Fluid Mech

zero shear stress

A fluid at rest is at a state of zero shear stress - when the walls are removed or a liquid container is tilted, a shear develops as the liquid moves and re-establishes a horizontal free surface - i.e. whichever way the fluid is tilted, it still claims a horizontal position

viscosity

A liquid's resistance to flowing

absolute pressure vs gage pressure vs vacuum pressure

Absolute pressure: The actual pressure at a given position. It is measured relative to absolute vacuum (i.e., absolute zero pressure). Gage pressure: The difference between the absolute pressure and the local atmospheric pressure. Most pressure-measuring devices are calibrated to read zero in the atmosphere, and so they indicate gage pressure. Vacuum pressures: Pressures below atmospheric pressure.

principle of similarity

3 necessary conditions for complete similarity between a model and prototype: 1. geometric similarity - can be scales, but shape must be the same 2. kinematic similarity - velocity at any point in the model flow must be proportional to velocity at corresponding point in prototype -> corresponds to scaling factor 3. dynamic similarity - all forces are proportional to scale (force-scale equivalence)

moment of intertia - for a rectangular prism

I = bd^3/12 - depends on which axis youre applying it on (b and d vary)

Ixx,O and Ixx,C

Ixx,O is moment of inertia about 0 axis Ixx,C is moment of inertia about centroidal axis

mass vs weight

Mass is a measurement of the amount of matter something contains, while Weight is the measurement of the pull of gravity on an object.

pressure equation

P=pgh g = 9.81 m/s^2 p = 1000 kg/m^3 thus we see that pressure varies with height ** eg. pressure is 4 metres of head P=(1000)(9.81)(4) - assumes water ** eg. pressure is 4 metres of head in air P=(density of air)(9.81)(4) - density varies according to air

ideal gas eqn

R may be related to the specific gas

expansive soil

Soil that expands when water is added and shrinks when it dries out. This change in soil volume can cause shifting and cracking in structures.

surfactant

Surface Acting Agents: compounds that lower the surface tension (or interfacial tension) between two liquids, between a gas and a liquid, or between a liquid and a solid - has a hydrophilic and hydrophobic part, and the hydrophilic part sticks out of the interface in order to reduce the surface tension between the air and water

capillary effect

The rise or fall of a liquid in a small-diameter tube inserted into the liquid

absolute pressure

The total pressure exerted on a system (gage pressure), including atmospheric pressure let "zero" of pressure be the atmospheric pressure - "gage pressure of 0 or absolute pressure of 101.3"

shaft work

The transfer of mechanical energy is usually accomplished by a rotating shaft, and thus mechanical work is often referred to as shaft work

surface tension

scenario: - liquid drops behave like small spherical balloons filled with liquid, where the surface act as a stretched membrane under tension bcs on top its surrounded by air (known as the interface) - due to attractive forces btwn molecules

stagnation point

a point in a flow field where the local velocity of the fluid is zero

define fluids

a substance that deforms continuously when subjected to shear stress - includes liquids and gases

plane surface

a surface which lies evenly with the straight lines on itself - 2 dimensions

Po

absolute pressure above liquid

accuracy vs precision

accuracy: how close experimental value is to accepted value - accuracy error: value of one reading minus the true value (repeatable, fixed errors) precision: how closely measured values agree with each other - precision error: value of one reading minus the average of the readings (unrepeatable, random errors)

accuracy vs precision

accuracy: value of one reading minus true value precision: value of one reading minus average of readings

example of water and air

air angle > 90 = non wetting fluid water angle < 90 = wetting fluid

density variation btwn air, and water

air: highly compressible, large density variation water: low compressibility

streamlines, pathlines, streaklines all...

all look the same if you have a steady flow

moment of inertia eqns

along x and y axes

mass flow rate

amount of mass flowing through a cross section per unit time

contact angle

angle that the tangent to the liquid surface makes with the solid surface at the point of contact - depends on fluid and surface involved

rate

anything that varies/time

bernoulli equation

approximate relation btwn pressure, velocity, and elevation - applicable in: regions of steady, incompressible flow where frictional forces are negligable - in example: orange part is the wing of a plane

ideal mechanical advantage of the hydraulic lift

area ratio A2/A1 - a larger area would make for a larger force to maintain the same pressure - force is directly proportional to area

moment of inertia increases as....

