Non-Newtonian Fluids and the 2nd Law of Thermodynamics
This is a modal window.
Das Video konnte nicht geladen werden, da entweder ein Server- oder Netzwerkfehler auftrat oder das Format nicht unterstützt wird.
Formale Metadaten
| Titel |
| |
| Serientitel | ||
| Anzahl der Teile | 20 | |
| Autor | ||
| Mitwirkende | ||
| Lizenz | CC-Namensnennung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. | |
| Identifikatoren | 10.5446/35356 (DOI) | |
| Herausgeber | ||
| Erscheinungsjahr | ||
| Sprache | ||
| Produktionsort | Leipzig |
Inhaltliche Metadaten
| Fachgebiet | ||
| Genre | ||
| Abstract |
|
Leibniz MMS Days 201810 / 20
00:00
StochastikAssoziativgesetzAerothermodynamikFluidMultiplikationsoperatorTurek, StefanFluidResultanteStatistische HypotheseWasserdampftafelAggregatzustandBeobachtungsstudieRechenbuchComputeranimationBesprechung/Interview
00:57
ModelltheorieAnalysisAerothermodynamikFluidAssoziativgesetzFluidWasserdampftafelStationärer ZustandKartesische KoordinatenNichtnewtonsche FlüssigkeitNichtlineares GleichungssystemKreisbewegungComputeranimation
01:47
ModelltheorieAnalysisAerothermodynamikFluidModelltheorieNumerische MathematikVierzigComputeranimation
02:20
DruckverlaufDivisionModelltheorieVariableNichtnewtonsche FlüssigkeitGeschwindigkeitFluidGradientMaxwellsche GleichungenSummengleichungZeitabhängigkeitGeschwindigkeitModelltheorieIterationImpulsPhysikalischer EffektNichtnewtonsche FlüssigkeitNebenbedingungArithmetisches MittelFunktionalHochdruckInverser LimesMereologieMomentenproblemNichtlineares GleichungssystemDruckverlaufHelmholtz-ZerlegungAdditionPunktSymmetrische MatrixGradientDruckspannungSummengleichungEvoluteMinkowski-MetrikFluidVariableKategorie <Mathematik>Derivation <Algebra>Sigma-AlgebraSkalarproduktKörper <Algebra>Divergenz <Vektoranalysis>MassestromMultiplikationsoperatorZweiComputeranimation
05:09
ModelltheorieScherbeanspruchungFluidGeometrieGeschwindigkeitScherbeanspruchungPunktDruckspannungRichtungMassestromMatrizenrechnungModulformComputeranimation
05:56
ModelltheorieScherbeanspruchungFluidGeometrieOrdnung <Mathematik>DrucksondierungEinflussgrößeZusammenhängender GraphPunktDifferenteComputeranimation
06:37
ModelltheorieRandwertFreie GruppeFluidGeschwindigkeitScherbeanspruchungModulformImpulsFreie OberflächeDerivation <Algebra>KoordinatenMomentenproblemTermThetafunktionKonstanteNichtlineares GleichungssystemNichtlinearer OperatorZusammenhängender GraphDruckverlaufNachbarschaft <Mathematik>Körper <Algebra>Flüssiger ZustandDruckspannungSummengleichungRadiusArithmetischer AusdruckDivergenz <Vektoranalysis>NichtunterscheidbarkeitKonditionszahlKlassische PhysikRandwertKreisbewegungOrdnung <Mathematik>MengenlehreGeradeGerichteter GraphRandverteilungQuadratzahlKlasse <Mathematik>Sortierte LogikComputeranimation
10:04
ModelltheorieTransformation <Mathematik>FluidDruckverlaufDivisionVariableNichtnewtonsche FlüssigkeitNichtlineares GleichungssystemZeitabhängigkeitGeschwindigkeitGradientMaxwellsche GleichungenEinfach zusammenhängender RaumDruckspannungDiagonale <Geometrie>ModelltheorieFunktionalDruckverlaufRadiusComputeranimation
10:29
DruckverlaufModelltheorieVariableGeschwindigkeitNichtnewtonsche FlüssigkeitNichtlineares GleichungssystemZeitabhängigkeitFluidGradientTransformation <Mathematik>Einfach zusammenhängender RaumRangstatistikTensorMatrizenrechnungInvarianteDerivation <Algebra>Physikalische TheorieRationale ZahlFluidZweiGeschwindigkeitTransformation <Mathematik>MatrizenrechnungVektorraumDerivation <Algebra>Nichtnewtonsche FlüssigkeitGesetz <Physik>Boltzmann-GleichungInverser LimesKoordinatenPhysikalisches SystemPhysikalismusRangstatistikNichtlineares GleichungssystemZusammenhängender GraphPunktKlasse <Mathematik>GradientSummengleichungArithmetischer AusdruckHierarchie <Mathematik>DifferenteKontrast <Statistik>ZehnObjekt <Kategorie>Klassische PhysikMultiplikationsoperatorArithmetisches MittelGruppenoperationIndexberechnungKinetische GastheorieMereologieTensorKörper <Algebra>MassestromSchlussregelGruppendarstellungBoltzmann-KonstanteComputeranimation
