3. Rheological Behavior of Fluids

Summary

This script explores the complex behavior of non-Newtonian fluids, contrasting them with Newtonian fluids through various experiments. It delves into the concept of rheology, the study of deformation in response to force, and highlights how non-Newtonian materials, such as high polymer solutions, exhibit unique properties like memory, nonlinearity, and yield stress. The video demonstrates how these fluids can bounce like solids, support weight, and display shear thinning or thickening behavior, challenging traditional fluid dynamics models and showcasing their practical implications in industries like food processing and manufacturing.

Takeaways

• 🔍 Rheology is the study of the relationship between force and deformation in continuous media, with a focus on fluids in this script.
• 🌊 The concept of an inviscid fluid is introduced, where there are no shear stresses and pressure is isotropic.
• 📚 The Navier-Stokes equation is derived from the stress-deformation relation for Newtonian fluids, which are independent of the observer.
• 🚫 Traditional models are inadequate for materials with large molecules, such as high polymer solutions, which exhibit non-Newtonian behavior.
• 🔨 The script demonstrates a high polymer solution's ability to support a shearing stress up to a critical yield value, highlighting non-Newtonian characteristics.
• 🔄 The memory fluid model is presented as a more general model that accounts for the history of deformation, showing nonlinear stress functions.
• 🧪 Experiments are conducted to show that stress in viscoelastic materials depends on the history of deformation, with examples like honey and high polymer solutions.
• 📉 The script explains how nonlinearity in non-Newtonian fluids affects both shear and normal stresses, leading to unique flow behaviors.
• 📈 Pseudoplastic or shear-thinning behavior and dilatant or shear-thickening behavior are described, showing how viscosity changes with flow rate.
• 🌀 Non-Newtonian fluids can exhibit normal stress differences, leading to phenomena like climbing up a rotating shaft, different from Newtonian fluids.
• 🛠️ The script concludes with complex flow experiments, such as extrusion and rotational flow, to illustrate the unique characteristics of non-Newtonian fluids.

Q & A

What is rheology in the context of this script?

Rheology, in the broadest sense, is the study of the relationship between force and deformation in continuous media. In this script, the focus is specifically on fluids.

What is the basic assumption behind the stress-deformation relation in the model of an inviscid fluid?

In the model of an inviscid fluid, the basic assumption behind the stress-deformation relation is that there are no shear stresses, or equivalently, that the pressure is isotropic.

What is the significance of the Navier's Stokes equation in the context of fluid dynamics?

Navier's Stokes equation is significant as it is derived from the stress-deformation relation for a Newtonian fluid and the momentum equations. It helps in understanding the motion of viscous fluid substances.

What is a Newtonian fluid and how does it behave?

A Newtonian fluid is a fluid in which the shear stress is linearly related to the rate of shear strain. It has no shear stresses with no relative motion, and the pressure is isotropic. However, in relative motion, the stresses are linear functions of the instantaneous velocity gradients.

Why are traditional models inadequate for materials containing large molecules?

Traditional models are inadequate for materials containing large molecules because they do not account for the complex behavior of these materials, such as their ability to bounce like an elastic solid or to hold their shape like mayonnaise, which are not explained by simple Newtonian models.

What is a memory fluid and how does it differ from a Newtonian fluid?

A memory fluid is a more general model of fluid behavior where the history of deformation is significant. Unlike Newtonian fluids, the stress in a memory fluid is a nonlinear function of the history of the deformation gradient. This means that the stress depends not only on the current state but also on the past states of deformation.

What is the yield value in the context of materials like clay suspension?

The yield value is a critical stress value that a material can support up to a point without flowing. Materials like clay suspension do not flow until a certain yield value is exceeded, indicating a transition from a solid-like to a fluid-like state.

What is viscoelastic behavior and how does it relate to the stress history?

Viscoelastic behavior refers to the properties of materials that exhibit both viscous and elastic characteristics. It relates to the stress history in that the material's response to stress depends on the duration and history of the applied stress, showing a memory effect that influences its flow and deformation.