as the reference axis is moved parallel to itself farther from the centroid

concept of continuum example

at a microscopic level, the molecules all move in DIFFERENT direction at a macroscopic level, the molecules all tend to move in the SAME direction as the arrow

derivation of bernoulli eqn notes

bernoulli eqn is same for two points along the same streamline

stretching liquid film with u-shaped wire

bottom diagram is top view

how do solids resist an applied shear stress?

by deforming - deform continuously under the influence of shear stress

limitations to bernoulli equation

can only be used in: 1. steady flow 2. frictionless flow 3. incompressible flow - only if mach number is less than 0.3 4. along or across a streamline cant be used in: 1. shaft work occurs - not applicable in a section where devices such as a pump, turbine, fan etc.. can destroy the streamline (use energy instead) 2. heat transfer occurs

finding central pressure for submerged rectangular plate

centroid of plate is at b/2 along axis of plane, the depth is s+b/2 knowing theta, you can find the hc (vertical value) multiply by gamma to find resultant force Fr = [Po+pg(s+b/2)sin(theta)ab ** a is into the plane (thickness of plane) this goes by the equation mentioned earlier for yp (Ixx,c)/[yc+Po/(pgsin(theta)A - moment of inertia of a rectangle about a centroidal axis is ab^3/12 - A=ab at the top FR = [Po+pgb/2]ab - this is at the top bcs s = 0

closed system

closed (control mass) - fixed amount of mass - no mass can cross boundary

parameter

combined set of dimensional variables, non-dimensional variables, and constants

bernoulli eqn aka

conservation of mechanical energy principle

shear stress

consider the deformation of a rubber block placed between 2 parallel plates - shear stress is on the rubber AND an equal but opposite force acts on the upper plate - doesnt impact fluids as they deform continuously regardless of how small they are

streakline

consists of all particles in a flow that have passed through a common point - collection of fluid elements released at successive times from a given location, and visualized at one instant in time https://www.youtube.com/watch?v=UqHd3pnEKtA

control surface

control surface: boundaries of a control volume - can be real or imaginary

turbine

converts the mechanical energy of a fluid to shaft work

why are there different angles for some non wetting fluids?

could be caused by different fluid type OR different surface type

meniscus

curved free surface of a liquid in a capillary tube

angular displacement or deformation expression

da = V dt bcs an increment of length is the velocity multiplied by an increment of time (speed=distance/time) du/dy = V/l = change of velocity with y

fluid mechanics

deals with behavior of fluids at rest (fluid statics) and in motion (fluid dynamics)

define mechanics

deals with both stationary and moving bodies under the influence of forces

fluid statics

deals with fluids at rest - can be gaseous or liquid - only applies in gravity fields (gravity exists)

fluid kinematics

deals with motion of fluids without considering the forces and moments which create the motion - fluids subject to shear, and pressure imbalance

surface tension equations

delta X = distance that its stretched by 2b = length upon which surface tension is acting

specific weight eqn

density multiplied by gravity

independent and dependent variables slide

density, pressure, velocity, and temperature are all dependent variables - independent variables are time and location

secondary units

derived from primary units

reynolds number vs flow

determines if your fluid will flow according to laminar, transitional, and turbulent flow according to density, velocity, diameter, and viscosity of the fluid

froude number

dimensionless ratio between fluid inertial forces and fluid gravitational forces

viscosity of liquids

due to cohesive forces between molecules - at higher temps = more energy = oppose large cohesive forces = decreased viscosity

Gases and viscosity

due to more molecular collisions causes a greater resistance of the gas to flow - viscosity increases with temperature

why can a mosquito and paperclip float on water?

due to surface tension

hydraulic conductivity

ease with which a fluid (usually water) can move through pore spaces or fractures

mechanical efficiency (n_mech)

effectiveness of the conversion process between the mechanical work supplied or extracted and the mechanical energy of the fluid is expressed by the pump efficiency and turbine efficiency

buckingham pi theorem -> method of repeating variables (CAN ADD TO CHEAT SHEET) -> can also write how to select repeating variables

eg. you have 3 variables - if those variables have 2 primary dimensions, k=1

pi method

ensures that all similarity criteria are met

law of dimensional homogeneity

every additive term in an equation must have the same dimensions

find location of particle given x, t and to, yo, xo

exam question

sutherland correlation

expresses viscosity of gases

extensive vs intensive properties

extensive - properties related to total mass of a system - capital case letters - eg. mass, weight intensive - properties independent of amount of fluid - lower case letters - eg. pressure, mass density (p)

extensive versus intensive

extensive: depends on mass (mass, momentum, volume, energy) - B intensive: not reliant on mass - b=B/m

gauge pressure

finds pressure in a container using the concept of continuum - this number is extremely close to the pressure found when looking at individual molecules

open-channel flow

flow of liquids in a duct in which the liquid is partially filled and there is a free surface - eg. rivers, irrigation channels

continuum assumes...

