As part ofthe course on rheology andthe introduction module of itin

this particularlecture we will look atsome of the unusual flow phenomenawhich take place in

complex materials andalso equally important is how do we visualize those phenomenato be

caused due to microstructure so lets start with a situation which seem to be similar inrespect

for examplewe have a single sphere which is falling through a pool of liquid andof course

we learn aboutthis problemwhenever we look at a terminal velocityof a sphere falling in liquid and

we learn about stokes law andlets consider that thissingle spherethere are two casesone inwhich

thesphere is falling in a newtonian fluid and then compare the responsewith the non newtonian fluid

andwhat we can see isif the viscosities of the newtonian and non newtonian fluids are similar

then the falling rate in a newtonian fluid will be similar to the falling rate in non newtonian fluid

this is becausewe know that the settling velocity related directly to the viscosity of the fluid and

if the two viscosities are similar then in this case the falling rates will also be similar

now if the same two fluids basically the non newtonian fluid as well as a newtonian

fluidrather than lets say having a sphere falling through the pipe now if we justremove the bottom

of the if we remove the lets say bottom and allow fluid to flow through the bottom of the pipe then

what happens is overall fluid flows through and in this case againthe newtonian fluid as well

as non newtonian fluid will flow through but what is more likely thenwe will observe that the flow

rate of the newtonian fluid will be much lower than the flow rate of non newtonian fluids

in other words flow rate of non newtonian fluid will be higher or it would seem as if

the viscosity of non newtonian fluid would be lower so thereforewhat we are seeing here are

predominant examples of shear flow so in both the casesits predominantly shearwhich isinvolved and

just to characterize and quantify the intensity of the shear flow we can think of shear rate for

example here the velocity would be whatever is the velocity of terminal velocity of the

sphere and then it would go to zero near the wall and therefore you have a velocity

gradient or shear rate which is very small because in general the terminal velocity of

a small sphere is expected to be very low on the other sidewe have a basically tubular

flow of the fluid and in general the velocity here will be very high in the center and overall

flow rate will be very high and thereforetowards the tube where the velocity is zero to and that

velocity which is maximum basically what you will have is a shear rate which is very high so what

we can again summarizes in this situation shear rate is low while in this situation shear rate

is very high so the two situations that we have seen andwhere the similar fluids give completely

different responses because in one case they give a falling rate of the sphere which is

similar in the other case the flow rates of the fluid are very different and those are caused

because of the difference in the shear rates so effectively what we have ishigh shear rate

viscosity is lower and low shear rate viscosity is higher andjust to understand why in a complex

material such behavior can be observed what we can do is look atan example of colloidal

dispersion now colloidal dispersions arenothing but collection of particles andthese particles are

agglomerated in a network form with each other atreasonably high amount of colloidalparticles

and thensurrounded by a liquid so therefore you can see in thishow

there is a network of these colloidal particles and so this would be the situation at a low

shear rates also at rest and so when a sphere is falling throughthis kind of a very high network

of a colloidal particles then the sphere is likely to experience a higher viscosity and

there because of the larger clusters which these colloidal particles make on the other

hand if we now lets say apply a very high shear rate to this colloidal dispersion what ends

up happening is theselarge colloidal clusters actually start breaking up and now what we see

is large number of smaller isolated clusters so in because of the smaller clusters at this

higher shear rate the fluid viscosity would appear to be lower and thereforewhat we have is low

shear rate would correspond to large clusters of particles while high shear rate would correspond

to small clusters of particles and in turn large clusters would imply high viscosity and small

clusters would imply low viscosity now looking back at our two examples that we saw earlier what

we cansee isthe microstructure is very important to explain the results in both the cases in one

case we have microstructure of the material at rest when there was no shear rate at all

when we apply very small or low shear rate there is a small disturbance of that

equilibrium structure retaining large clusters of colloidal particles however when we have it under

flow and with large shear rate those clusters break down so therefore when we visualize the