14:43
FluidGradientModelltheorieKonstanteParametersystemMaxwellsche GleichungenModelltheorieAerothermodynamikDerivation <Algebra>UngleichungGesetz <Physik>Arithmetisches MittelFundamentalsatz der AlgebraGruppenoperationRechenschieberTermKoeffizientDruckspannungInelastischer StoßKonditionszahlMassestromMultiplikationsoperatorRhombus <Mathematik>EntropieBeweistheorieResultanteParametersystemJensen-MaßZweiComputeranimation
17:46
ModelltheorieTensorMaxwellsche GleichungenGradientGeschwindigkeitModelltheorieAerothermodynamikDerivation <Algebra>Gesetz <Physik>TermVerschlingungKonstanteNichtlineares GleichungssystemKlasse <Mathematik>KoeffizientEvoluteAlgebraische StrukturDruckspannungMultiplikationsoperatorComputeranimation
18:31
Rationale ZahlModelltheorieInvarianteDerivation <Algebra>Nichtlineares GleichungssystemMatrizenrechnungKonstanteParametersystemEntropieTensorMaxwellsche GleichungenGradientZusammengesetzte VerteilungModelltheorieNumerische MathematikStereometrieScherbeanspruchungAerothermodynamikDerivation <Algebra>UngleichungGesetz <Physik>KonstanteSterbezifferDruckspannungZweiDynamisches SystemEinfügungsdämpfungMultiplikationsoperatorMechanismus-Design-TheorieComputeranimation
20:16
ModelltheorieScherbeanspruchungParametersystemEntropieTensorGradientMaxwellsche GleichungenModelltheorieScherbeanspruchungMassestromFluidGeometrieGeschwindigkeitModulformUngleichungGesetz <Physik>GruppenoperationSterbezifferGradientDruckspannungVorzeichen <Mathematik>DifferenteMultiplikationsoperatorDerivation <Algebra>HyperbelverfahrenTermPunktPoisson-KlammerMultifunktionJensen-MaßZweiKappa-KoeffizientComputeranimation
22:04
KonstanteParametersystemModelltheorieEntropieTensorMaxwellsche GleichungenGradientScherbeanspruchungGeschwindigkeitGesetz <Physik>Körper <Algebra>DifferenteEinsAerothermodynamikComputeranimation
22:35
KonstanteParametersystemModelltheorieEntropieTensorMaxwellsche GleichungenGradientModelltheorieModulformZahlensystemGesetz <Physik>TermNumerische MathematikNichtlineares GleichungssystemComputeranimation
23:14
ModelltheorieMaxwellsche GleichungenGradientParametersystemTensorFluidRationale ZahlMatrizenrechnungInvarianteDerivation <Algebra>Physikalische TheorieNichtlineares GleichungssystemKonstanteEntropieDerivation <Algebra>Physikalisches SystemTermFluidGeschwindigkeitNumerische MathematikAerothermodynamikUngleichungGesetz <Physik>MereologiePerpetuum mobileParametersystemKubikzahlQuadratzahlPunktDichte <Physik>GradientKoeffizientArithmetischer AusdruckDifferenteMultiplikationsoperatorZweiDynamisches SystemLeistung <Physik>SubstitutionComputeranimation
24:49
Besprechung/InterviewComputeranimation
Transkript: Englisch(automatisch erzeugt)
00:01
Well, indeed. First time I came into contact with this subject was in 1980. I was a young assistant, and I had to supervise a master thesis on experimental stuff on non-Newtonian fluids. And then I left this topic completely until 37 years
00:26
later. This was last year when I was in the audience during the MMS days in Anova. And I listened to Stefan Turek. He gave a talk on non-Newtonian fluids, a very interesting talk.