What is the phenomenon of shear thinning and how does it differ from shear thickening?

Shear thinning, or pseudoplastic behavior, is when the viscosity of a fluid decreases with increasing shear rate. In contrast, shear thickening, or dilatant behavior, is when the viscosity increases with increasing shear rate. These behaviors are indicative of non-Newtonian fluids that do not follow the simple linear relationship between shear stress and shear rate.

How do normal stresses affect the behavior of non-Newtonian fluids in flow?

Normal stresses in non-Newtonian fluids can lead to phenomena such as the climbing of fluid up a rotating shaft or the unequal distribution of stress across different parts of a flow system. These effects are due to the nonlinearity in the stress-deformation relation, which is a characteristic of non-Newtonian fluids.

What is the significance of the velocity profile in the flow of non-Newtonian fluids through tubes?

The velocity profile in the flow of non-Newtonian fluids through tubes can be very different from the parabolic profile typical of Newtonian fluids. This is because the flow rate and velocity field are governed by a viscosity function that can vary with the shear rate, leading to complex flow patterns and behaviors.

Outlines

00:00

🌊 Rheology and Newtonian Fluids

This paragraph introduces rheology, the study of the relationship between force and deformation in fluids. It focuses on the behavior of fluids, particularly Newtonian fluids, which exhibit a linear relationship between stress and velocity gradients. The Navier-Stokes equation is mentioned as a fundamental model for fluid dynamics. However, the paragraph also highlights the limitations of this model, especially when dealing with materials containing large molecules, such as high polymer solutions, molten plastics, egg white, paint, and mayonnaise. These materials do not follow the simple Newtonian model and exhibit complex behaviors like elasticity and viscosity.

05:01

🔍 Memory Fluids and Viscoelasticity

The second paragraph delves into the concept of memory fluids, which are non-Newtonian and exhibit a history-dependent stress response. Unlike Newtonian fluids, the stress in memory fluids is a nonlinear function of the deformation gradient's history. Experiments are described that demonstrate the viscoelastic properties of these fluids, showing both viscous and elastic characteristics. The paragraph also discusses how these properties affect the behavior of materials under stress, such as the fluid ball that behaves like an elastic solid during impact and the high polymer solution that shows a fading memory effect.

10:02

🌀 Non-Newtonian Behavior and Pseudo Plasticity

This paragraph explores the nonlinear behavior of non-Newtonian fluids, focusing on their time-dependent characteristics and how they differ from Newtonian fluids. Experiments are conducted to show that the stress in these fluids depends on the history of deformation. The concept of pseudo plasticity, or shear thinning, is introduced, where the viscosity decreases with increasing velocity gradient. Conversely, dilatancy, or shear thickening, is also discussed, where the viscosity increases with higher flow rates. The paragraph also explains how these behaviors affect the flow rate and velocity field in fluids, leading to different velocity profiles from the typical parabolic shape seen in Newtonian fluids.

15:04

📊 Normal Stresses and Non-Linearity

The fourth paragraph discusses the effects of non-linearity on normal stresses in fluids. It describes experiments that show how the stress exerted by a fluid can be greater on the inner cylinder compared to the outer cylinder, contrary to the behavior of Newtonian fluids. The paragraph also explains how these normal stress effects arise from the inequality in the stress deformation relation. The concept of a logarithmic relation between stress and distance from the axis of rotation is introduced, which is expected for all memory fluids. The paragraph concludes with a discussion on how these normal stress phenomena can be observed in different geometries and how they are related to the non-linear stress deformation relation.

