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Technical English - I 12th week

Fluid Mechanics

FLUID

STATICS

FLUID

DYNAMICS

FLUID

MECHANICS

Fluid Statics or Hydrostatics is the study of fluids at rest. The main equation required for this is Newton's second

law for non-accelerating bodies, i.e. ΣF=0

Fluid Dynamics is the study of fluids in motion. The main equation required for this is Newton's second law for

accelerating bodies, i.e. ΣF=ma

fluid / mechanics / statics / hydrostatics / rest / motion / acceleration

Density (): mass per unit volume (kgsn2/m4)

Specific weight (): weight per unit volume or specific gravity (kg/m3)

= g

Specific volume (v= 1 /): volume per unit mass

Relative density (s): the ratio of the density of the fluid to the density of water at +4 oC

mass / weight / density / specific gravity / specific volume / relative density viscosity / flowability / fluidity / resist / resistance / high flow / thick / thin / viscous

(From https://www.drive2.ru/l/4881470/)

Viscosity: The resistance of a fluid to flowing and movement A thick fluid has high viscosity (very viscous) A thin fluid has low viscosity

temperature / raise / increase / cold / hot / high flow / liquid / gas

(From http://syntheticperformanceoil.com/spo/motor_oil_viscosity.php)

Most liquids become less viscous as the temperature is raised.

The variation of viscosity with temperature

However, the viscosity of a gas will increase with temperature.

Stationary Plate

Moving Plate constant force F constant speed U H

viscositykinematic

viscositydynamic is stress,shear is where,dy

du

resist / shearing motion / moving plate / stationary / constant / shear stress

Due to interaction between fluid molecules, the fluid flow will resist a shearing motion. The viscosity is a measure of this resistance

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Surface Tension: Surface tension is the tendency of the surface of a liquid to behave like a stretched elastic membrane.

There is a natural tendency for liquids to minimize their surface area.

For this reason, drops of liquid tend to take a spherical shape in order to minimize surface area.

surface tension / tendency / stretched / elastic / membrane / spherical shape

Capillarity: Capillarity, or capillary motion, is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity.

IMPORTANCE OF CAPILLARITY IN CIVIL ENGINEERING. Moisture increases due to capillary rise. Water is a key factor in most cases of deterioration of structural elements, such as walls and columns. If the exterior and interior faces of a wall are connected by capillary passages, severe wetting at the interior of building may occur because of capillarity.

capillarity / narrow space / wall / deterioration / exterior / interior / moisture / wetting

https://www.slideshare.net/qwerty7696/fluid-properties-density-viscosity-surface-tension-capillarity

A static fluid can have no shearing force acting on it, and that

Any force between the fluid and the boundary must be acting at right angles to the boundary.

For an element of fluid at rest, the element will be in equilibrium - the sum of the components of forces in any direction will be zero.

The sum of the moments of forces on the element about any point must also be zero.

static / boundary / shearing force / equilibrium / component / moment and force

Pascal’s Law: The pressure at a point in a fluid at rest is independent of direction as long as there are no shearing stresses present.

Pressure at a point

pressure / point / rest / independent / direction / shearing stress

P2 – P1 = - g A (z2 – z1)

Thus in a fluid under gravity, pressure decreases with increase in height

z = (z2 – z1)

In other words, in an incompressible fluid at rest the pressure varies linearly with depth.

Pressure variation in a fluid at rest (Hydrostatic distribution) Variation of pressure vertically in a fluid under gravity

pressure variation / distribution / gravity / incompressible / linearly / depth

= g

p = h P1 = P2 Pressure in the horizontal direction is constant (h1=h2).

In other words, the pressure is the same for any two points at the same elevation in a continuous mass of fluid.

Equality of pressure at the same level in a static fluid

horizontal / level / direction / constant / continuous

P1 P2

h1 h2

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Absolute pressure (Pabsolute): Measured relative to a perfect vacuum (absolute zero pressure) it’s always positive. So it is equal to gauge pressure plus atmospheric pressure.

Gage pressure (Pgage): Measured relative to the local atmospheric pressure. So it is equal to absolute pressure minus atmospheric pressure. It can be negative or positive.

Head (h): The vertical height of any fluid of density ρ which is equal to this pressure.

= g

Pabsolute = Pgage+ Patm Pgage = h h = Pgage/

head / absolute pressure / gage pressure / vacuum

Atmospheric pressure can be measured using a barometer:

Vacuum p=0

Patm=10.33 t/m2

p=0

p=patm

L

atmospheric pressure / barometer / vacuum / glass pipe / piezometric tube / mercury

Force balance; Patm= ρgL

ρ is the density of the fluid, g is the gravitational constant

Piozemeter Tube U-tube Manometer

Inclined-tube Manometer

Manometers Mechanical Pressure

Measurement Instruments

Electronic Pressure

Measurement Instruments

Measurement of

Pressure

manometer / piezometer tube / u-tube / instrument / strain gauge / oscillation

For a horizontal submerged plane

the pressure, p, will be equal at all points of the surface. Thus the resultant force will be given by

F = pressure x area of plane = p A = ( h) A

If the surface is a plane the force can be represented by one single resultant force, acting at right-angles to the plane through the center of pressure.

submerged plane surface / resultant force / center of pressure / hydrostatic paradox

h

p

For a vertical or inclined surface

The net pressure force (Resultant acts through the center of pressure, CP)

Fr = γ hc A

hc: the vertical distance from the fluid surface to the centroid (center of gravity) of the area.

