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On turbulence in dilatant dispersions Article in Physica Scripta · July 2016 DOI: 10.1088/0031-8949/91/7/074003
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Royal Swedish Academy of Sciences
Physica Scripta
Phys. Scr. 91 (2016) 074003 (8pp)
doi:10.1088/0031-8949/91/7/074003
On turbulence in dilatant dispersions Helmut Z Baumert1 and Bernhard Wessling2 1 2
IAMARIS, Bürgermeister-Jantzen-Str. 3, 19288 Ludwigslust, Germany LaoWei Chemical Technology Consulting Ltd., Shenzhen, LuoHu District, People’s Republic of China
E-mail:
[email protected] and
[email protected] Received 28 April 2016 Accepted for publication 9 May 2016 Published 6 June 2016 Abstract
This paper presents a new theory on the behaviour of shear-thickening (dilatant) fluids under turbulent conditions. The structure of a dilatant colloidal fluid in turbulent motion may be characterized by (at least) four characteristic length scales: (i) the ‘statistically largest’ turbulent scale, l 0 , labeling the begin of the inertial part of the wavenumber spectrum; (ii) the energycontaining scale, ; (iii) Kolmogorov’s micro-scale, l , related with the size of the smallest vortices existing for a given kinematic viscosity and forcing; (iv) the inner (‘colloidal’) microscale, li , typically representing a major stable material property of the colloidal fluid. In particular, for small ratios r = li l ~ (1), various interactions between colloidal structures and smallest turbulent eddies can be expected. In the present paper we discuss particularly that for r = l 0 l (1) turbulence (in the narrow, inertial sense) is strangled and chaotic but less mixing fluid motions remain. We start from a new stochastic, micro-mechanical turbulence theory without empirical parameters valid for inviscid fluids as seen in publications by Baumert in 2013 and 2015. It predicts e.g. von Karman’s constant correctly as 1 2 p = 0.399. In its generalized version for non-zero viscosity and shear-thickening behavior presented in this contribution, it predicts two solution branches for the steady state: The first characterizes a family of states with swift (inertial) turbulent mixing and small l , potentially approaching li . The second branch characterizes a state family with r (1) and thus strangled turbulence, r » (1). Stability properties and a potential dynamic commuting between the two solution branches had to be left for future research. Keywords: turbulence, mixing, dispersions, dilatant fluids, stirring, colloidal dispersions/ emulsions (Some figures may appear in colour only in the online journal) (Wessling, personal lab experience). Shear-thickening (dilatant) fluids may be transformed even into a colloidal liquid up to a highly viscous gel or cream (Wessling 2000, 2007)—by purely mechanical means. Dilatant fluids exhibit increased viscosity upon increased shear stress. Already their manufacturing in laboratories or industry is often strongly controlled by turbulence characteristics like the energy-containing length scale, , and a corresponding time scale or, equivalently, by corresponding wavenumber and frequency. These two parameters may be tuned independently by modifying geometry or/and rotation rate of the equipment preparing or using such fluids. Clearly these questions play an important role in many industries and the economical features and effects connected with the properties of various industrial fluids and pumps or
1. Introduction General
While the behavior of frictionless, ideal (Eulerian) fluids under turbulent conditions is today well understood (see Baumert 2013 and the literature cited therein), this is not the case for Newtonian and still less for non-Newtonian fluids. At a first glance one might classify these problems as very complex but purely academic. However, the shear-thickening property may lead to extreme situations in large industry installations wherein these fluids are pumped so that the hydromechanic properties and the flow status really matter. Pumps may even suddenly block up to being fully destroyed by sudden quasi-solidification of the colloidal liquid 0031-8949/16/074003+08$33.00
1
© 2016 The Royal Swedish Academy of Sciences
Printed in the UK
Phys. Scr. 91 (2016) 074003
H Z Baumert and B Wessling
Here the flux ò (energy per time and volume, m2 s−3) is dissipated by the smallest feasible eddies at scale l (m) into heat; ν (m2 s−1) is the kinematic viscosity. It holds generally (but not always) that l l 0 . The higher ò, the smaller l and graininess, and the better the homogeneity in a vessel of hydrodynamically neutral dissolved materials. This rule is true only for classical Newtonian fluids and dissolved materials. In non-Newtonian, colloidal dispersions the situation is different. The internal interactions between different fluid-mechanical and colloidal variables are subtle: The latter actually represent fine-scale solid-state structures with features like elasticity, load limitations, inner friction and more features. In addition to the (until recently) very limited understanding of turbulence they represent a significant extra challenge.
