Merken
NonNewtonian Fluids and the 2nd Law of Thermodynamics
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Erkannte Entitäten
Sprachtranskript
00:01
well indeed 1st time I came into contact is the subject was in nineteen 80 of all the young assistant and I had to the
00:14
master sees this also on experimental stuff on nonNewtonian fluids and then I left this topic completely and to study so if studies 7 years later this was last year when I was in the audience in the you in Member States in an unlawful and I have listened to Mr. von to work he gave a talk on nonNewtonian fluids interesting talk and you in this talk I started some calculations and today I will present some of the desired I achieved from time to time in the last year that minutes stops at this experiment what you see on the lefthand side is
00:57
simply a colored water there's a glass filters there's that application of the water facilitation is set in my tool small magnets 1 is immersed in the fluid you can see it in the picture it's due to lie to you the and the other a magnet is below this play there is a small more draw and he said so I said so what tuition of the water to a stationary state is leached currently the others the occasions are nonstationary however on my examples concerns a stationary state and what you see here this is a classical behavior everybody knows this for new influence and every would see that this can also be perfectly were described as enormous toxic agents well he
01:48
has a different look the same experiment but a divalent subsystems this goes the other way around this is so believable in IndyCar the the early is a solvent fully usable to his supporting me the solution was in the solvent and you see it goes up in the middle and not at the outer walls of the glass though the question now is how how can we stands this and the modeling started in the forties interfaith to of the last potentially it us start is the numbers still
02:21
skis for us I consider incompressible fluids so the divergent constraint is always satisfied then here we have some momentum balance evaluate workers who determine in the incompressible case as the pressure and the velocity the HKMA denotes the stress stress is symmetric is assumed to be symmetric matrix A stress is decomposed into pressure and into the gonad explosed the exposed 1st must be pressed me and Newtonian means that the exercise stress is simply proportional tools a gradient was prick part of the gradient of the velocity of the symmetric part and it I'm not here is simply use the viscosity and it's cause is positive no nonNewtonian the Modest smallest the same the stresses print the decomposition is the same however now you see in the expressed 1st there is this quantity and their embodies spores a parties of C so you would the memory of the solvent and a salute and this is intended to be capable to the tool to model of the female when I was in the salute know as a whole so in this will in this lecture place a velocity gradient only for you form moment and is that the gradient of the velocities above by capital M. and is decomposed into enzymically part and an antiSemitic part and the W is also sometimes called to spin so now had NavierStokes is a fluent rather stressed depends on pressure and on the symmetric part of the velocity gradient and the 1st attempts to model of nonNewtonian behavior as a socalled 2nd conveyed through words and they assume warm that's of foreign aid is represented here in the constitutive function and in addition we have said what it means a kind of limit of the this is 1 possibility to determine Xeon here the other possibility is totally different this is my main point in this lecture as a 2nd possibility to abuse Cardinal Newtonian flow is to or post next type of balance type equations and these new evolution iterations for this and so and as a few depending on time and space and the need evolution equations you seen it was in this explicit if you had if you rotten bits is function run along than you can in that it here and that is so the creations as before OK now I come to the
05:11
experiment side of few people in the experiment on various the soul of vanished simple deal only at the office of flow for example in the experiment it's there is a definition what is called the school matrix you flaw in this committee which she of the velocity of the fluid goes in 1 election and it depends on the perpendicular the election this may is the mean point here and then 1 can undermined assumptions 1 can show that the stress assumes a little bit simpler form as before here's a tool Please the all entries and precesses midweek of course the s must be praise for users leave that we have here as 1 1 last as to tool this is realizing the people
05:58
think it is lies in the experiment but this is a divalent point you see is this is a typical bison back so Rysanek real it used to use more toward the 2 geometries are favored by people in the experiment UFC played cold and you see years these types of fabricated here bath so that's a fluid can climb up and 1 can measure this phenomenon that's a fluid goes up inside in the middle more or less and this is a play played ulimately more or less the same this is in order to to measure different components of it OK now let us