flow properties vary continuously throughout the fluid - in continuum, the smallest element of a fluid is a FLUID PARTICLE (contains significant number of molecules) and NOT a fluid molecule

boundary layer

flow region adjacent to the wall in which viscous effects (and velocity gradients) are significant

layers and wakes

fluid motion is governed by combined effects of pressure and gravity, rather than friction

newtonian fluids

fluids which the rate of deformation is proportional to its shear stress

Knudsen number

for Kn<0.01, flow field should be large or mean free path should be small

DIMENSIONAL ANALYSIS

for sure a 20 mark question on exam

hydrostatic force

force due to pressure of fluid at rest - eg. force exerted on wall of storage tanks, dams, and ships

define stress

force per unit area - pressure is also force/unit area "Pressure is often used with fluids (gases or liquids), whereas stress is more often used with solids. One major difference is that pressure only acts perpendicular to a surface, whereas stress can also be parallel to a surface as well as perpendicular to it. A stress parallel to a surface is called shear stress."

mechanical energy

form of energy that can be converted completely into mechanical work

what is gamma

g = pg pressure times gravity

absolute vs gage pressure

gage pressure = 0 means that its equal to atmospheric pressure

fundamental manometer equation

gamma is weight density

gas vs vapor

gas: when phase is above critical temperature vapor: when current phase is not far from state of condensation

indefinite volume means...

gases have a specific volume at a specific temperature and pressure - gases occupy all the spaces thats available to it - can compress or expand depending on the volume of the container

flow from a LARGE reservoir (torricelli's law)

height is the main thing that is driving the flow - water flow from a tank is unsteady bcs when h becomes smaller, so does the water pressure

mach number

how speed compares to speed of sound - fluid at a Ma number means that its moving with the speed of sound

video on reynolds tranport theorem

https://www.youtube.com/watch?v=PDq9YQh650g

WATCH THIS VIDEO

https://www.youtube.com/watch?v=QUgXf2Rj2YQ - started adding notes, make them more detailed

hydrostatics vs aerostatics

hydrostatics: fluid at rest is liquid aerostatics: fluid at rest is gas

pi method continued

if pi(2) for a model is equal to pi(2) for a prototype AND pi(3) of the model is equal to pi(3) of the prototype THEN pi(1) of the model and pi(1) of the prototype are equal - thus model and prototype are completely similar

velocity vs diameter

if velocity v1 comes into a small diameter, and velocity v2 comes out of a large diameter, then v2<v1

nondimensional equations

if we divide a dimensionally homogeneous equation by the shared dimensions, we get a nondimensional equation - normalized eqns is when the equations are in order unity, and when you divide out the dimension you get 1

importance of no slip condition

if you put one twig in the middle of a stream and one on the side, the one in the middle will move much faster bcs the one on the side experiences the "no slip condition" another example: swimming at the bottom of a stream is easier bcs its at the boundary layer and the stream rate is slower

absolute pressure

in air (left) - pressure change is negligable in liquid (right) - Pgage would be just pgh bcs Patm is not included (its 0)

centre of buoyancy vs centre of gravity

in the same vertical line (centre line- BG)

incompressible vs compressible flow

incompressible flow: density of fluid remains nearly constant throughout - liquids, gases at low speeds - density changes of gas flows are under 5% or when Ma<0.3 compressible flow: density changes of fluid is significant - gases at high speeds - density changes of gas flows are above 5% or when Ma>0.3

internal vs external flow

internal flow: flows in which fluid is completely bounded by solid surface - dominated by viscosity - eg. pipe or duct external flow: flows in which fluid is unbounded over solid surfaces - viscous effects are limited to boundary layers near solid surfaces and to wake regions downstream of bodies - eg. flow over a plate, wire, spherical object

viscosity is due to...

internal frictional force that develops btwn different fluid layers as they are forced to move relative to eachother

mechanical energy does not change during its flow if....

