Fluid Dynamics Final

Transport phenomena

momentum, heat, mass, fluid mechanics

flux

rate of transfer or flow of a physical quantity per unit area per unit time

fluid

substance that deforms continuously under the action of a shear stress (gases/liquids)

continuum

continuous distribution of matter

The smallest volume of interest in a continuum is large enough \_____

to make statistical averages meaningful

Continuum NOT applicable

for small volumes of gas at rest because time dependent

Continuum variation in properties is smooth

to make differential calculus useful for analysis

Density

mass/volume

Fluid property

density, specific heat, viscosity, surface tension

flow property

velocity, shear stress, concentration flux, temp flux

stress

force/area

2nd order tensor

an ordered array which is completely determined by specifying its 9 components

Surface forces

friction, pressing, shear stress

Body Forces

gravity, magnetism, electrical potential

compressible fluid

fluid where density varies under normal conditions

incompressible fluid

fluids where density is constant

surface tension

work associated with creating a new surface

capillary pressure

a pressure difference resulting from surface tension

Contact Angle is 0

for a clean glass tube

Newton's First Law

a body will remain at rest or in motion unless acted upon by an external force

Newton's Second Law

the sum of the forces on an object is equal to mass x acceleration

Newton's Third Law

to every action there is an equal and opposite reaction

Inertial reference frame

the coordinate system is fixed with respect to the earth

non-inertial reference frame

uniformly translating coordinate system

manometer

device for measuring pressure

Pascal's principle

the pressure in a fluid at rest must be the same at all points with the same elevation

Absolute pressure

gauge pressure + atmospheric pressure

buoyant force

vertical component of the pressure force integrated around the entire surface of a solid

Archimedes principle

a floating body displaces a volumes of fluid whose weight is equal to its own

field

a quantity defined as a function of position and time throughout a given region

steady flow

flow that is independent of time

unsteady flow

flow dependent on time

streamline

a line everywhere tangent to the velocity vector at a given instant in a flow field

pathline

the actual path of the fluid

Streamlines and pathlines

are exactly corresponding in steady flow

system

a collection of entities enclosed in a set of identifiable boundaries

control volume

region in space through which fluid flows

open system

allows mass transfer across boundaries

closed system

does not allow mass transfer across boundaries

extensive property

depends on an element of matter (mass, energy, momentum)

intensive property

independent of amount of matter (pressure, temperature, color)

Inertial Delta P \=

rho g

Non intertial Delta P \=

rho (g-a)

Inertial F\=

0

Noninertial F\=

ma

mass flow rate \=

pho v A

volumetric flow rate

v A

Soap surface pressure delta P \=

4 sigma / R

1st Law of Thermodynamics

Energy can neither be created nor destroyed

Positive Work

done by the system

Negative Work

done on the system

Bernoulli assumptions

1. Steady flow
2. incompressible
3. inviscid
4. isothermal
5. no change in internal energy
6. no shaft work

Toricelli eq conditions

small inlet, large outlet

head loss

change in internal energy

viscosity

property of a fluid to resist the rate at which deformation takes place when a fluid is acted upon by a shear stress

Newtonian fluid

constant viscosity, linear relationship between shear stress and shear rate

Non-newtonian fluid

does not obey Newton’s Law of viscosity, shear stress and shear rate not linearly related

pseudoplastic

materials that decrease in viscosity with increased shear, shear-thinning

dilatent

materials that show an increase in viscosity with increase shear, shear-thickening

bingham plastic

requires a finite yield stress before flow begins

no slip boundary condition

fluid at the boundary will move at the same velocity as the boundary

laminar flow

flow is gentil with parallel streamlines

fully developed flow

steady flow, no entry/exit effects, flow does not vary along the axis of the flow

shear stress profile in circular pipe

approximately 0 in the middle and maximum at the walls

HP equation assumptions

1. laminar
2. steady
3. FDF
4. continuous
5. newtonian
6. no slip
7. incompressible

Partial time derivative

local derivative, assumes fixed position observer

total time derivative

d/dt, moving observer

substantial time derivative

D/Dt, observer moving with system, time rate of change of a fluid or flow variable along the path of fluid element

Continuity equation

DP/Dt = -rho( del dot v), describes the rate of change in the density as seen by an observer moving with the fluid

Continuity eq for incompressible fluid

del dot v = 0

Vector form of NS eq

rho(Dv/Dt)=rho(g)-delP+mu(del^2 v)

Assumptions for NS eq

incompressible, constant viscosity, laminar/turbulent

Euler’s eq

for inviscid flow, rho(Dv/Dt) = rho(g)-delP

vorticity

rotation of a fluid element, a measure of the moment of momentum of a small fluid particle about its own center of mass

vorticity (mathematical)

the curl of the velocity vector

irrotational flow

regions where flow properties have no net rotation, inviscid regions of flow far away from solid walls

kinematics

a branch of dynamics dealing with motion in time and space disregarding mass and force

Velocity potential

exists in 3D, flow must be irrotational

Orthogonal

The velocity potential and stream function are this to each other

source flow

radially upward from an origin

sink flow

directed toward an origin

geometric similarity

exists between 2 systems if ratios of significant dimensions are equal

kinematic similarity

if geometric sim exists and all velocity ratios are equal

dynamic similarity

if all relevant dimensionless numbers are identical between 2 systems

foundational units

mass, length, time

Froude number

inertial force/gravity force: v^2/gL

Euler number

pressure force/inertial force: P/rho v^2

Reynold’s number

rho v L/mu

2300

Re threshold for circular pipes

drag force

caused by shear stresses at the surface of a solid object moving through a viscous fluid

BLT hypothesis

the effects of fluid motion at high Re are limited to a thin layer of fluid near the boundary

BLT assumptions

a thin boundary layer at high Re, incompressible flow, 2D flow over flat plate

3x10^6

Re threshold for turbulent flow on a flat plate

2x10^5

Re threshold for laminar flow on a flat plate

wall turbulance

contact of the flowing stream with solid walls or boundaries

free turbulance

contact of 2 layers of fluid with different velocities

eddies

groups of fluid particles of various sizes

Leq

length of pipe that produces head loss equivalent to the head loss in a particular fitting

elevation head

fluid elevation above an arbitrary level, proportional to liquid’s PE

velocity head

height of a column of liquid that can be supported when a liquid is forced to stop, proportional to KE