pump-like devices are simply huge. This was the background of the research we did based on a novel micro-mechanical picture of turbulence which for the first time ever gave the fundamental constants of turbulent motions. We then finally could identify the quantitative rules and laws governing the shear-thickening behavior of the fluids in question. Non-newtonicity
When manufacturing colloidal dispersions or when using them, quality issues like homogeneity of the fluid or poorly reproducible results of the application of said dispersions in a physical or chemical process matter. One is often inclined to increase the input energy and turbulence degree by increasing the rotation rate of the manufacturing equipment, or the geometry of the containers. However, this quite often results in undesired effects, and an increase of shear does not necessarily improve the quality, quite often in contrast. Manufacturing of colloidal fluids and their use are fundamental technological steps in nearly all industries and connected with huge financial efforts. Therefore these questions deserve attention also from a theoretical point of view. In particular, due to the complexity of the many involved natural processes, they react on external measures often in a counter-intuitive way. This paper presents a targeted study based on a new theory of turbulence for non-trivial fluids in stirred vessels.
Historical note
The arts of hydraulics and mixing processes, the targeted treatment of advanced dispersions or suspensions in agriculture and food processing are well known in form of phenomenologies (oral history, alchemistry, cook-book receipes) since more than thousands of years, namely from great ‘hydraulic empires’ like ancient China, Egypt, and Mesopotamia. More analytical mathematical sciences of dynamic fluid motions were started only ‘just recently’ by Leonhard Euler (1707–1783), supplemented somewhat later with a more explicit theory of vortices and circulation by Herrmann von Helmholtz (1821–1894). When more general physical fluctuation theories like generalized Brownian motions, Langevin processes and Fokker–Planck equations became available, a more systematic statistical treatment of turbulence was initiated by Kolmogorov (1903–1987) and his unique school. This school namely initiated a breakthrough in the understanding of turbulence but was misunderstood or simply forgotten because it took place at an inter-disciplinary seminar during the January wartime days of 1942 in the city of Kazan/USSR whereto the Academy of Sciences has been evacuated. In addition to Kolmogorov who played the role of a conductor, the physicists Kapitza and Landau participated and the mathematician Obukhov as well. The summary report was written by Kolmogorov (1942), accompanied by relevant remarks and comments of Landau. The setup of this team was the personal guarantee that most modern theoretical ideas on large stochastic many-particle ensembles (e.g. notions like self-consistency and Fokker–Planck equations) were present in the seminar room. Note that in the mathematical literature Fokker–Planck equations often are called Kolmogorov-forward equations because Kolmogorov gave them an axiomatic foundation (Kolmogorov 1931). The documented oral comment by Landau should be read with attention. It is reported and documented by many authors. Here it is cited from p 219 in Falkovich (2011) (emphases by the authors):
Homogeneity, graininess