its pressure, density, velocity and elevation all remain constant

how to actually find viscosity

its the slope of the graph of rate of deformation (du/dy) vs shear stress (T) - low viscosity is air bcs it has a smaller slope

lagrangian vs eularian

lagrangian: fixed mass (system) - can change size and shape, but always follow same mass - mass cant cross the boundary eularian: control volume (fixed region of interest) - can change size and shape and mass might cross boundary - interested in region rather than mass

Laminar vs. Transitional Flow Turbulent Flow

laminar - highly ordered - smooth layers of fluid - eg. flow of high-viscosity fluids such as oils at low velocities transitional - flow that alternates between laminar and turbulent turbulent - highly DISordered - occurs at high velocities & is characterized by velocity flunctuations - eg. flow of low viscosity fluids such as air at high velocities

streamline

line that is tangent to the instantaneous velocity field at every point - black lines on the left graph dy/dx=v/u dy/dz=v/w dx/dz=u/w https://www.youtube.com/watch?v=TOUylg7Eyec

liquids vs gases compressibility

liquids - incompressible - eg. pressure of 210 atm causes density of liquid at 1 atm to change by 1% only gases - highly compressible - eg. pressure change of 0.01 atm causes a change of 1% in density of atmospheric air - HOWEVER, it can be considered incompressible if density changes<5% (Ma<0.3) ie. speed is less than 100m/s

example

location 2 has a lower pressure than location 3 bcs one is a pressure from upstream side, and one is the pressure from the downstream side - HOWEVER elevation is actually only regarded when we look at potential energy - Z2 and Z3 are SIMILAR not the same - use mechanical energy change equation, everything cancels out except the potential energy term

midterm question: what should be the minimum diameter of the tube so that the surface tensions are negligible?

look at equation where h=2(sigma s)cos(theta)/pgR set height of rise to be tiny (not 0), and find value of r - every other value follows values of mercury - i think theta = 0

hydrostatic forces on submerged CURVED surfaces

look at horizontal and vertical projections of surface (a and b) - these projections are straight bcs it projects onto a specific plane pressure on 'a' is pgh pressure on 'b' varies - also need to consider weight of water bounded by curved region

macroscopic (classical) vs microscopic (statistical) approach -> not a relevant slide

macroscropic (classical): - no previous knowledge of individual molecules and a direct method is provided to analyze the engineering problems - eg. pressure is measured with a pressure gage microscopic (statistical): - based on average behaviour of large groups of individual molecules - eg. pressure is measured by interaction of individual gas molecules

centre of pressure

magnitude of the force and its point of application

solid body floats if

mass density of body is less than that of fluid - weight of fluid is equal to that of body

mass density

mass of substance per unit volume - symbol: p (rho) - units: kg/m^3 - dimensions: M/L^3

venturi meter (EXAM EXAM EXAM)-> will need to use both BERNOULLI AND CONTINUITY EQN

measures avg velocity or flow of an incompressible fluid through a pipe - as velocity increases, pressure decreases - first eqn is insufficient bcs theres one equation and 4 variables - thus continuity equation allows to find v1 in terms of v2-> 3 variables left in equation - thus manometer is used to measure pressure change

manometer

measures pressure using liquid column in vertical tubes

third eqn used to calculate mols (at STP)

mol = volume (L)/22.4

parallel-axis theorem

moment of inertia wrt a parallel centroidal axis plus the product of the area and the square of the distance btwn the two axes Iyc or Ixc is the centroidal axis ** can find moment of inertia of an axis as long as its parallel to the inertia of a known axis

conservation of momentum principle

momentum of a system is constant only when net force acting on it is zero - ie. law of inertia

visual continued

multiply pressure by area then integrate to find force on entire shape

capillaries

narrow tubes or confined flow channels

natural (unforced) vs forced flow

natural flow - any fluid motion thats due to natural means - eg. buoyancy effect: warmer & lighter fluid rises whereas cooler and denser fluid falls - eg. anomalous expansion of water: property of water whereby it expands instead of contracting when the temperature goes from 4°C to 0°C, and it becomes less dense. The density becomes less as it freezes because molecules of water normally form open crystal structures when in solid form. (water at bottom of lake remains at 4 degrees bcs water is densest and sinks) forced flow - fluid forced to flow over surface or in a pipe by external means - eg. pump or fan

negative vs positive pressure

negative: less than atm positive: more than atm

stresses in fluid statics

no relative motion occurs between fluids thus no shear stresses are trying to deform it - however normal stress exists which is the pressure due to the weight of the fluid