06:39
consider a simple case we assume all of for this cylindrical coordinates we assume a velocity field of science Khot kind in the cylindrical coordinates the accordion at lines are crafted for these reasons the classical divergence operator there must be substituted by Cu variant of the motives and of course it is cool this velocity feared is not really rallied below and New year in the neighborhood of the magnet who rich that's a fluid into what it should be I literate far away from this OK now in this case it is clear that the excess risk depends only on the radius ancestry as as the devil margins condition is satisfied identically and the sleek creations of the of the momentum balance is given here in this simplified case a simple velocity field and now our plan is to sort of but at 1st this sort of it for them to onion case and in the note on it is it is clear as S has a very simple form if node i organ the components we have only the components as 1 tool as seat on the and he has this quantity is called the shield it this is the velocity remember this is an abbreviation of of this expression here and now we have boundary condition and on go into details at the audible only the velocity should be the at the flea boundedly the pressure is given by the order pressure the pressure in the fluid is given by the order pressure and the magnet applies the moment to the liquid and the moment this is the definition in the on the experimental side is related to the shear stress appealing here and that this quantity is given no obvious source occasion how to solve a simple at 1st so this and you see this depends only on the radius for that reason P must be linear antiterror however it must be continuous so there is no dependence of those as a pressure cannot depend on the the because otherwise we would have a mighty solutions so the was reduces to this simple set of equations at 1st so this you see the best artsy dot is a constant this relates the S to the moment when this so if this equation for the velocity and the get alters the dependence and this is a major quotation plus a potential fuel but canvassing eyes orderly use of the class now this is important for the behavior you see the PDR the derivative of suppression if you go from that you know from inner side to the other side that is positive because it is given by the square that by thus we can only describe this a and here's the free surface you can describe isn't it wasn't clear OK is all let us go back this is a simple form was obtained because the pressure was really simple in the general case where few additional terms only in that equation and remember this was responsible for the behavior so in other words nonNewtonian behavior
10:04
can only 1 year to 1 in
10:10
behavior can only be described by a dialog expresses because we must clear humanity as and as the twoseater because only this can prevent that's the pressure as a function of radius is more notable so now next we consider the nonNewtonian model and the
10:30
memories starts at I introduce already the 2nd plate fluid and now I status simple constitutive model for the 2nd plate
10:38
fluid before it loses I must talk a little bit on that of of of that utterance formulations is is related to the time derivatives that has appeared via a CD can describe everything in the coordinated system in this 1 on which was that long so the same point in the coordinate system can be described from different observe a point of view was and the motion of a Commission hasn't motion of a transformation in the classic in classical physics is the time t in both systems is the same and the socalled OK medium conformational relates the point being in this system to the same point in that system by this linear expression always some orthogonal matrix that may depend on the B that links the 2 origins of the system the guy is takes off angular velocities between the 2 systems and it is as defined in this way and now it is important object that we have here are socalled objective Kenzo's of rank and and we must check if I have a few was compliments that are denoted here by this for research in its here than if the complement satisfies this equation here this was formed a transformation then you say it is my objective tens of and all the course should point is of hiring mainly in Valiant time derivatives see that as a song the matrix here is that satisfies this law so in the the out there was a 2nd observer sees a component of the matrix according to this definition no we need time derivatives but it is clear if I've found the time derivative of the when this is not simply the time derivative of of the star it is not simply the time derivative of the why because he always depend on time so we have 2 different Ch and CC creations are