20:06

🚀 Complex Flow Phenomena in Non-Newtonian Fluids

The final paragraph presents more complex experiments involving non-Newtonian fluids, highlighting their unique flow phenomena. It discusses how non-Newtonian fluids can exhibit expansion when emerging from an orifice, unlike Newtonian fluids. The paragraph also touches on the extrusion process in plastic manufacturing, where the dye must be designed smaller than the final product due to the flow characteristics of non-Newtonian fluids. The paragraph concludes by emphasizing the importance of being aware of time-dependent and non-linear characteristics when dealing with new fluids, especially those containing high polymers and suspensions.

Keywords

💡Rheology

Rheology is the scientific study of the flow of matter, primarily in liquids but also in solids, to understand the relationship between force and deformation in continuous media. In the context of the video, rheology is crucial for analyzing how fluids behave under different conditions, such as in flow fields and under various stress conditions.

💡Inviscid Fluid

An inviscid fluid is an idealized fluid that assumes no viscosity, meaning it has no internal resistance to flow. The video uses this concept to discuss the basic assumptions behind the stress-deformation relation in fluid dynamics, where the pressure in an inviscid fluid is isotropic, implying uniform pressure in all directions.

💡Newtonian Fluid

A Newtonian fluid is a type of fluid whose stress is linearly proportional to the rate of deformation. The video highlights that water and glycerin are examples of Newtonian fluids, where the pressure is isotropic in the absence of relative motion, and the stresses are linear functions of the instantaneous velocity gradients.

💡Navier's Stokes Equation

Navier's Stokes equations are a set of equations that describe the motion of fluid substances. The video mentions these equations in the context of formulating the stress-deformation relation for a Newtonian fluid, which, when coupled with momentum equations, helps in understanding fluid flow dynamics.

💡Viscoelastic Fluid

Viscoelastic fluids exhibit both viscous and elastic characteristics. The video uses this term to describe materials like high polymer solutions, which can bounce like an elastic solid during short impacts but also flow like a fluid over time. This behavior is crucial in understanding materials that do not follow the simple Newtonian model.

💡Yield Value

The yield value is a critical stress level that a material can support before it starts to flow. The video illustrates this concept with clay suspension, which does not flow until a certain weight is applied, indicating the presence of a yield value in some non-Newtonian fluids.

💡Memory Fluid

A memory fluid is a type of non-Newtonian fluid where the stress is not only dependent on the current deformation rate but also on the history of deformation. The video explains that unlike Newtonian fluids, memory fluids show nonlinearities in both shear and normal stresses, making them more complex to model and predict.

💡Pseudo Plastic

Pseudo plastic, or shear thinning, behavior is exhibited by fluids that become less viscous as the rate of shear increases. The video demonstrates this with an experiment where the flow rate of a polymer solution increases more than proportionally to the increase in pressure head, indicating a decrease in viscosity with higher velocity gradients.

💡Die Lent

Die lent, or shear thickening, is the opposite of pseudo plastic behavior, where the viscosity of a fluid increases with increasing shear rate. The video mentions this behavior in the context of a suspension, where the viscosity is higher at higher rates of flow, contrasting with the behavior of Newtonian fluids.

💡Normal Stresses

Normal stresses are forces that act perpendicular to the direction of flow in a fluid. The video discusses how non-Newtonian fluids can exhibit unequal normal stresses in different directions, leading to phenomena like the climbing of fluid up a rotating shaft, which is different from the behavior of Newtonian fluids.

💡Non-Linearity

Non-linearity in fluid dynamics refers to the relationship between stress and deformation that does not follow a straight line, as opposed to the linear relationship in Newtonian fluids. The video emphasizes that non-linearity affects both shear and normal stresses in non-Newtonian fluids, leading to complex flow behaviors that cannot be easily predicted.

Highlights

Rheology is the study of the relationship between force and deformation in continuous media.

Focus on the behavior of fluids, particularly outside the viscous boundary layer where water behaves almost like an inviscid fluid.

The inviscid fluid model assumes no shear stresses and isotropic pressure, leading to the equation of motion.

Glycerin and water are examples of Newtonian fluids, which have no shear stresses and isotropic pressure without relative motion.