pressure

normal force exerted by a fluid per unit area

pressure

normal stress in a fluid at rest - same in all directions

normal stress vs shear stress

normal stress: normal component of a force acting on a surface per unit area shear stress: tangential component of force acting no a surface per unit area

open system

open (control volume) - properly selected region of space - both mass and energy can cross the boundary - encloses a device that involved mass flow such as a compressor, turbine, or nozzle

measuring a pressure differential using a u-tube manometer

pa-pb = pressure difference between containers A and B (differential pressure)

flow around a curved body

particles decelerate as they approach B (point of stagnation) and this energy is transferred into pressure - since theres no elevation difference btwn them gz=0 for both points - assume that VB is 0 since its a point of stagnation - pressure came from energy that got converted into pressure

hydrostatic force on a circular surface

passes through centre of circle, since the pressure forces are normal to the surface and they all pass through the centre

pump (fan)

receives shaft work (usually from an electric motor) and transfers it to the fluid as mechanical energy (less frictional losses)

reynolds transport theorem

relationship between the time rates of change of an extensive property of a system and for a control volume - helps transfer between lagrangian and eularian approach

will water flow faster through sand or clay?

sand bcs it has larger particles, and the pores are larger so the water is able to maneuver through it more easily

pressure is a _____ quantity

scalar - has magnitude but not direction

upper vs lower meniscus (wetting vs non-wetting fluid)

upper: concave meniscus - wetting fluid: adhesive to other surfaces - eg. water lower: - non-wetting fluid: cohesive to itself - eg. mercury

buoyant force

upward force that a fluid exerts on a body immersed in it - caused by increase of pressure with depth in a fluid

what if you dont know froude's number?

use dimensional analysis

which distance do you measure when you locate the centroid?

vertical distance - pressure changes vertically

viscous vs inviscid regions of flow

viscous flow region: flows in which frictional effect is significant inviscid flow region: viscous forces are negligibly small compared to inertial or pressure forces

mass density of: - water - air - paraffin oil + temp and pressure at which they exist

water: 1000 kg/m^3 mercury: 13546 kg/m^3 air: 1.23 kg/m^3 paraffin oil: 800 kg/m^3 pressure is 1.013 x 10^5 N/m^2 temperature is 288.15 K **however, density varies

2 forces acting on buoyant force

weight buoyancy force -> equal to weight of immersed body

specific weight

weight of a fluid per unit volume - symbol: Y (gamma)

floating bodies

weight of entire body is equal to the buoyant force

ambient pressure

whatever the pressure happens to be in the situation - if liquid is exposed to air then the ambient pressure is Patm - if liquid is exposed to vacuum then the ambient pressure is 0

no slip condition pt 2

when solid is placed between 2 solid plates, it doesnt slip, but rather deforms

final exam question

when you find acceleration from velocity vectors hes gonna ask you to identify time and convection components and discuss the results

metacentre

where centre of buoyancy crosses centre of gravity line

is continuum a reasonable assumption?

yes, in most engineering problems - HOWEVER, it depends on the Knudsen number

how can you tell if its uniform flow?

your u and v vectors are not a function of x and y - eg. u = A/1+t, v=Bt^2

problem: finding line of action (location of centroid along plane)

yp=location of centroid first moment of area about x axis: integral of y dA second moment of area about x axis: integral of y^2 dA

I sections

- high moment of inertia - bending stresses are lower

gases vs compressibility

- highly compressible - eg. a pressure change of just 0.01 atm causes a change of 1% in the density of atmospheric air HOWEVER - if the density changes are under ~5% (Ma<0.3) then the gas flow can be approximated as incompressible - compressibility effects of air at room temp can be neglected at speeds under 100 m/s

2 surface tension units

1. force per unit length - N/m 2. energy per unit area - J/m^2

application areas of fluid mechanics

1. household appliances - fridge 2. turbomachines - pumps 3. military - aircraft 4. automobile - external aerodynamics 5. medicine - glucose monitor 6. electronics - convective cooling of generated heat 7. energy - boiler 8. oil and gas - pump

7 primary dimensions

1. mass 2. length 3. time 4. temperature 5. electric current 6. amount of light 7. amount of matter

mechanical energy change represents....