two cases that we discuss in one case we have a sphere which is falling through basically a

strong network of colloidal particles and therefore this particle experiences high

viscosity and therefore the if newtonian and non newtonian fluids are of same viscosity we would

see that the particle falls at the same rate now when the same fluid the colloidal particle

system is subjected to high shear rate it ends up being in small clusters and low viscosity

therefore the flow rate is very high newtonian fluid on the other hand in both the cases has

same microstructure same molecular structure and therefore would have the same high viscosity in

both cases so it would have a small falling rate for sphere in this case and correspondingly it

would also have a very small flow rate so what we have seenin this discussion so far is thatwhen

fluids the complex materials are subjected to shear flow we canah quantify thedeformation

that the fluid is experiencing during flowing with shear rates andproperties of fluid seem to

be dependent on the shear rate very strongly andthe example that we saw was that of a shear

thinning fluid because the viscosity was lower at higher shear rate and more importantly we

also saw thatthe response of the material could be correlated back to the microstructure that

the material has now lets move on and look at another exampleah again now these flow phenomena

that we are seeing for complex materials one of the most important thing to remember is that they

are in fact qualitatively very different for example ifwe take a beaker in which there is a

fluid and then i put a rod and then i have the rod stirred with a motor if i rotate this rod

with a motor then what we would expect is at the center the fluid to go down and we would see that

the fluidair interface takes a shape which is concave upwards andthis iswhat we would observe

inwhen we lets say are stirring tea or milk using very high rates in inah day to day life

on the other hand if we take a non newtonian fluid and do a similar exercise what happens

is the fluid actually starts climbing onto the rod which is being used to stir and therefore

this phenomena is called rod climbing so in this case even though the two fluids are being

subjected to same situation their response is qualitatively different andthe importance of

rheology in general is that if you know small amountno amount of modification of newtonian

flow behavior is going to lead to this kind of qualitatively very different response

so clearly a different class of a fluid behavior is required for us to explain and observe

such phenomenon anotherexample which is a very practicalissue andwhen we are making a objects

out of polymer and these could be fibers for example polyester fiber which are used in clothing

or upholstery or it could be a grocery bagfilms thin films in all of these cases what is done is

we take a small capillary in whichthrough which fluid is forced out so generally we apply a very

high pressureat this end and thenthrough pressure we push the fluid out and this is what happens in

fact when we open a tap at home and then we have a newtonian fluid water flowing out of the tap

because the other end of the pipe line there is higher pressure quite often due to gravity because

the tank is located somewhere upstairs anddue to that pressure difference the fluid flows through

and we generally observe that the jet of the fluid is smaller than the diameter of the tap

andof course this is as long as we maintain a flow rate which is lower and such that the jet

doesn’t breakup and forms a spray in which case of courseit start spraying everywhere so we are

still looking at a jet nicely following and usually the diameter of the jet would be same

or so similar order of magnitude and slightly smaller in fact its about eighty to ninety

percent of a depending on the conditionsof the diameter of the tube on the other hand

if we take a non newtonian fluid and again carryout the same exercise and this is what

is done when a fiber is made or a sheet is made we basically have a whats called a die and there is a

die opening through which a fluid is extruded and thereforethis material which is extruded is

called extrudate andthedevice through which this extrusion happens is called a die and given that

the fluid when it comes out its diameter is much larger thenthe diameter of the tube this

phenomenon is referred to as die swell or extruded swell so again this is an example of an unusual

flow phenomena in the sense that the qualitative response itself is diametrically its opposite to

what you would see in case of newtonian fluid in case of newtonian fluid we see the extrudate

being smaller than the diameter of the tube or the die in the case of a a non newtonian

fluid we see that the diameter is higher andagain we cantry to understandwhat is the origin

of this phenomena by looking at the microstructure and so lets now look at a lets say macromolecular

solution so what we have here is a macromolecule which is a very long molecule andgenerally it