00:41
And during this talk, I started some calculations. And today, I will present some of the results I achieved from time to time in the last year. Let us start with the experiment. What you see here on the left-hand side is simply colored water. There is a glass filled with water.
01:02
There is a rotation of the water. The rotation is set in by two small magnets. One is immersed in the fluid. You cannot see it here in the picture. It's too light here. And the other magnet is below this plate. There is a small motor. And it sets the rotation of the water
01:24
until a stationary state is reached. Currently, the equations are non-stationary. However, all my examples concern the stationary state. And what you see here, this is a classical behavior. Everybody knows this for Newtonian fluids.
01:41
And we will see that this can also be perfectly well described by the Navier-Stokes equations. Now, here's the same experiment, but a different substance. This goes the other way around. This is polyisobutylene in decaline. Decaline is a solvent. Polyisobutylene is a polymer solution within the solvent.
02:03
And you see it goes up in the middle and not at the outer walls of the glass. So the question now is how can we understand this? And the modeling started in the 40s and the 50s of the last century. Let us start with the Navier-Stokes case at first.
02:22
I consider incompressible fluids. So the divergence constraint is always satisfied. Then here we have some momentum balance. The variables to determine in an incompressible case is the pressure and the velocity. Sigma denotes the stress.
02:41
Stress is symmetric. It's assumed to be symmetric. Matrix, the stress is decomposed into pressure and into the, call it extra stress. The extra stress must be trace fee. And Newtonian means that the extra stress is simply proportional to the gradient, to the symmetric part of the gradient of the velocity.
03:03
Symmetric part. And eta naught here is simply the viscosity and it is clear viscosity is positive. Now, non-Newtonian, the model is more or less the same. The stress is symmetric. The decomposition is the same. However, now you see in the extra stress there is this quantity n here.
03:23
And n embodies all the properties of the solute. Remember, we have a solvent and a solute and this is intended to be capable to model the phenomena within the solute.
03:42
Now, a crucial role in this lecture plays a velocity gradient only for you for remembering. It is a gradient of the velocity is abbreviated by capital L. L is decomposed into an symmetric part and an anti-symmetric part. And the W is also sometimes called the spin.
04:02
So, now Navier-Stokes is a fluid where the stress depends on pressure and on the symmetric part of the velocity gradient. And the first attempt to model non-Newtonian behavior are the so-called second grade fluids.
04:20
And they assume that the full L is represented here in the constitutive function. And in addition, we have the L dot. It means the time derivative of the L. This is one possibility to determine the n here. The other possibility is totally different. This is my main point in this lecture.
04:41
The second possibility to describe non-Newtonian flow is to propose Maxwell type or balanced type equations and these mean evolution equations for this n. So n is a field depending on time and space and we need evolution equations. Here you see everything is explicit.
