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)