not usable in physics because we must have occasions between vectors Stenzel's and so on and this is the reason that people have introduced time derivatives that fear this law here time derivative of bridges God in Valiant I'm deliberative must fulfill this law and disease so you can check many of these umbrella known upper convective lower convective time delimited quotation you see the difference in the devil and is only the minus he is a full alias used so for velocity gradient he only the spinning is used as a divalent between the upper convective and the quotation derivative balance creations in you totally divalent time derivatives you see at the dot he is a damn it you s minus SWS is totally different from this expression here however in in contrast to the quotation of to his all media appears everything is OK this is an invariant time derivative however there is a price to pay for this but the world this is the time derivative of air the only golf the confirmation appears in the creations was this is having their people from the dating of gasses noses I mean the Boltzmann equation because in the meanfield limit of the Boltzmann equation 1 gets a limited when he also is omega of the transformation of the bodies are invariant under limited rational mechanics this is the Gould whose stocks tool represent or Newtonian flow by constitutive equations before this 1 you see the 1 there is a hierarchy of tens was the first one is the the new could pound the next 1 don't classes and you see this is exactly that want to see a code law convective and it's further on values is a typical constitutive law s is given by this is is some of his storks don't received z of all as he is called a T off the solvent and 2 additional term and there are sweet constant constituted by it amount of how 1 and from 2 another question is this and this is the unknowns in the title of Michael what is the what is the meaning of the entropy principle to this quantitatively creation the meaning is clear it has been exploited already in 1974 by Deligne and Fosdick and they got all the fundamental be principle age the whole year must be positive clear intuitively for everybody yeah I've found 1 is this coefficients must be because in the and a further with ideas of how 1 class of fractal must be users of his idols the 2nd also modern and I don't use the whole feel this is without doubt and now the disaster analyzes this is the experiment already performed in 1960 9 that I fell 1 which must be positive according to the 2nd law of thermodynamics is smaller than the of in the experiment there's no doubt experiment gives I found 1 is smaller than the along this coefficient and I found this is not important is because India was is not important because the behaviors at the nonNewtonian flow climbs up in the middle is only due to this effect the diseases and in in the 2 so what is now what has happened to you see something must be along these creations and in their from 1970 faults that it's legal to all the scientists the 1st hope say I'm not interested in experiments they ignore the experimenter fun facts and this did you resists by because it is clear that the 2nd also modern predicts these inequalities 2nd cool discusses well obviously is there is an mental on it to the womb there's also at all for many discussions and a selfcooled blames S is considered to be finished and IPOs personally believe belong to this and this is 1 also here that we have mixed with conditions next without collisions at this is the next model that I'm considering this is this is a typical makes tidy creation also obviously constant but considered if Parliament on the added if and eaten what here here's a God here is related to and then we have to a relaxation times of relaxation for stress is a diamond indicates the time derivative knowing what are switched on derivatives I presented you this a few slides before a lot of so you can introduce that was saying you like here and these considered if equations must now be considered risk respect to the 2nd floor of thermodynamics the same as here and after the depth this we really relates the coefficients little with because it is clear that the Scotch off this class here is totally different from this year's an evolution nucleation this week constants and here is not an illusion nucleation user stresses explicit given in terms of the motion mainly off derivatives of the velocity this is different he knows the 2nd law of someone anonymous by the way in in this talk 1 year before a given by stiff onto link use this model we see AP a convict do
18:33
