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YEAR 2 Sedimentology L1 Part 2 Newton's Laws, fluid stresses and viscosity

Terms in this set (19)

Newtons Laws

what do these define and how many are there ?
Newton has a 1st 2nd and 3rd law

These are the laws of mechanics = any element of fluids will follow this =

Here applying to motion of fluid but newtons laws also true for solids
What is Newtons 1st law?

hint - what 2 thing will a fluid element remain at ?
until when?

why's is it that this is possible?
A fluid element will remain
AT REST or in a STATE OF UNIFORM MOTION (without acceleration) in a straight line until acted on by an external force.

Because although there may be other forces acting on it but if they SUM TO ZERO = then at rest
What is Newtons 2nd law?

hint - the rate of change of momentum is prop to what?
how does this relate to the force applied and direction?

what do you need to remember about force?
The rate of change of momentum of a fluid element is proportional to the force applied and takes place in the direction of that force

Force = as a VECTOR = so will always have DIRECTION
What is Newtons 3rd law?

What is this a consequence of ?

Ie imagine book on table = the weight of this book -
going into down table = so we know must be equal and opposite force acting up for that book to be stationary = what is this force ?

Give an example of a N.......... R.......... F......
3rd law = The MAGNITUDE of a force and its reaction are EQUAL and are in opposite directions.
(every action has an equal and opposite reaction? )
3rd law is a consequence of Newtons 1st law

This force opposing weight = is called the NORMAL reaction force
Momentum is the product of what ?

so for a fluid element of constant mass - Newtons 2nd law relates what? - and to what?
what do we call this force?

what is the equation for the inertial force?
MOMENTUM = product of MASS and VELOCITY

Newtons 2nd law relates the change of velocity occurring in a given time (ie the ACCELERATION) to
The applied force (the INERTIAL force)

Rate of change of momentum = region of constant mass - take time derivative of momentum = equal to time derivative of velocity = come up with F = ma
F in F= ma has a name = this is the inertial force
position, velocity, acceleration

Velocity is the derivation of what?

Acceleration is the derivation of what ?
Velocity = the derivation of POSITION. respect to time

Acceleration = derivation of VELOCITY with respect to time

(need to look more into derivatives etc in the physics refresher he put on blackboard)
what does a velocity time graph look like ?

what does an acceleration time graph look like? - ie if constant acceleration
velocity is on the y axis and time is on the x axis

for constant velocity = will be a linear (diagonal) line from the origin

acceleration time graph = acceleration on y time on x

horizontal line form the value of acceleration if constant
Forces on fluids and stresses

An object in a fluid will experience forces due to what?

In general the MAGNITUDE of the force will depend on what?

How does this help us define stress - what equation =

Since force is a VECTOR quantity - what does this mean for stress?

what two stresses do we usually need to distinguish between ?
due to the MOLECULAR and BULK motions of the fluid

mag of force = depends on size of object = specifically its Surface Area

SO can define stress as the force PER UNIT AREA

STRESS = force exerted/area of boundary

force = vector quantity so stress has DIRECTION
(remember... VECTOR! with DIRECTION ANNDDD MAGNITUDEEEEE)

need to distinguish between NORMAL and SHEAR stresses
Pressure and stress = similar but have a KEY distinction between the two

PRESSURE

Imagine a square with lots of fluid molecules surrounding it

These fluid particles COLLIDE with the edges of the square - what does this transfer and what is the result of this?

How is this force directed towards the boundary? - with a magnitude that is what?

The resulting n........ s.... is called what?

and how can we actually;;lab define this pressure?
These collisions transfer MOMENTUM - generating FORCE on the boundary

The force is directed normal to the boundary - with a magnitude that is PROPORTIONAL to the AREA of the BOUNDARY SURFACE - ie if double the are = double the molecules

The resulting Normal stress = called the FLUID Pressure,P

Pressure = FORCE oer UNIT AREA that acts in ALL DIRECTIONS

- Occurs for both fluids at rest (ie STATIC pressure that the diver feels) and in motion
If there is a bulk fluid motion, what does this mean for the molecules?

ie imagine bulk flow dir is to the right above the square = so there are MORE COLLISIONS on the upstream boundary than the downstream boundary = what does this mean for PRESSURE?

how would we describe this?

If more collisions on the upstream side has created a pressure difference(gradient) = what does this mean as a consequence ?

SO pressure gradients can lead to what?
If there is a bulk fluid motion, the molecular motions have a preferred direction.​

Means he pressure is HIGHER on the upstream boundary.​

would say there is a PRESSURE GRADIENT

pressure difference(gradient) = means there's a difference in FORCE (FORCE IMBALANCE)- so there must be a CHANGE IN MOTION
- as stated by Newtons laws

So pressure gradients can lead to MOTION!
Shear Stress

If there is a gradient of VELOCITY within the fluid - what can be generated?

back tot he square analogy - have molecules moving in left to right flow - FAST flow upstream at top of square boundary
SLOW flow downstream at bottom of square boundary

In both cases, Collisions with the boundary transfer what?
- What does this lead to ?

If there is a velocity gradient then there can be what?
- as higher velocity molecules have what?

so this diff in force or stress form top to bottom would mean what for this perfect square?

how do we know this is shear stress not normal stress?

so to summarise , Gradient in velocity = what? = which creates what?