will adopt the conformation which is coil like and therefore we can encasethe overall macromolecule

in a hypothetical sphere andof course the radius of this sphere would be called the radiusof

gyration of this particular macromolecule and we are talking about a macromolecular solution

so therefore surrounding fluid is the solvent so we have a situation where we have a polymer

solution or a macromolecular solution being sheared and we can look at two different cases

let’s say where there is a high shear rate and there is a low shear rate so in case of

a shearshearing the material what we can do visualizes the lets say the top surface of

this domain is moving faster than the bottom surface we can think for example that the top

surface is moving with a velocity andwhile the bottom surface lets say stationary what that

does is basically as you go higher and higher the velocity would be higher and higher

so if you look at the macromolecule zoom in this is of course a considerably zoomed in version in

which we are able to see the macromolecule what we will see is the macromolecule which

is closer to the top surface will have a higher velocity and the macromolecular portions which

are in the lower part will have lower velocity so therefore there is a velocity gradient on

the scale of the macromolecule and what we would end up seeing is the solvent will drag

these portions of macromolecular little faster whilethe moleculesmacromolecular portionshere

will actually be dragged with a lower rate and if the same situation can be seen at the

higher shear rate what we would see is the difference between the velocity at

this end of the macromolecule and will be very different compared to the velocity here and in

general the upper part of the macromolecule may get dragged little a higher so in effect what

we are seeing is this macromolecule which in general could be describedas a random a coil

object seems to have stretched a bit as well as oriented a bit so we can see that

it is stretched and as well as oriented so therefore whats happening is whenever we are

subjecting these macromolecules to shear what we have is stretching as well as orientation

so stretching and orientation are inherentphenomena which take place

at the macroat the molecular scale when we have macromolecules under shear of course the result

of this stretching andorientation is also an inherent elasticity which is built into these

macromolecular systems because as soon as we lets say we stop the shear here there is a

tendency of this macromolecule to go back to the sphericalconformation a random conformation that

it had when there was no flow so thereforewhenever we stop the flow there is a tendency to return to

unstretched and randomly oriented macromolecule and which is anyway what is a predominant

mechanism for elasticity in these systems so now lets look at a given that we have

understood that how a macromolecule stretches when its put in a shear flow what we can see here

is that a macromolecule is there in shear flow and clearly there are shear stresses

involved because we are sharing the molecule in along these planes we have shear stresses which

act on this macromolecule but because of the stretching that this macromolecule experiences

as we saw what we end up also having in these materials is the presence of normal stresses

so even though at the boundaries we are only applying a shear stress because of the normal

stress present in the material at the boundaries we canagain measure whats called a normal force

so due to this we can see thatthe fluid response is again qualitatively very different compared

to what a newtonian fluid would be in case of newtonian fluid since we are applying a shear

stress we will only observe a shear stress in the material in case of macromolecular solution

even though we are applying shear stress we observe normal stresses also andjust to go

back and lookthat this is the coordinate system that we can use where x is the

direction of the flow and y is the direction of the shear and z is the third direction

so therefore in this situation when we are looking at these normal stresses we could

denote them as tau xx which is a normal stress in x direction on x plane here by x plane we

mean this surface whose unit normal is in the same direction as x direction and therefore in

the macromolecular under shear in this case we havethe stress components which are not only

shear stress but also these normal stresses so the normal stresses that arise which are due to

the stretching of the macromolecule anddue to its orientationthey are very important

in explaining the unusual flow phenomena that we saw earlier namely extruded swell

or die swell as well as rod climbing effect so lets now try to see how do these stresses

actually explain at least qualitatively how the rod climbing and die swell phenomena are observed

later on in the course we will actually solve the governing equations related to such situations

and observe the response to be conforming to what are experimental observations or what are

qualitative differences between a non newtonian fluid which has elasticity in it when we compare

its response to newtonian fluid so now lets look at the first example of rod climbing