05:00
If this function were known, then you can invert it here and then you solve the equations as before. Okay, now I come to the experimental side of you. People in the experiment always assume a very simple geometry of the flow. For example, in the experiment,
05:21
there is a definition what is called viscometric shear flow. In viscometric shear flow, the velocity of the fluid goes in one direction and it depends on the perpendicular direction. This is mainly the main point here and then one can, under mild assumptions,
05:40
one can show that the stress assumes a little bit simpler form as before. Here the two entries are zero and these entries, the S is symmetric of course. The S must be trace-free. This is the reason that we have here S one one plus S two two. This is realized in the, people think it is realized in the experiment,
06:00
but this is the different point. Don't you see here, this is a typical Weisenberg, so-called Weisenberg rheometer. You see here the motor. The two geometries are favored by people in the experiment. You have the plate cone and you see here these pipes are fabricated here
06:21
above so that the fluid can climb up and one can measure this phenomenon that the fluid goes up inside in the middle, more or less. And this is a plate geometry more or less the same. This is in order to measure different components of this. Okay, now let us consider a simple case.
06:41
We assume now for this cylindrical coordinates, we assume a velocity field of this kind. In the cylindrical coordinates, the coordinate lines are curved and for this reason the classical divergence operator must be substituted by covariant derivatives.
07:02
And of course it is true this velocity field is not really valid below and near in the neighborhood of the magnet, which sets the fluid into rotation. We are a little bit far away from this. Okay, now in this case it is clear the extra stress depends only on the radius
07:21
and the three, the divergence condition is satisfied identically. And the three equations of the momentum balance is given here in this simplified case with a simple velocity field. And now our plan is to solve,
07:41
but at first we solve it for the Newtonian case and in the Newtonian case, it is clear the S has a very simple form. We have no diagonal components. We have only the components S one, two, S r, theta and here this quantity is called the shear rate. This is the velocity.
08:01
Remember this is an abbreviation for this expression here. And now we have boundary condition. I don't go into details. At the outer boundaries, the velocity should be zero. At the free boundary, the pressure is given by the outer pressure. The pressure in the fluid is given by the outer pressure and the magnet applies a moment to the liquid
08:24
and the moment, this is the definition on the experimental side, is related to the shear stress appearing here and this quantity is given. Now we solve the equation. How to solve is simple.
08:40
At first we solve this and you see this depends only on the radius. For that reason P must be linear in theta. However, it must be continuous so there is no dependence of, the pressure cannot depend on theta because otherwise we would have multivalued solutions. So everything reduces to this simple set of equations.
09:03
At first we solve this. You see the S r theta is a constant. This relates the S to the moment. Then we solve this equation for the velocity and we get out this dependence and this is a rigid rotation plus a potential flow
09:20
that can be seen. R is the outer radius of the glass. Now this is important for the behavior. You see the P dr, the derivative of the pressure if you go from inner side to the outer side is positive because it is given by V squared by R. Thus we can only describe this behavior
09:41
and here's the free surface. You can describe this and everything is clear. Okay, so let us go back. You see this simple form was obtained because the pressure was very simple. In the general case we have here additional terms only in that equation and remember this was responsible for the behavior.
10:01
So in other words, non-Newtonian behavior can only, non-Newtonian behavior can only be described by diagonal stresses because we must calculate the Sr and S theta theta because only this can prevent that the pressure
10:21
as a function of the radius is monotone. So now next we consider the non-Newtonian model and remember we start, I introduced already the second crate fluid and now I start with a simple constitutive model for the second crate fluid. Before I do this I must talk a little bit
10:40
on observer transformations. This is related to the time derivatives that has appeared here. See we can describe everything in the coordinate system in this one or we choose that one. So the same point in the coordinate system can be described from different observer point of views
11:01
and the most general equation, the most general transformation in classical physics is the time T in both systems is the same and the so-called Euclidean transformation relates the point P in this system to the same point in that system
11:20
by this linear expression. O is an orthogonal matrix that may depend on T. B links the two origins of the system. The omega is a matrix of angular velocities between the two systems and it is defined in this way. And now the important object that we have here
11:43
are so-called objective tensors of rank N and we must check if I have a field with components that are denoted here by this F with indices here. Then if the component satisfies this equation here
12:03
with the Os from the transformation then we say it is an objective tensor. And now the crucial point is arriving, namely invariant time derivatives. C, let us assume the matrix here satisfies this law. So in the other observer, the second observer,
12:22
sees a component of the matrix according to this definition. Now we need time derivatives but it is clear if I form the time derivative of T then this is not simply the time derivative of T star, it is not simply the time derivative of T, why?