Newtonian fluids exhibit linear stress functions of the instantaneous velocity gradients during relative motion.

Navier's Stokes equation is derived from the stress-deformation relation for Newtonian fluids, coupled with momentum equations.

Some materials, like high polymer solutions, do not follow the Newtonian model and exhibit non-Newtonian behavior.

Non-Newtonian materials can support a shearing stress up to a critical yield value, as demonstrated by clay suspension experiments.

Memory fluids, unlike Newtonian fluids, have stress that is a nonlinear function of the history of deformation gradients.

Viscoelastic behavior of materials is demonstrated through experiments showing stress depending on the history of deformation.

Honey and high polymer solutions are examples of viscoelastic fluids, exhibiting both viscous and elastic characteristics.

Non-linearity in stress-deformation relation results in unusual normal stress effects, such as cake batter climbing a mixing shaft.

Pseudo plastic or shear thinning behavior is observed where viscosity decreases with higher velocity gradients.

Dilatant or shear thickening behavior is the opposite, where viscosity increases with higher rates of flow.

Non-Newtonian fluids can have a viscosity function that governs flow rate and velocity field, differing from the parabolic profile of Newtonian fluids.

Normal stress differences in non-Newtonian fluids can lead to phenomena like fluid climbing around a rotating center tube.

Non-linearity affects not only shear stresses but also normal stresses, leading to unique flow behaviors in various geometries.

Extrusion processes for plastic articles must account for non-Newtonian fluid behavior, as the dye must often be designed smaller than the final product dimensions.

High polymer solutions demonstrate unique flow patterns, such as spiraling inward at the equator and outward at the pole, unlike Newtonian fluids.

When dealing with new fluids, especially those containing high polymers and suspensions, it's crucial to consider time-dependent and non-linear characteristics.