1. mechanical work supplied to fluid (if delta e mech >0) 2. mechanical work extracted from the fluid (if delta e mech<0)

3 types of manometers

1. piezometer tube 2. u-tube manometer 3. inclined tube manometer

disadvantages of piezometer

1. pressure in container must be greater than atmospheric pressure 2. pressure must be relatively small 3. must be a liquid

diagram showing normal and shear stress at surface of a fluid

** for fluids at rest: shear stress is zero and pressure is the only normal stress

example of two buckets of water

- bucket 1 flows much faster in the hole due to higher pressure

know derivations from 3 videos for final exam

- cant write it on cheat sheet

viscosity example

- constant parallel force F is applied to the upper plate, while the lower plate is fixed - upper plate moves at constant velocity (V) due to F - shear stress (T) acting on a fluid layer is T=F/A-> A is contact area btwn plate and fluid - bottom plate follows no slip condition velocity profile - u(y) means that u changes according to y bcs on the bottom where y=0, u=0, while on top, y=l, and u=V - velocity in the middle will be V/2 using the equation u(y)=(y/l)V , at y=l/2 EQN IS SIMILAR TRIANGLES

one, two and three dimensional flows

- constant velocity across the profile in the beginning - however as it moves to the right, the velocity starts to decrease - profile on the right is under the no-slip condition bcs fluid isnt really flowing - went from 2d to 1d flow - V= V(r,z) is the axes - r = radius, z = location

eulerian description

- control domain through which fluids flow - need to define field variables - doesnt consider what happens to individual fluid molecules - more convenient

mechanical energy formulas

- divided out everything by mass

solving for h (capillary rise)

- eqn also works for non-wetting liquids (theta>90, which makes h negative) - the larger the capillary, less fluid rises - the denser the fluid, the less fluid rises

specific properties

- extensive properties per unit mass - eg. v=V/m (specific volume) - eg. e=E/m (specific total energy) image shows that mass and volume are dependent on mass while the other properties are not

lagrangian description

- follows the path of individual objects and their velocities - used for small objects such as billiard balls on a pool table

absolute pressure at any point on the plate

- h is vertical distance of point from free surface - y is the distance of point from x-axis (from point O) - Pavg = [gamma(h)/2]A - Pc is pressure at the centroid

centroids of areas

take moment of small area wrt x and y axes - integrate it and find force over entire surface

total derivative

takes partial derivatives of f with respect to t

buoyancy

the ability or tendency to float in a fluid

mass equation (continuity equation)

the dot on top implies that its a mass flow RATE

liquids vs compressibility

the flow of liquids is typically incompressible since the densities of liquids are essentially constant (aka incompressible substances) - eg. increase pressure of liquid to 210 atm = density of liquid water at 1 atm to change by only 1%

pathline

the line traced out by a given particle as it flows from one point to another - show history of particle movement - u= change in x wrt t=dx/dt - v= change in y wrt t=dy/dt https://www.youtube.com/watch?v=ZRUYHQCQLBk

atmospheric pressure

the pressure caused by the weight of the atmosphere - keeps fluids from bursting and flowing outwards - 1 atm = 101.3 kPa

moment of inertia

the resistance to rotation

pressure at any point in a fluid is....

the same in all directions

if the curved surface is above the liquid....

the weight of the liquid and vertical components of the hydrostatic force act in the opposite directions

goal of dimensional analysis

to obtain result in terms of mach number, freud number, or reynolds number, drag coefficients

exam question

top two plates will be stationary, while a plate in the middle will move

pitot-static tube

tube which combines all three pressures - pitot tube is inside, piezometer is on the outside measures static and stagnation pressure Za and Zb = same height = 0 stagnation pressure at B = 0

lagrangian and eulerian

two methods to describe motion of fluids

what is u and v

u = velocity in x direction v= velocity in y direction

uniform vs non-uniform flow

uniform flow - no change of fluid properties with location over a specified region - check different regions and every region has the same speed non-uniform flow - if at any given instant, the fluid properties change with location over a specified region *** velocity vs location (spacial dimension)

dimensional analysis

- help in design of experiments - develop scaling laws in prototype and model performance - predicts trends

Galilean Thermometer

- made of sealed glass cylinder containing a clear fluid - weight are suspended in the liquid - as temperature varies, the weights move up bcs the density is changed -> suspended weights rise and fall to stay at the position where density is equal to that of surrounding fluid

pressure example

- more pressure occurs on a smaller surface area rather than a larger one

careless drilling leads to...