kind of it this was a mixture model that he short OK now you're are here and also more than it's 2nd all summer dynamics alone excitation models do not so dynamically not convinced there's a 2nd clause hemodynamics except goal is the is the next the model is a quotation but nobody in in nematics uses a quotation of evidence that go occasionally deliberative is in agreement with the 2nd loss hemodynamics that has at 1st consider this and then I come back to this problem in this case I can have unequal inequalities the Nazis the constants are in war all of that time and the inequalities as this the heat on this positive tor s of annexation time and 24 stress and fall this year it must be positive this is very important the 2 relaxation times are independent of each other as they are related by the ratio few of the of the discovered is this is the risk or the tumor a member of the Pugh World the solvent name is because the India and the die as service quality of the mixture of the important 1 is this said relaxation time for the shield is smaller than the relaxation time for the stress people use assists in mechanics and solid mechanics this is in this value posting this landmark there it goes the other way around and this is smaller than this however in a fluid so that the relaxation time of the she laid is smaller and this is important because you
20:18
know I would like to compare this
20:21
model with Cemex retired model and for that I use as a simple shear flow as
20:26
before I use this velocity geometry I introduce a compound which is the sheer rate here and now I calculated that was saying and remember this is the inequality from the 2nd law for the shield for the 2nd grade fluid and in all forms experiment a bond of this won't don't live long now this is different and now you see the difference is let us consider it for us the shear stress the it is more or less the same ignores is composed wrapped around here and then you see it a couple of times it's the same as he except that we have here the at deal in the simple model and here's the Utah now a correction of a come back to this collection later now he has enormous stress effect and you see here the eye of how 1 is positive and here's the tall dB goes to these smaller than as this has a correct sign so the experiment can be described by this is can be predicted nonNewtonian throws goes up in the middle of this model and furthermore you see here that's if I call this viscosity eta times suspected to food like it then you see the viscosity depends also on the shear and this is also observed in the experiment and you see that there is some minus you because Tor D is smaller than than is so profit by by the way are not quite perfect these i've alarm I've had to postpone its 2 that's a quotation model the modern receipt quotation the time
22:06
derivative that gives S 1 1
22:09
plus as to 2 equal to the however this is also due to the fact that the velocity field is so simple this is an assumption that we will we assumption from the beginning he has this is the is a consequence of the 2nd law of thermodynamics so you cannot escape you you can escape consider different velocity fields and this will the more divalent OK now the other ones I said already let us go back to they are not compatability in
22:38
this form however won't mind can prepare the deficiency 1 can repair guide for what other types of that monitors and I can't as the upper
22:51
convective and law convective as I said before this is given by this is the work of our people in nematics however no you see there must be there are different look more terms are extended Drancy creation is not so simple is here by the way is the creation is not simple it seems to be in this notation it is linear however remember
23:14
the they are nonlinearities to correction
23:19
terms here in the kind of evidence is not a linear system but now as the
23:25
nonlinearity becomes stronger
23:27
because the 2nd law of someone dynamics require this additional terms and remember here n is the difference between S and said as is the stressed these the as we part of the velocity gradient is I will just and here for simplicity so Everly where various and yeah I can introduce I can substitutes and that is minus the this is simply the important point is this is the free energy of the nonNewtonian fluid the important point is I'm no more part because the 2nd also more dynamics give the expression of the free energy density and the expression is here it starts with AMD's Karanth next term is and to the server power and you see you all the coefficients that has a beer have appeared here in in the main part let me say that let me call this may part also appears here there are no new coefficient the inequalities are the same as before and this is my hope also for all people pretty numerics there instead of C been completed both makes with moderates 1 should deal you with this 1 and optimal I had more time to check every officer consequences that these additional don't have central
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Beobachtungsstudie
Aerothermodynamik
Fluid
Fluid
Rechnen
Analysis
Aggregatzustand
00:57
Aerothermodynamik
Fluid
Wasserdampftafel
Stationärer Zustand
Mathematisches Modell
Klassische Physik
Zahlenbereich
Kartesische Koordinaten
Extrempunkt
Vierzig
Mathematisches Modell
Assoziativgesetz
Fluid
02:20
Geschwindigkeit
Matrizenrechnung
Impuls
Nebenbedingung
Punkt
Momentenproblem
Fluid
Hochdruck
Summengleichung