Transcripts

rheology in its broadest sense is a

study of the relationship between force

and deformation in continuous media in

this film we shall focus our attention

on fluids we have marked a portion of a

flow field outside the viscous boundary

layer here water behaves almost like an

inviscid fluid in the model of the

inviscid fluid the basic assumption

behind the stress deformation relation

whether the fluid is a Calibri 'm or in

motion is that there are no shear

stresses or equivalently that the

pressure is isotropic this statement

when combined with the balance of

momentum equation leads to orders

equation of motion in this flow glycerin

shows evidence of shear stresses water

and glycerin are two of many fluids that

behave in a manner called Newtonian with

no relative motion the Newtonian fluid

has no shear stresses the pressure is

isotropic in relative motion however the

stresses are linear functions of the

instantaneous velocity gradients if we

formulate this stress deformation

relation for the atonium fluid so that

it is independent of the observer and

couple with momentum equations we obtain

the navier's stokes equation there are

many situations however for which these

models are quite inadequate this is

especially the case when you are dealing

with materials containing large

molecules this ball for example bounces

quite vigorously like an elastic solid

yet it really is a fluid we'll come back

to it later here's the viscous material

which I can pour into this beaker

no I think I'll juice this speaker

instead I can even cut it to length this

material is a solution of a high polymer

some other materials that do not follow

the simple Newtonian model are molten

plastics egg white paint and mayonnaise

mayonnaise holds its shape if you let it

alone yet it spreads with ease here is a

more controllable experiment

illustrating the same phenomenon there

is no stop clock at the bottom and this

vessel is open to the atmosphere on top

and yet this clay suspension does not

flow I can put this simple piston here

place this small weight on it still no

flow now a larger weight it flows but

very slowly a much larger weight the

flow is fairly rapid

I take the weight off the flow stops in

a state of equilibrium this material can

support a shearing stress up to a

critical value called a yield value even

with materials not having yield values

the Newtonian fluid cannot explain many

flow phenomena for many of these a more

general model that of the memory fluid

does provide an explanation for the

memory fluid the pressure is isotropic

in the equilibrium state where no

changes of either the stress or the

deformation are taking place in relative

motion however in contrast to Newtonian

fluid the history of the deformation is

significant instead of a linear function

the memory fluid reveals nonlinearities

in fact non-linearity affects both the

shear stresses and the normal stresses

so forth a memory fluid the stress is a

nonlinear function of the history of the

deformation gradient for this model to

flow problems are solved by expressing

this assumption mathematically and using

the momentum equations first we will

look experiments demonstrating that

stress does depend on the history of the

deformation this material exhibits both

viscous and elastic characteristics

honey has a high viscosity but no

elasticity

remember the fluid ball it was like an

elastic solid during the short impact of

bouncing an elastic solid has the

preferred configuration to which it will

return when stresses are removed a

viscoelastic fluid behaves similarly if

the stress has been applied only for a

short time but in contrast to the

elastic solid the fluids memory is not

perfect this time-lapse sequence shows

that if the stress acts for a time long

compared with the relaxation times of

the fluid the fluid forgets its previous

state in reality it took 45 minutes for

the force of gravity to change the ball

into a puddle this is a container of

high polymer solution I'll drop in the

steel ball again once more in slow

motion this liquid too is viscoelastic

here's some more polymer solution I can

put this cylinder into it

and turn it a full 360 degrees when I

let go quickly

it returns almost half the distance

this time I'll hold it for a few seconds

since the fluid has a fading memory when

I do let go the recovery is a much

smaller fraction of the original

deformation we can make quantitative

measurements of viscoelastic behavior in

this type of apparatus the test liquid

is defined in this annulus the outer

cylinder is held stationary and we can

apply a torque to the inner one with

this weight which we can hang there a

trigger here releases the mechanism to

remove the torque there is a slip toggle

there we can record the response of the

inner cylinder that is its angular

rotation as a function of time with the

pen on a chart moving at constant speed

the weight of the recording pen is

counterbalanced by this small weight we

have of course designed this experiment

to minimize extraneous effects arising

from the inertia of the fluid the

inertia of the moving parts and from

friction the first experiment is with

the Newtonian fluid designated by the

letter n a constant velocity is attained

almost at once since inertial effects

are small

now they apply torque is zero again and

the cylinder stops immediately

this time the fluid and the annulus is

viscoelastic a constant velocity is not

immediately obtained even though the

stress is constant the deformation and

the rate of deformation are changing

with time now the weight is disengaged

let's watch that again

it takes a while for the viscoelastic

fluid to acquire the steady-state motion

appropriate to the constant torque when

we remove the torque we see the cylinder

actually reverse its direction although

no torque is being applied the material

is deforming because of the elastic

character of the fluid not all the

energy used to produce the flow was

dissipated some was stored and then

recovered we have looked at some

experiments showing that the stress

depends on the history of the

deformation to illustrate nonlinear

behavior we shall do experiments in

which the time-dependent characteristics