A: the area of the surface

A y

I x x

A y

I y y

c

xyc c p

c

xc c p

+ =

+ = xc and yc: coordinates of the center of gravity of surface

Ixc: the moment of inertia of the surface about the x axis

Ixyc: the product of inertia of the surface about the x and y axes passing through the center of gravity of surface

inclined surface / resultant / centroid / center of gravity / moment of inertia

Weight of water above sloping surface W= V

inclined surface / resultant / centroid / center of gravity / pressure prism

For a vertical or inclined surface (The other alternative approach)

F Fx

H

H/3

b

Fy

Lx/3

H

b

PRESSURE PRISM

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Horizontal forces

The resultant horizontal force of a fluid above a curved surface is:

Fx = Resultant force on the projection of the curved surface onto a vertical plane.

Fx acts horizontally through the center of pressure of the projection of the curved surface onto an vertical plane.

curved surface / resultant force / horizontal forces / vertical forces / projection

Vertical forces The resultant vertical force of a fluid above a curved surface is:

Fy = Weight of fluid directly above the curved surface.

and it will act vertically downward through the center of gravity of the mass of fluid.

Resultant force and the angle the resultant force makes to the horizontal

22

yx FFF x

y

F

F1tan

Fx

Fy

The principle of Archimedes: The buoyancy acting on a submerged object is equal to the weight of the displaced fluid due to the presence of the object. This law is valid for all fluid and regardless of the shape of the body. It can also be applied to both fully and partially submerged bodies.

Archimedes Principle / buoyancy / submerged / displaced fluid

Archimedes screw pump

If a body returns to its equilibrium position when displaced it is said to be in a stable equilibrium position. Conversely it is in an unstable equilibrium position if, when displaced (even slightly), it moves to a new equilibrium position. For the completely submerged body, as long as the center of gravity falls below the center of buoyancy, the body is in a stable equilibrium position with respect to small rotations.

stable / unstable / equilibrium / center of gravity / metacentre point

A general class of problems involving fluid motion in which there are no shearing stresses occurs when a mass of fluid undergoes rigid-body motion.

ax

a

dx

dy

g

axatan

Linear motion: Rigid-body rotation:

g

rPP o

2

22

pressure variation / fluid motion / rigid-body motion / shearing stresses / rotation

Compressible or Incompressible flows

Uniform or non-uniform flows

Steady or unsteady flows

Laminar or turbulent flows

One-, two-, and three-dimensional flows

Compressible or Incompressible: All fluids are compressible -

even water - their density will change as pressure changes. Under steady conditions, and provided that the changes in pressure are small, it is usually possible to simplify analysis of the flow by assuming it is incompressible and has constant density.

Classification of flow types

classification / compressible / incompressible / uniform / non-uniform / steady unsteady / laminar / turbulent

Uniform flow: If the flow velocity is the same magnitude and direction at every point in the fluid it is said to be uniform.

Non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform.

Steady: A steady flow is one in which the conditions (velocity,

pressure and cross-section) may differ from point to point but DO NOT change with time.

Unsteady: If at any point in the fluid, the conditions change with time, the flow is described as unsteady.

magnitude / uniform / non-uniform / steady / unsteady / condition / describe

(From https://www.meted.ucar.edu/)

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Steady uniform flow: Conditions do not change with position in the stream or with time. An example is the flow of water in a pipe of constant diameter at constant velocity.

Steady non-uniform flow: Conditions change from point to point in the stream but do not change with time. An example is flow in a tapering pipe with constant velocity at the inlet - velocity will change as you move along the length of the pipe toward the exit.

Unsteady uniform flow: At a given instant in time the conditions at every point are the same, but will change with time. An example is a pipe of constant diameter connected to a pump pumping at a constant rate which is then switched off.

Unsteady non-uniform flow: Every condition of the flow may change from point to point and with time at every point. For example waves in a channel.

stream / tapering / pipe / pump / pumping / wave / channel

Laminar and turbulent flow: In laminar flow the motion of the particles of fluid is very orderly with all particles moving in straight lines parallel to the pipe walls.

Turbulent flow

Re > 4000

“high” velocity

Dye mixes rapidly and completely

Particle paths completely irregular

Average motion is in the direction of the flow

Cannot be seen by the naked eye

Changes/fluctuations are very difficult to detect. Must use laser.

Mathematical analysis very difficult - so experimental measures are used

Most common type of flow.

Transitional flow

2000 > Re < 4000

“medium” velocity

Dye stream wavers in water - mixes slightly.