Iteration
Nichtnewtonsche Flüssigkeit
Mathematik
Gleichungssystem
Gradient
Bilinearform
Massestrom
Symmetrische Matrix
Gradient
Mathematisches Modell
Numerisches Modell
Inverser Limes
Gasdruck
Scherbeanspruchung
Gleichungssystem
MinkowskiMetrik
Addition
Lineares Funktional
Physikalischer Effekt
Variable
Maxwellsche Gleichungen
HelmholtzZerlegung
Summengleichung
Arithmetisches Mittel
Druckverlauf
Mereologie
Evolute
Nichtnewtonsche Flüssigkeit
Geschwindigkeit
Numerisches Modell
05:56
Geschwindigkeit
Nachbarschaft <Mathematik>
Randverteilung
Impuls
Punkt
Momentenproblem
Randwert
Fluid
Klasse <Mathematik>
Gleichungssystem
Bilinearform
Term
Gerichteter Graph
Mathematisches Modell
Arithmetischer Ausdruck
Knotenmenge
Numerisches Modell
Zusammenhängender Graph
Scherbeanspruchung
Gerade
Divergenz <Vektoranalysis>
Radius
Nichtlinearer Operator
Scherbeanspruchung
Klassische Physik
Zylinder
Summengleichung
Randwert
Druckverlauf
Freie Oberfläche
Quadratzahl
Menge
Sortierte Logik
Konditionszahl
Körper <Physik>
Ordnung <Mathematik>
Geometrie
10:04
Mathematisches Modell
Lineares Funktional
Radius
Druckverlauf
Diagonale <Geometrie>
Numerisches Modell
Transformation <Mathematik>
Nichtnewtonsche Flüssigkeit
Mathematik
Variable
Geschwindigkeit
Numerisches Modell
10:29
Matrizenrechnung
Punkt
Gleichungssystem
Mathematik
Nichtnewtonsche Flüssigkeit
Oval
Gradient
Extrempunkt
Gesetz <Physik>
Massestrom
Gradient
Arithmetischer Ausdruck
Numerisches Modell
Gasdruck
Kontrast <Statistik>
Turm <Mathematik>
Verschlingung
Klassische Physik
Variable
Invariante
Konstante
Rechenschieber
Arithmetisches Mittel
Rhombus <Mathematik>
Aerothermodynamik
Tensor
Konditionszahl
Koeffizient
Evolute
Derivation <Algebra>
Nichtnewtonsche Flüssigkeit
Koordinaten
BoltzmannGleichung
Geschwindigkeit
Subtraktion
Fluid
Stoß
Physikalismus
Klasse <Mathematik>
Gruppenoperation
Matrizenrechnung
Derivation <Algebra>
Transformation <Mathematik>
Term
Mathematisches Modell
Ungleichung
Rangstatistik
Inverser Limes
Zusammenhängender Graph
Hierarchie <Mathematik>
Fundamentalsatz der Algebra
Zehn
Transformation <Mathematik>
Physikalisches System
Vektorraum
Schlussregel
Summengleichung
Objekt <Kategorie>
Fluid
Gerichtete Größe
Rangstatistik
Geschwindigkeit
Numerisches Modell
18:31
Stereometrie
Einfügungsdämpfung
Gradient
Maxwellsche Gleichungen
Konstante
Zusammengesetzte Verteilung
Dynamisches System
Mathematisches Modell
Ungleichung
Numerisches Modell
Parametersystem
Scherbeanspruchung
MechanismusDesignTheorie
Modallogik
Numerisches Modell
20:18
Geschwindigkeit
Subtraktion
Sterbeziffer
Fluid
Scherbeanspruchung
Gruppenoperation
Gradient
Bilinearform
Massestrom
Gesetz <Physik>
Maxwellsche Gleichungen
Gradient
Mathematisches Modell
Numerisches Modell
Ungleichung
Vorzeichen <Mathematik>
Parametersystem
Scherbeanspruchung
Geometrie
Numerisches Modell
22:04
Geschwindigkeit
Subtraktion
Gradient
Gesetz <Physik>
Eins
Maxwellsche Gleichungen
Mathematisches Modell
Numerisches Modell
Aerothermodynamik
Parametersystem
Entropie
Körper <Physik>
Scherbeanspruchung
22:35
Mathematisches Modell
Zahlensystem
Numerisches Modell
Parametersystem
Entropie
Gradient
Bilinearform
Term
Gesetz <Physik>
Maxwellsche Gleichungen
23:14
Geschwindigkeit
Subtraktion
Punkt
Fluid
Matrizenrechnung
Mathematik
Gradient
Term
Gesetz <Physik>
Gradient
Dynamisches System
Mathematisches Modell
Arithmetischer Ausdruck
Ungleichung
Numerisches Modell
Entropie
Substitution
Leistung <Physik>
Physikalisches System
Maxwellsche Gleichungen
Dichte <Physik>
Invariante
Fluid
Koeffizient
Mereologie
Parametersystem
Derivation <Algebra>
Perpetuum mobile
Metadaten
Formale Metadaten
Titel  NonNewtonian Fluids and the 2nd Law of Thermodynamics 
Serientitel  The Leibniz "Mathematical Modeling and Simulation" (MMS) Days 2018 
Autor 
Dreyer, Wolfgang

Mitwirkende 
LeibnizInstitut für Oberflächenmodifizierung e.V. (IOP)

Lizenz 
CCNamensnennung 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. 
DOI  10.5446/35356 
Herausgeber  WeierstraßInstitut für Angewandte Analysis und Stochastik (WIAS), Technische Informationsbibliothek (TIB) 
Erscheinungsjahr  2018 
Sprache  Englisch 
Produktionsort  Leipzig 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik 
Abstract  This lecture addresses the peculiar behavior of NonNewtonian fluids. Particularly the Weissenberg effect in a rotating shear flow is discussed in detail. To this end we consider models of Maxwelltype. In the literature there are three alternatives to describe the evolution of the NonNewtonian part of the stress by Maxwelltype models. They differ by use of corotational, upper convective and lower convective time derivatives, respectively. We show that solely the classical Maxwelltype model with a corotational time derivative is admissible from a thermodynamic point of view. However, thermodynamics also permits the two other time derivatives but only within a generalized setting, where Maxwelltype models are furnished by supplementary terms. We explicitly provide these new terms and discuss its influence on the HighWeissenberg number problem. 