What can we relate the ease of deformation o this material to ?

meaning we can quantify what?
Then shear stress, 𝛕 can be generated

Collisions with the boundary transfer some MOMENTUM in the direction of the flow

Leads to a STRESS directed along the boundary

If there is a velocity gradient then there can be a force imbalance

higher velocity molecules have greater force assoc w them

Means that creates diff in stress from top to bottom = this stress diff acting in dir in flow - ie top would get more deformed than bottom. = called SHEAR stress​

As molecules acting/exerting force PARALLEL to the sides of square , not perpendicular (90*/NORMAL) to them

Gradient in velocity = imbalance in SHEAR stresses between top and bottom. = creates DEFORMATION

Can relate the ease of deformation of this material to the SHEAR STRESSthat's exerted - so means we can quantify VISCOITY​
VISCOCITY

We have seen that bulk properties are transported by fluid motion.​
This includes the fluid momentum, which leads to an important physical property of fluids: VISCOSITY

Imagine similar situation to square, but not square = now its a horizontal LINE representing a surface with fast flow ABOVE it and SLOW flow below

so there is a pressure gradient form left to right and a difference in MOMENTUM

So what occurs across the surface (the line) where there is a gradient of VELOCITY? - what must be assoc with this?

As a result of this velocity gradient - what occurs as a consequence?
what new thing is formed due to this? - and how
Transfer of momentum occurs across the surface where there is a gradient in velocity.​
A FORCE must be assoc with this?

as prev said = transfer of momentum occurs = this has the effect of reducing the velocity gradient.​

Reduction in velocity gradient - do this by slowing down some of the fast molecules and speeding up some of the slow ones = creating molecules of an INTERMEDIATE velocity
Velocity pt 2

The effect of momentum transfer is that the slower moving fluid is accelerated by what? and equally the the faster moving fluidf is decelerated by what?

So the gradient of velocity characterises what?
​
The just mentioned accelerations MUST be due to what ?
what is this f... called?


Gradient of velocity is the derivation of gradient in y direction(across the flow) -
and the thing that relates this gradient and the shear stress applied is what?

whats the equation?

So The magnitude of the stress is proportional to what?
The effect of momentum transfer is that the slower moving fluid is accelerated by the faster moving fluid, or EQUALLY the faster moving fluid is decelerated by the slower moving fluid.​

gradient of velocity characterises the DEFORMATION
​
Must be due to a FORCE acting at the surface where there is a velocity gradient
Force = called the VISCOUS SHEAR STRESS


is the DYNAMIC VISCOSITY , 𝝁


𝛕 = 𝝁 x d𝒖 / d𝑦


Magnitude of the stress applied is proportional to the LOCAL VELOCITY GRADIENT and the DYNAMIC VISCOSITY
Dynamic Viscosity

What's the difference between something with a HIGH dynamic viscosity and a LOW dynamic viscosity?

Comon fluids ie air - relationship of stress and deform - expressed by what?

Fluids w a constant dynamic viscocty are what ?

Are all fluids Newtonian?
HIGH dyn viscocity = take a LARGE AMOUNT of shear stress to get the deformation​
​
Lower dyn viscocity - need LESS SHEAR STRESS ​
​
Ie why harder to stir bucket of cement vs water ​

Common fluids ie air - relationship of stress and deform - expr by CONTSANT μ (mui)

Fluids w constant viscocty = NEWTONIAN​

Not all are tho - ie toothpaste. = its viscosity is not constant through deformation = is non newtonian​
Dynamic Viscosity & Rheology

The study of fluid deformation is called what?

The rheology of a fluid refers to what?

​​
The rheology of a fluid is measured using what?
The study of fluid deformation is called Rheology.​

The rheology of a fluid refers to the form of the viscosity and viscous stresses.​
​
​
The rheology of a fluid is measured using a rheometer.
Non-Newtonian Rheology​

For a NEWTONIAN fluid (dyn viscosity is constant) - what happens when you increase the shear stress ?

Key element we're interested in = Newtonian vs plastic materials​

What happens to plastic materials if you exert a small amount of stress? - what if you add even MORE stress?
- so when do u get something to deform plastically?

what is this minimum amount of stress applied to get plastic deformation known as ?

what is this behaviour known as (has special name ) - and why is it key for us?

what are DILATANT materials?
Incr the shear stress = the rate of deform increases LINEARLY (see graph)

Plastic materials - exert small amou nt of stress = get no deformation - Add more = STILL no deformation

only get deformation when CRITICAL amount of shear stress shown by dashed line on graph (not on this one) is exceeded​

called YIELD STRESS - so plastic deform only happens when applied stress exceeds yield stress

known as BINHAM behaviour - Key for us as some particles In water form mixtures which have this bingham behaviour​

materials in which the dynamic viscosity increases as the shear rate increases
ie in geology = particle suspensions or fluidised solids
Viscosity and the no-slip condition​

Imagine we have a diagram with a solid stationary boundary at the bottom = ie a riverbed
and fluid flow parallel and touching it running left to right shown by arrows (see pic)

So if the riverbed is stationary (has zero horizontal velocity ) - what must the velocity of the fluid parcels in contact with the boundary be ?

As we move away from the boundary - the successive layers of fluid PARALELL to the boundary (shown by arrows) move with what?

This rate increases depends on what?

so what is this overall condition known as ?
Velocity of fluid @ boundary MUST be equal to the velocity of the boundary = so in this case, the fluid particles in contact = WILL ALSO BE STATIONARY


As we move away from the boundary - the successive layers of fluid PARALELL to the boundary (shown by arrows) move with INCREASING velocities -

Rate increase depends on VISCOSITY of fluid

Known as the NO SLIP CONDITION
THE NO SLIP CONDITION

What is this a consequence of?

so what does the no slip condition 𝒖 equal on a solid stationary boundary?
Consequence of the fluid viscosity and Newtons Laws

on solid stationary boundary, 𝒖 = 0

- can be applied to river system
going back to few slides ago
- what I sit that we have as we move away from the boundary (regarding velocity) =

What is the shape of this dictated by?
Have a VELOCITY GRADIENT away from boundary

The shape of which is DICTATED by a property of fluid ie how weekly it deforms
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