- only have static pressure in the centre, bcs drilling was proper

equation for pressure at bottom of a tank

- pressure doesnt vary on the same horizontal plane - gamma is density*gravity ***fluid forces act PERPENDICULAR to surface

where are piezometer used

- put a tube into into the groundwater and the pressure will rise up depending on the pressure underground

stress for fluids at rest

- shear stress is zero (no motion occurs) - pressure is only caused by normal forces

you give me a certain amount of weight of gold on earth, and I return the same amount as that weight on of gold on the moon ? is this a good deal for you or not?

- since gravity there is less, you would need more mass to compensate for the decreased weight - thus youre getting a good deal bcs of the extra mass of gold youre given

golden crown experiment

- their masses are equal - indicates that their volumes differ so they densities are different

inclined-tube manometer

- used to measure SMALL pressure changes - get rid of gases bcs their pressure is negligible

velocity profile

- velocity gradient exists

dont need to know integration process but know how formula works

..

range of efficiencies

0% all mechanical/electrical energy input turns into thermal energy and the device functions as a resistance heater 100% none of the mechanical/electrical energy input turns into thermal energy (no friction or other irreversibilities)

Kinetic Molecular Theory (ideal gas behavior)

1. actual volume occupied by gas molecules is negligibly small compared to total volume of gas - eg if you put 2 hydrogen atoms in a room, then the actual volume taken up by the two molecules is volume of room - boyles law: pressure and volume of a gas are inversely related if temperature remains constant-> P=1/V (T=const) - charles law: direct relationship btwn volume and temperature of a gas if pressure remains constant -> V=T (P=const) 2. no force of attraction or repulsion exists between gas molecules *** this theory doesnt hold true at high pressure and low temp

problem

1. find centroid (middle of shape) 2. find hc (vertical distance) 3. multiply by pg

example

2 big variables exist i) velocity - dependent on viscosity and density ii) length Fd is the drag force on the car - depends on length of car and velocity of car

"of head" units

2 metres of head means that pressure of that place is equal to pressure when there is two metres of fluid (water unless specified otherwise) on top of the object

celcius to kelvin

273+*C

final exam question

4-1 - KNOW HOW TO PLOT IT

barometer

An instrument that measures atmospheric pressure 1. fill tube with mercury 2. fill bath with mercury 3. put inverted tube in bath of mercury 4. air pressure will push some mercury down into the bath in diagram - pressure where bath meets tube is 101.3 KPa (760 mm of mercury) - since mercury is 14 times the density of water, the height would also be 14 times the height of water WATCH https://www.youtube.com/watch?v=EkDhlzA-lwI

hydrostatic resultant force for inclined plane (pressure prism)

CP = centre of pressure

conservation of energy

Energy cannot be created or destroyed - can only be transferred to or from a closed system by heat or work - control volumes also involve energy transfer via mass flow

FBD pressure at a point

F=PA - FBD on bottom left shows P1 (dy)(dz) bcs its force = pressure * area - pressure ALWAYS acts normal to the surface - since all forces equal zero (fluid is at rest) we can equate all the forces and solve for pressure - theta is negligibly small, so all pressures end up equaling each other - component that is "1/2pg(dx)(dy)(dz)" is the weight of the fluid moral: pressure at a point is the same in all directions

newton's second law (ie. linear momentum equation)

F=ma rate of change of momentum is equal to net force acting on a body

viscosity example in a steady laminar flow (continued)

IF AND ONLY IF flow is steady and laminar, then the flow is considered linear - fluid velocity between plates varies linearly between 0 and V , so the velocity profile and velocity gradient are as follows - integrated with respect to y

flow in a closed conduit

piezometer is used - no dynamic pressure bcs theres no velocity in the upward direction (only to the right) - pressure at A is due to velocity, and so it causes fluid to rise up (static pressure) 3 pressures exist - pressure from weight of fluid (d) - static pressure from velocity (h) - dynamic pressure (l)

point vs path functions

point function: describes a particular state without depending on the path taken to reach this state - displacement path function: value depends on the path taken to get between two states - distance

point vs path functions and their differentials

point functions have exact differentials path functions have inexact differentials

example

point is that potential energy at the top is converted to pressure at the bottom

most abundant salt in canada

potassium (potash) - produce 30% of world's potassium (potash)

Pascal's law

pressure applied to a confined fluid increases the pressure throughout by the same amount

piezometer tube

pressure comes from closed container and from atmosphere - move from left to right (remains the same until height changes) - pressure decreases as you move up - po is atmospheric pressure