With respect to its hydrodynamical aspects the degree of homogeneity of a fluid may be characterized by (among other features) Kolmogorov’s microscale, l . For material dissolved within the fluid it is generally assumed that the smaller its value, the lower what we may call its ‘temporary graininess’. The latter means here the size of fluid regions within which concentrations significantly deviate for some time from the concentration in the ambient fluid. The a.m. microscale, l , may serve as a spatial proxy of such hydrodynamic graininess of dissolved materials. The situation is much more difficult in dispersions. However, also here the microscale l matters when r (1) and r (1) such that the scale ratios affect the turbulent state of the fluid as well as the setup of colloidal aggregates and thus modify their physico-chemical functions. For a colloidal fluid under turbulent conditions, Kolmogorov’s microscale l is a most important characteristic because if l is in the scale size of the dispersed phase of the colloid or their superstructure, respectively, then the vortices will interact with the dispersed phase and/or their superstructure. This can easily be understood, because for manufacturing dispersions, one is already intuitively using stronger turbulence the smaller the final particle size should be. According to Kolmogorov, l follows ⎛ n 3 ⎞1 4 l = ⎜ ⎟ . ⎝ ⎠
(1 )
2
Phys. Scr. 91 (2016) 074003
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As their kinematic viscosity is not a material constant, smart fluids have always non-Newtonian character. They all are at least two-phase systems, whereas at least one dispersed phase is mostly nanoscopic in size (i.e., less than 100 nm diameter). Hence these are colloidal systems, either colloidal dispersions (the dispersed phase being a solid) or emulsions (the dispersed phase being a liquid). In contrast to widespread assumptions, such colloidal systems exhibit complex structures (Wessling 1993). Non-Newtoninan fluids and turbulence
As long as the flow of a non-Newtonian fluid or gas is laminar, the consequences of external controls on the kinematic viscosity remain more or less predictable (Wessling 1995). This is going to be changed when the flow becomes critical and eventually turbulent. In this case the distinctive turbulent scales as well as the ‘inner’ scales of the dispersion/ emulsion (, l 0 , l , li ) get statistical chances to interact. The non-Newtonian behavior of colloidal dispersions/ emulsions and suspensions, especially the shear-thickening (dilatant) behaviour which can very often be observed, is the object of broad research work (Brown and Jaeger 2012, 2014, Wagner and Brady 2009). However, several aspects are widely overlooked or ignored: first the fact, that colloidal dispersions and emulsions exhibit a complex three-dimensional inner structure (Wessling 1993) which plays a role in shear thickening (Melrose and Ball 2004); then the fact that colloidal systems are generated under thermodynamically non-equilibrium conditions which means that they can not be treated as if ‘close to equilibrium’ (which many authors do, e.g. Wagner and Brady 2009); last but not least the fact that quasi-turbulent flow can not describe the behaviour under turbulent conditions. Even the Reynolds number in non-Newtonian fluids looses much of its classical meaning. Traditionally it is defined as the dimensionless ratio of inertial to viscous forces
Figure 1. The effective kinematic viscosity in shear-thickening
(green), shear-thinning (red) and Newtonian fluids (blue) following an Ostwald–de-Waele relation (3) where for illustrative purposes n0 = 0 .
‘... As to the equations of turbulent motion, it should be constantly born in mind ... that in a turbulent flow the vorticity is confined within a limited region; qualitatively correct equations should lead to just such a distribution of eddies...’ His view of turbulence may loosely be interpreted as an entangled ensemble of spatially distinct, individual vortices in autonomous motions and local interactions, just as it has been postulated independently much later in form of a particle theory of turbulence by Baumert (2005). Note Landau’s emphasis on a qualitatively correct picture. I.e. Landau has been aware that the equations published by Kolmogorov after the Kazan seminar were correct only in a structural sense. But with respect to the various pre-factors of the terms probably not: the micro-mechanical model (simple vortex-gas approach: 2013 Baumert 2005) for the vortex–vortex interactions was still missing.