of the fluids are negligible here we

have two identical reservoirs with two

identical outlet tubes one contains a

Newtonian fluid the other a

non-newtonian the levels are the same

at first the non-newtonian fluid flows

faster however after a while it is

overtaken by the Newtonian fluid the

basic phenomenon occurring here is more

easily seen if the pressure head remains

practically constant these are two

identical burette containing the same

Newtonian fluid the pressure head in one

is twice that in the other the ratio

rates the flow is also two-to-one as we

expect from poor sods law

but if we charge both burette with the

fluid exhibiting nonlinear behavior and

apply the same ratio of pressure heads

the result is quite different with this

polymer solution when the head is

doubled the rate of flow is more than

doubled it appears that the viscosity is

less at the higher velocity gradient

this type of study flow behavior is

called pseudo plastic or shear thinning

there are some fluids that create a

contrary situation for this suspension

the viscosity is higher at the higher

rates of flow this is called dye latent

or shear thickening behavior

and the studied laminar flow of

incompressible Newtonian fluids through

tubes a single material constant the

coefficient of viscosity governs the

volume rate of flow and the velocity

field here we get the familiar parabolic

velocity profile however for more

general fluids the flow rate and the

velocity field are governed by a

viscosity function for such a fluid the

velocity profile can be very different

from parabolic

we have seen how non-linearity affects

your stresses it also has an unusual

effect on the normal stresses here are

some glycerin and here is one of my

wife's favorite recipes

why does the cake batter climb the

mixing shaft in contrast to the behavior

of Newtonian fluid we can get a better

idea of what happened there if we look

at a simpler experimental configuration

this shaft is mounted to rotate inside

the larger glass tube in a coaxial

geometry the shaft is a hollow tube with

the hole through the wall the force per

unit area exerted by the fluid in the

direction normal to this cylindrical

surface will be indicated by the level

of the fluid inside this manometer tube

the stress normal to the outer cylinder

will be shown by the level of the fluid

in this side our manometer tube attached

over a hole in the outer cylinder when

the central shaft is rotated a shear

flow is established in the annulus the

polymer solution climbs up around the

rotating Center tube as the cake batter

did because the fluid is so viscous it

took about an hour for the levels of

fluid inside the tubes to indicate the

steady-state stress difference

note that the stress exerted by the

fluid normal to the cylinder walls is

greater on the inner cylinder and on the

outer this is in contrast to the

behavior of a Newtonian fluid where the

centrifugal field alone governs the

pressure distribution here is another

apparatus where we can observe a related

phenomenon this time the material shared

between two parallel disks one of which

can rotate while the other is stationary

the distribution of stress normal to the

stationary disc will be shown by the

level of the fluid in these glass tubes

of course this level will be uniform in

the tubes if there is no rotation

remember this time we are not concerned

with the flow and the annulus between

the vertical walls but with the shear

flow in the narrow space between the

flat plates

if the fluid is Newtonian the height of

the fluid in the tubes at steady state

is a little less near the center than

toward the outside for a non-newtonian

fluid at the same speed the force normal

to the stationery plate may be

considerably greater toward the center

although the steady-state stress

distribution between the plates is

obtained almost immediately again it

took several hours for the levels in the

tubes to reach their final positions

indeed quite high stresses can occur

near the center and this principle has

been used in the design of a pump for

molten plastics if the bottom disc is

replaced by a very shallow cone the

stress normal to the stationery disc

varies linearly with the logarithm of

the distance from the axis of rotation

this logarithmic relation is expected

for all memory fluids these normal

stress effects come directly from the

non-linearity in the stress deformation

relation this diagram represents steady

simple sharing between infinite parallel

plates in addition to the shear stresses

there are also normal stresses for the

Newtonian fluid the normal stresses in

these three directions are all the same

with non-linearity in general there will

not be the same we have just seen some

experiments in cylindrical geometries it

is the inequality of the normal stresses

on this intro test modem element that

gives rise to the normal stress

phenomena which we have just observed we

have looked at some simple flow

situations now let's look at two more

complicated experiments this tank

contains two fluids each of which is in

its own compartment but both of which

can be subjected to the same air

pressure one of the fluids exhibits

Newtonian

behavior the other non-newtonian the

compartments have identical small

orifices through which the liquids can

be forced notice that with the

non-newtonian fluid there is a

considerable expansion of the stream as

it emerges when plastic articles are to

be made by the extrusion process the dye

must often be designed smaller than the

dimensions desired in the finished

product here a sphere is rotating in a

high polymer solution dye introduced at

the left reveals the flow spiraling

inward at the equator and outward at the

pole for a Newtonian fluid on the other

hand where centrifugal forces are

dominant we would look for flow inward

at the poles and outward at the equator

we have Illustrated some phenomena that

can occur with non-newtonian fluids the

degree to which they occur depends on

the particular material and the

particular flow situation in any case

when dealing with new fluids especially

those containing high polymers and

suspensions one should be on the lookout

for time-dependent and non-linear

characteristics