HGL and EGL diagram (EXAM)

pressure goes down bcs the area of the tube becomes larger and the velocity decreases - static and velocity pressure decrease EXAM: when pressure head goes down, velocity goes down - friction losses occur so values are changing

variation of pressure with depth

pressure increases with depth bcs theres more weight of fluid on top of the subject - gamma is the weight density (density multiplied by gravity) - pgh = gamma(h) - capital P is absolute pressure

flow in an open channel

pressure is atmospheric - bent tube is inserted into stream to observe height to which liquid rises - velocity is greatest at top and centre but we put the tube in the middle of the height to get an average value - fluid is not moving at point b bcs that would cause the fluid to exit the tube. water is not moving anywhere in the tube, at first when it moves in then it has a velocity, but every other moment the fluid just stays there - h due to dynamic pressure is due to velocity stopping at point B and converting into pressure

u-tube manometer EXAM EXAM EXAM

pressure is the same at the interface line so you can extend the green line to the red line - equals 0 bcs of gage pressure - can ignore gas bcs its pressure is so small that its negligible

pgage

pressure that gage reads when 0 is set at atmospheric pressure - increases with depth

primary units

primary units - all eqns are made of these units

linear momentum

product of mass and velocity of a body - aka momentum

4 types of reactions to forces

pseudoplastic: requires a lot of stress in the beginning to overcome static friction bingham plastic: apply shear stress constantly and it only starts to move later dilatant: deforms excessively in the beginning under a small amount of force

concept of continuum

question: is it possible to track all molecules at a microscopic scale? answer: no, and its also unnecessary. instead, we treat engineering problems at macroscopic scale concept of continuum: we treat fluids as a continuum, and do not concern with the behavior of individual molecules

specific gravity (relative density)

ratio of density of a substance to density of some standard substance at a specified temperature - comparing the density of something to density of water (1000 kg/m^3) - dimensionless - if SG is>1, its denser than water - if SG is<1, its less dense than water

kinematic viscosity

ratio of dynamic viscosity to density

no slip condition

scenario: fluid between 2 parallel plates thats subjected to shear stress due to motion of the upper plate - no relative motion between fluid and boundary (ie. fluid in contact with upper plate moves at speed U, while fluid in contact with lower plate is stationary) - fluid deforms (undergoes rate of strain theta) due to shear stress T

fluid rate of deformation is directly related to...

shear stress - velocity gradient is also directly related to shear stress - right equation is deformation rate, left equation is velocity gradient

shear flow of newtonian fluids in 1D

shear stress can be expressed by linear relationship in image *** notice that this is the same eqn as before, but is exact and not proportional

stress and inertia relation

sigma = MY/I

incompressible flow (special case)

simplifies the case - usually occurs for liquids conservation of volume (flow rate) - air is compressed inside tank to allow for larger flow rate out of tank

pressure in straight line example

since they are at the same depth, pressure is the same at all points PH=/=PI bcs the fluids have different densities

velocity and acceleration field

since velocity is a vector, there are three components to it - acceleration just takes the derivative of velocity

proportionality of stress vs strain

solids - stress is proportional to strain - eventually stops deforming at some fixed STRAIN ANGLE fluids - stress is proportional to strain rate - never stops deforming, however it approaches a constant RATE OF STRAIN

density and specific volume

specific volume is volume per mass

stable vs neutrally stable vs unstable

stable - centre of gravity (G) is directly below centre of buoyancy (B) neutrally stable - G and B coincide unstable - G is directly above B ADD PIC

continuity equation

states that volumetric flow rate at two points is equal

statics vs dynamics

statics: branch of mechanics that deals with bodies at rest dynamics: branch of mechanics that deals with bodies in motion

steady uniform vs steady non-uniform vs unsteady uniform vs unsteady non-uniform flow

steady & uniform - same with time and location steady but non-uniform - flow is same at every point in time at ONE specific point, however it varies at different locations unsteady but non-uniform - also changes with location as well as time unsteady & uniform - not consistent with time however its the same with location dv/dt = change in velocity/change in time dv/ds = change in velocity/change in location

steady vs unsteady flow

steady flow - no change of fluid properties (velocity, pressure) at a point in time unsteady flow - fluid properties change at a point with time *** velocity vs time (temporal)

hydraulics

studies liquids flowing in pipes, ducts, and open channels

system, surroundings, boundary

system: quantity of matter or space chosen to study (can be open or closed) surroundings: mass or region outside the system boundary: real or imaginary surface that separates the two (can be fixed or movable)