12:41
Because the Os depend on time so we have to differentiate and these equations are not usable in physics because we must have equations between vectors, tensors and so on. And this is the reason that people have introduced time derivatives that fulfill this law here.
13:01
Time derivative, which is called invariant time derivative must fulfill this law and these are you can check. And many of these are well known, upper convective, lower convective time derivative, co-rotational, you see the difference. Here the difference is only the minus sign, here the full L is used, the full velocity gradient,
13:22
here only the spin is used. This is the difference between upper convective and the co-rotational derivative. Balance equations induce totally different time derivatives, you see, T dot, here's the W, S minus SW, S is totally different from this expression here.
13:41
However, in contrast to the co-rotational derivative, here's omega appears. Everything is okay, this is an invariant time derivative, however there is a price to pay for this rule. This is the time derivative where the omega of the transformation appears in the equations. But this is very natural.
14:01
People from the kinetic theory of gases know this, I mean the Boltzmann equation, because in the mean field limit of the Boltzmann equation one gets this time derivative. And here also the omega of the transformation appears. All these are invariant time derivatives. Rational mechanics, this is the group who starts to represent non-Newtonian flow
14:25
by constitutive equations prefer this one. You see the A1, there is a hierarchy of tensors. The first one is the D, the metric part. The next one, the A dot plus this, and you see this is exactly that what is here called law or convective, and it runs further on.
14:43
And now you see this is a typical constitutive law, S is given by this, this is another Stokes term with the eta zero as the viscosity of the solvent. And two additional terms here, and there are three constitutive parameters,
15:00
eta naught, alpha one, and alpha two. And now the question is, and this is announced in the title of my talk, what is the meaning of the entropy principle to this constitutive equation? The meaning is clear, it has been exploited already in 1974 by Dan and Fostick, and they got out
15:24
from the entropy principle eta zero here must be positive, clear intuitively for everybody. Here alpha one is this coefficient must be bigger than zero and the further result is alpha one plus alpha two must be zero.
15:40
This is the result of the second law of thermodynamics. I don't give the proof here, this is without doubt. And now the disaster arises, this is the experiment already performed in 1969. The alpha one which must be positive according to the second law of thermodynamics is smaller than zero
16:04
in the experiment, there is no doubt the experiment gives alpha one is smaller than zero, this coefficient. And alpha one, this is not important, is bigger than zero, this is not important because the behavior that the non-Newtonian flow climbs up in the middle is only due to this fact.
16:24
We will see this in detail. So what is now, what has happened here? You see, something must be wrong with the equations and from 1974 there appeared three groups of scientists. The first group say I'm not interested in the experiment,
16:42
they ignore the experimental facts and they still deal with this but because it is clear the second law of thermodynamics predicts these inequalities here. Second group discusses whether really there is an entropy in non-equilibrium.
17:01
This is also a matter of many discussions and the third group blames this constitutive equation. And I personally belong to this and this is one of the reasons that we have Maxwell equations, Maxwell type equations. This is the next model that I'm considering. You see, this is a typical Maxwell type equation,
17:21
also with three constant, a constitutive parameter. They are different, we have the eta naught here, here's the eta, we will relate the two and then we have two relaxation times, the relaxation for stress. Here's a diamond indicates the time derivative. Now you might ask which time derivative? I presented here a few slides before, a lot of.
17:43
So you can introduce everything you like here. And these constitutive equations must now be considered with respect to the second law of thermodynamics, the same as here. And after we did this, we will relate
18:00
the coefficients a little bit because it is clear the structure of this stress here is totally different from this. Here's an evolution equation with three constants and here is not an evolution equation, here the stress is explicitly given in terms of the motion, namely of derivatives of the velocity. This is different here. Now the second law of thermodynamics.
