven flow in ocean

- kinematic viscosity

y

As engineers, Newton’s Law of Viscosity is very useful to us as we can use it to evaluate the shear stress (and ultimately the shear force) exerted by a moving fluid onto the fluid’s boundaries.

a t boundarydu

dya t boundary

Note is direction normal to the boundaryy

Viscometer

Coefficient of viscosity can be measured empirically using a viscometer

Example: Flow between two concentric cylinders (viscometer) of length

Moving fluid

Fixed outer cylinder

Rotating inner cylinder

,T

r

R

h

x

z

y

r - radial coordinate

O

L

Inner cylinder is acted upon by a torque, , causing it to rotate about point at a constant angular velocity and causing fluid to flow. Find an expression for

T T k

O

Because is constant, is balanced by a resistive torque exerted by the moving fluid onto inner cylinder

T T k

T T kres res ( )

The resistive torque comes from the resistive stress exerted by the moving fluid onto the inner cylinder. This stress on the inner cylinder leads to an overall resistive force , which induces the resistive torque about point

x

y

z

O

T T res

T

res

T

F res

res

R T

T res

F res

T

Compressibility

• All fluids compress if pressure increases resulting in an increase in density

• Compressibility is the change in volume due to a

change in pressure where V is volume and p is pressure.

• A good measure of compressibility is the bulk modulus (It is inversely proportional to compressibility often denoted K sometimes B).

Edp

d

1

( )specific vo lum e

p is pressure

Material β (m²/N or Pa−1)

Plastic clay 2×10–6 – 2.6×10–7

Stiff clay 2.6×10–7 – 1.3×10–7

Medium-hard clay 1.3×10–7 – 6.9×10–8

Loose sand 1×10–7 – 5.2×10–8

Dense sand 2×10–8 – 1.3×10–8

Dense, sandy gravel 1×10–8 – 5.2×10–9

Rock, fissured 6.9×10–10 – 3.3×10–10

Rock, sound <3.3×10–10

Water at 25 °C (undrained) 4.6×10–10

Vertical, drained compressibility's

Compressibility• From previous expression we may write

• For water at 15 psia and 68 degrees Farenheit,

• From above expression, increasing pressure by 1000 psi will compress the water by only 1/320 (0.3%) of its original volume

( ) ( )

fina l in itia l

in itia l

fina l in itia lp p

E

E psi 320 000,

• Thus, water may be treated as incompressible (density is constant) ( )

• In reality, no fluid is incompressible, but this is a good approximation for certain fluids The degree of compressibility of a fluid has strong implications for its dynamics.

Vapor pressure of liquids• All liquids tend to evaporate when placed in a closed container

• Vaporization will terminate when equilibrium is reached between the liquid and gaseous states of the substance in the container

i.e. # of molecules escaping liquid surface = # of incoming molecules

• Under this equilibrium we call the call vapor pressure the saturation pressure

• At any given temperature, if pressure on liquid surface falls below the the saturation pressure, rapid evaporation occurs (i.e. boiling)

• For a given temperature, the saturation pressure is the boiling pressure

Properties of Liquids: Surface Tensionhttp://www.visionlearning.com/library/module_viewer.php?mid=57Water Strider Video

Surface TensionSurface Tension-a force that tends to pull adjacent parts

of a liquid’s surface together, thereby decreasing surface area to the smallest possible size.

~The higher the attraction forces (intermolecular forces), the higher the surface tension. Surface tension causes liquid droplets to take a spherical shape.

The surface of any liquid behaves as if it was a stretched membrane. This phenomenon is known as surface tension

Surface tension is caused by intermolecular forces at the liquid’s interface with a gas or a solid.

more bugs that think they’re all that and a bag of chips: the Water Strider

Surface Tension Surface tension depends on the nature of the

liquid, the surrounding media and temperature. Liquids that have strong intermolecular forces

will have higher values of surface tension than liquids that have weak intermolecular forces.

A. Beading of rain water on a waxy surface, such as a leaf. Water adheres weakly to wax and strongly to itself, so water clusters into drops. Surface tension gives them their near-spherical shape, because a sphere has the smallest possible surface area to volume ratio.

B. Formation of drops occurs when a mass of liquid is stretched. The animation shows water adhering to the faucet gaining mass until it is stretched to a point where the surface tension can no longer bind it to the faucet. It then separates and surface tension forms the drop into a sphere. If a stream of water was running from the faucet, the stream would break up into drops during its fall. Gravity stretches the stream, then surface tension pinches it into spheres.

C. Flotation of objects denser than water occurs when the object is nonwettable and its weight is small enough to be borne by the forces arising from surface tension. For example, water striders use surface tension to walk on the surface of a pond. The surface of the water behaves like an elastic film: the insect's feet cause indentations in the water's surface, increasing its surface area.

D. Separation of oil and water (in this case, water and liquid wax) is caused by a tension in the surface between dissimilar liquids. This type of surface tension is called "interface tension", but its chemistry is the same.

E. Tears of wine is the formation of drops and rivulets on the side of a glass containing an alcoholic beverage. Its cause is a complex interaction between the differing surface tensions of water and ethanol; it is induced by a combination of surface tension modification of water by ethanol together with ethanol evaporating faster than water.

Even a piece of steel can do this trick if it is small (steel ~ 8x water)

4 H2O molecules separated

in space from each other

have partial + and – charges

what would they do???

but what’s surface tension, really?

4 H2O molecules they clump

together + and –

charges snuggle up close

potential energy of system has dropped

Surface Tension water in bulk has

many binding partners

water at surface has less, has exposed charges left over

potential energy of water at surface is higher

deforming droplet to increase surface area takes work

Contact Angles

here’s a droplet on a surface -

Contact Angle

here’s a slice of it –

tangent to droplet edge is “contact angle”

why is theta theta?

Contact Angle

balance of forces

surface tension pulls up

gravity & adhesion pulls down

what are the other two?

Remember this?

water at surface has less binding partners

energy at surface is higher

What if -

what if the circles are aluminum atoms in a solid?

what if the space above it is liquid ethanol?

Contact Angle

F = dE/dX

surface/air & surface/water interfaces also have “surface tension”, in ergs/cm2

moving water edge back and forth incurs energy costs/profits

but units of F are energy/distance, not area?! what’s the deal?

Obtuse contact Angles

hydrophobic surface

“gravity & adhesion” is now “gravity & repulsion”

if no gravity, drop leaves

http://citt.ufl.edu/Marcela/Sepulveda/html/en_tension.htm