that we saw where the fluid actually climbed up on the rod andwe can use the cylindrical

coordinate system where r is this direction and z is of course the vertical direction

ah if we look at it from this side from the what we see is basically sphericthe circular a polar

coordinates and what we have is fluid moving like this because of the rotation of the rod and we can

look it lets say one small element here and then look at what may be the presence of stresses in

that fluid element similarly in the other example of die swell we had the tube from which the fluid

was being forced out and again in this case we can look at the zr plane r being this direction and

the velocity being in the z direction we can try to see what are the stresses which are present

so for example what we will do is try to look at a small element fluid element here and then try to

see what are the stresses which are present so as i mentioned earlier because of the shearing thats

there due to the rod rotating this component of stress which is tau r theta will always be present

in case of a newtonian fluid this is the only stress component which would be there but given

that with the shearing here the macromolecules are getting stretched and oriented what will

happen is they will also get the normal stresses so therefore both tau theta theta and tau rr both

the normal stresses in theta direction as well as r directions will also be there and its the

normal stress difference which is defined based on these quantities which leads the fluid to actually

go in the z direction and similarly in this case also we have againtheshear stress which is a tau

rz andwhen we havethe fluidflowingwhether its newtonian or non newtonian because of

shear this component will always be present but because of the normal stress differences

between tau rr and tau zz we actually have the normal stress difference being present

and therefore a tendency for fluid to dilate now one difference we can see in these two cases

is this difference between tau rr and tau theta theta in this case is in this direction andbut

we have confined the fluid based on this wall and the result of this will end up this fluid

rising here while in this case the result of this normal stress difference is in terms of

having the fluid x two a swell in the a radial direction while in this case the fluid actually

ends up climbing in the z direction but later on in the course we will actually

see the governing equations which show how some of these normal stresses arise in the material

andtherefore we will be able to quantitatively and qualitatively describe such phenomena

insummary what we have looked at arebasically macromolecules under shear and we have understood

the basic need for looking at the material and its microscopic mechanisms so in macromolecules under

shear we can think of these macromolecules as if they aremade up of a beads which are connected by

a spring so this particular mechanical analog can actually orient when the shear is applied because

this bead which exchange exchanges friction with the surrounding will move little faster

compared to this bead and therefore this particular dumbbell will have a tendency

to orient itself as well as it will stretch so you can see thatat higher and higher shear

rates thestretching as well as orientation will be higher one other issue that we

should alwaysremember is that this picture of macromolecule the way we have drawn here

is only one snapshot of it in generalbecause of the thermal energy being sufficient the

macromolecular segments are undergoing thermal motion and similarly the solvent molecule is

also undergoing thermal motion so there is random collisions of solvent molecules andthe

macromolecules and there is segmental motion so when we look at this bead and spring picture

what we have is actually thesebeads and springs by themselves will also be vibrating so the between

these two there will be vibration going on as well as the orientation of thethe dumbbell will

keep on changing even underwhen there is no flow but when we have flow what will happens is these

beads on an average are get oriented along the flow directionon an average they gets stretched

in the flow direction so therefore we should remember the overall fluctuating picture at

the microscopic scale but the on averagefeature is what is important to describe the bulk properties

such as viscosity or normal stress differences andagain just to restatewhat we had seen earlier

thatstretching and orientation happens because of the shear rate being imposed on the material

andconversely because of this stretching and orientation we can have elastic recovery also

so what we have seen so far areaspects of shear flowwhich is characterized using shear rate and

microstructure of a colloidal system as well as molecular stretching inmacromolecular systems are

very important to understand what were the unusual flow phenomena and of course we can quantify

and the characterize the overall phenomena using the shear and normal stresses during the flow

so in the nextpart of the lecture what we will see isadditional examples ofthese flow

phenomena and how they are related again to the microstructure and if there are what we will also

see is an addition to shear flow additional types of flows may also be encountered