18:22
By the way, in this talk, one year before, a given by Stefan Turek, he used this model with the upper convective time derivatives. This was a Maxwell model that he showed. Okay, now we are here and now thermodynamics.
18:41
Second law of thermodynamics, you learn, Maxwell models do not agree, are thermodynamically not consistent with the second law of thermodynamics except the Maxwell model with the co-rotational. But nobody in numerics uses the co-rotational derivatives.
19:01
The co-rotational derivative is in agreement with the second law of thermodynamics. Let us at first consider this and then I come back to this problem. In this case, I can have inequalities where these three constants are involved more or less of that type and the inequalities are this.
19:22
The eta is positive, tor S of relaxation time and tor D for stress and for the shear rate must be positive. This is very important. The two relaxation times are not independent of each other. They are related by the ratio of the viscosity.
19:41
This is viscosity, remember, of the pure solvent, namely the decaline here, and the eta is the viscosity of the mixture. The important one is this. The relaxation time for the shear rate is smaller than the relaxation time for the stress. People who uses this in mechanics, in solid mechanics,
20:03
this is very interesting, this remark. It goes the other way around. This is smaller than this. However, here in the fluid, the relaxation time for the shear rate is smaller. This is important because now I would like to compare
20:20
this model with the Maxwell type model and for that I use this simple shear flow as before. I use this velocity geometry. I introduce the kappa, which is the shear rate here, and now I calculate everything and remember, this is the inequality from the second law
20:41
for the shear rate, for the second grade fluid, and we know from the experimental point of view this is wrong, totally wrong. Now this is different and now you see the differences. Let us consider at first the shear stress. You see it is more or less the same. Ignore this kappa squared term here and then you see eta kappa times this matrix,
21:02
the same as here except that we have here the eta zero in the simple model and here we have the eta. Now the correction, I come back to this correction later. Now here is the normal stress effect and you see here the alpha one is positive and here's the tau d because tau d is smaller than tau s.
21:23
This has a correct sign so the experiment can be described by this. This can be predicted. Non-Newtonian flows goes up in the middle by this model and furthermore we see here that if I call this viscosity,
21:40
eta times this bracket here, this full bracket, then you see the viscosity depends also on the shear rate and this is also observed in the experiment and you see that there is a minus sign here because tau d is smaller than tau s. So perfect. By the way, not quite perfect. This alpha one, alpha two corresponds to
22:00
that the co-rotational model, the model with the co-rotational time derivative gives s one one plus s two two equal to zero. However, this is also due to the fact that the velocity field is so simple. This is an assumption, a priori assumption from the beginning. Here this is a consequence
22:20
of the second law of thermodynamics. So here you cannot escape. Here you can escape. Consider different velocity fields and this will come out different. Okay, now the other ones. I said already, let us go back to, they are not compatible in this form. However, one can repair the deficiency.
22:44
One can repair it for other types of models and here I consider the upper convective and the lower convective. As I said before, this is favored by people in numerics. However, now you see there must be,
23:01
there are different, more terms. There are extended terms. The equation is not so simple as here. By the way, this equation is not simple. It seems to be in this notation, it is linear. However, remember there are non-linearities due to the correction terms here in the time derivatives.
23:21
It's not a linear system. But now the non-linearity becomes stronger because the second law of thermodynamics require these additional terms. And remember here, n is the difference between s and the d, s is the stress, d is the metric part of the velocity gradient.
23:41
I wrote this n here for simplicity. So everywhere we have the n here, I can introduce, I can substitute the n by s minus d. This is simple. The important point is this is the free energy of the non-Newtonian fluid. The important point is there are no more parameter because the second law of thermodynamics
24:01
give an expression of the free energy density and the expression is here. It starts with n squared. Next term is n to the third power. And you see all the coefficients that have appeared here in the main part. Let me call this main part also appears here.
24:22
There are no new coefficient. The inequalities are the same as before. And this is my proposal for people in numerics. Instead of the incompatible Maxwell models, one should deal with this one. And up to now, I had no time to check every of the consequences that these additional terms have.
24:42
Thank you very much.