Laminar flow

Re < 2000

“low” velocity

Dye does not mix with water

Fluid particles move in straight lines

Simple mathematical analysis possible

Rare in practice in water systems.

laminar / turbulent / transitional / filament dye / particle / path / fluctuation / detect

Although in general all fluids flow three-dimensionally (3D), with pressures and velocities and other flow properties varying in all directions, in many cases the greatest changes only occur in two directions or even only in one. In these cases changes in the other direction can be effectively ignored making analysis much more simple.

Two-dimensional flow: The flow parameters vary in the direction of flow and in one direction at right angles to this direction. Example: flow over a weir toe.

One-dimensional flow: The flow parameters (such as velocity, pressure, depth etc.) at a given instant in time only vary in the direction of flow and not across the cross-section. Example: the flow in a pipe.

one-dimensional / 2D / 3D / velocity / pressure / depth / weir / toe

Streamlines: The lines that are tangent to the velocity vectors throughout the flow field.

Streamtube: The imaginary tubular surface formed by streamlines along which the fluid flows.

Pathline: The line traced out by a given particle as it flows from one point to another.

streamlines / streamtube / pathline / imaginary / tabular / immersed object

Lagrangian Method: This method involves following individual fluid particles as they move about and determining how the fluid properties associated with these particles change as a function of time. That is, the fluid particles are identified and their properties determined as they move.

Eulerian Method: The fluid motion is given by completely prescribing the necessary properties (pressure, density, velocity, etc.) as functions of space and time. From this method we obtain information about the flow in terms of what happens at fixed points in space as the fluid flows past those points.

Lagrangian and Eulerian Approach / prescribing / properties

The rate of mass stored = the rate of mass in - the rate of mass out

outin mmdt

dm

Vm inm outm

volumeV :

Conservation of mass – the continuity equation

governing equations / conservation of mass / continuity equation / rate

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Flow rate

Mass flow rate:

The mass of the fluid flow through the control surface per unit time

Volume flow rate – Discharge:

The discharge is the volume of fluid flowing per unit time. Multiplying this by the density of the fluid gives us the mass flow rate.

flow rate / mass flow rate / volume flow rate / discharge

2

2

221

2

11

22gz

Vpgz

Vp

Bernoulli’s equation

Flow work + kinetic energy + potential energy = constant

Integration of Euler’s equation

The Bernouilli Equation

Bernoulli’s equation has some restrictions in its applicability, they are: Flow is steady; Density is constant (which also means the fluid is incompressible); Friction losses are negligible; The equation relates the states at two points along a single

streamline, (not conditions on two different streamlines).

dx

dz

dL

p1

p2

steady flow / incompressible flow / negligible / friction losses

The Bernouilli Equation – Conservation of energy

Pressure Kinetic Potential Total

energy per + energy per + energy per = energy per

unit weight unit weight unit weight unit weight

Hzg2

Vp 2

Pressure Velocity Potential Total

head + head + head = head

pressure energy / kinetic energy / potential energy / total head Pitot tube / Venturi meter / tank / sharp edge orifice

Conservation of momentum – (Newton’s second law)

Newton’s 2nd Law can be written:

The rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force.

)(mVdt

dF

Force = rate of change of momentum

F Q ( u 2 u 1 )

momentum / resultant force

Conservation of momentum – (Newton’s second law)

The force in the x-direction Fx Q u 2 x u 1x

and the force in the y-direction Fy Q u 2 y u 1y

We then find the resultant force by combining these vectorially:

2y

2xresultant FFF

and the angle which this force acts at is given by

x

y1

F

Ftan

momentum / resultant force

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Application of the Momentum Equation 1. Force due to the flow of fluid round a pipe bend. 2. Force on a nozzle at the outlet of a pipe. 3. Impact of a jet on a plane surface. 4. Force due to flow round a curved vane.

pipe bend / nozzle / water jet / curved vane

The force exerted by the fluid on the solid body touching the control volume is opposite to FR. So the reaction force, R, is given by R = - FR

Dimensionless Number

Symbol

Formula

Numerator

Denominator

Importance

Reynolds number

NRe

Dvr/m

Inertial force

Viscous force

Fluid flow involving viscous and inertial forces

Froude number

NFr

u2/gD

Inertial force

Gravitational force

Fluid flow with free surface

Weber number

NWe

u2rD/s

Inertial force

Surface force

Fluid flow with interfacial forces

Mach number

NMa

u/c

Local velocity

Sonic velocity

Gas flow at high velocity

Drag coefficient

CD

FD/(ru2/2)

Total drag force

Inertial force

Flow around solid bodies

Friction factor

f

tw/(ru2/2)

Shear force

Inertial force

Flow though closed conduits

Pressure coefficient

CP

Dp/(ru2/2)

Pressure force

Inertial force

Flow though closed conduits. Pressure drop estimation

dimensionless number / Reynolds, Froude, Weber, Mach numbers / coefficient

Search about the hydrostatic paradox; and decide on which one of the following is correct?

PA = PB = PC = PD = PE or PA PB PC PD PE