Re =
U´L n
(2 )
where U (m s−1) is a velocity scale for the mean flow, L (m) a characteristic (outer) length scale (e.g. a distance from a solid wall or body) and ν (m2 s−1) is the kinematic viscosity. This means that Re depends linearly on the macroscopic state of flow via the rms velocity, U. However, if also ν depends on the flow state then things become more difficult. Relatively simple classes of smart fluids are the so-called shear-thickening (dilatant) and the shear-thinning (pseudoplastic) fluids. In the former case viscosity increases with increasing shear rate so that the fluid becomes ‘thicker’. The most prominent example is the fluid in body-armor wests of policemen. Certain water–sand mixtures may behave in similar form. Prominent examples for shear-thinning fluids are lava, certain mineral oils, blood and whipped cream, and namely the notorious quicksand. In all these cases the turbulent state is understood (if at all) only in an empirical sense. It is reasonable to assume that shear-thickening is caused by a dynamically evolving, highly complex 3D network of
2. Smart fluids/colloidal systems Today so-called smart fluids become increasingly relevant in various industries, from medicine over small-scale bio-technology to electronics, oil and gas production, construction engineering, waste-water treatment, and defense. A central property of a smart fluid is its kinematic viscosity, which may explicitely depend on time or not, which may be tuned by shear or radiation, by electric or magnetic fields like in magneto-rheological fluids and in plasmas. 3
Phys. Scr. 91 (2016) 074003
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dispersed particles/droplets in nanoscale up to the extreme situation that pumps can even be blocked (and destroyed) by a sudden solidification of the colloidal liquid (Wessling, personal lab experience) or the transformation of a colloidal liquid into a gel (Wessling 2000, 2007). To call such a network a ‘spiderman’ network is a strongly simplified and somewhat meager picture as most spidermen nets are 2D and more or less static and widely independent from the environmenal status. The kinematic viscosity of many non-Newtonian fluids may be described as a power-law function of local shear rate, S, by the following slightly generalized Ostwald–de-Waele ansatz, wherein n0 = n0 (T ) and g = g (T ) are (generally temperature-dependent) empirical parameters: n (S ) = n 0 + g ´ (S S*)n . (3 ) Figure 1 shows three example fluids with (not completely fictive) parameters like g = 10-6 m2 s-1, reference shear S* = 0.19 s−1, n0 = 0.43 m2 s-1; n = {-0.5, +0.65, +1.5}. In the following we exclusively deal with shear-thickening fluids (n > 0 ) and use the notion of turbulence in a narrow sense defined later. However, the theoretical foundations go beyond.
Figure 2. Kinetic-energy spectra for two cases of the Reynolds-
number proxy r ~ Re = l 0 l . Turbulence in the narrow sense is concentrated in the dimensionless wavenumber range g = 1 ... g . In the case of the red curve, this range extends until g ¥.
For the case r = ¥ we have f2 (k, ¥) = 1 (see relation (9) below) and it has been shown (Baumert 2013) 1 that a = 3 (4p )2 3 = 1.80 .
3. Turbulent kinetic energy (TKE) TKE presents itself in form of smooth, continuous spectra. The total TKE is thus the integral over the relevant range in wavenumber and/or frequency space. In the general case and in the narrow sense (i.e. covering only the intertial range), the spectrally integrated TKE is given as follows: =
òk
k
E (k ) d k =
0
w
òw
Eˆ (w ) dw .
b ðω; ρÞ Turbulent frequency spectrum: E
In analogy to the above the generalized form of the frequency spectrum reads as follows:
Eˆ (w , r ) = b ´ fˆ1 (w ) ´ 2 ´ fˆ2 (w , r ) . w
(4 )
0
Here the values k0 and k denote the integration limits for the inertial (‘Kolmogorov’) spectral range. They will be discussed further below. The values w0 and w are corresponding frequencies.
For the case r = ¥ we have fˆ2 = 1 and it has also been shown (Baumert 2013) that b = 2. The frequency spectrum has been given here only for reasons of completness. Further below we concentrate exclusively on the wavenumber spectrum.
Turbulent wavenumber spectrum: E ðk ; ρÞ
Within the inertial (‘Kolmogorov’) range the wavenumber spectrum is universal and follows the 5/3 law, i.e. E µ k-5 3. Not so for small and high wavenumbers where specific semi-empirical laws apply ( f1 and f2 , respectively, both dimensionless, i.e. with the properties3 [f1] = [f2] = 1). To leave room for a generalized treatment we thus write the general wavenumber spectrum as follows: E (k , r ) = a ´ f1 (k ) ´
2 k5
3 3
Dimensionless wavenumber spectrum
To introduce a dimensionless form of the wavenumber spectrum, we first remind the classical wavenumber as k = 2p l and define on this basis a dimensionless wavenumber, g, as follows: g = k k 0 = l 0 l,
´ f2 (k , r ) .
(7 )
(5 )
where k0 and l 0 are defined further below. In the new terminology and with (5), the dimensionless wavenumber spectrum is
Note that f2, responsible for the smallest scales, essentially depends on the Reynolds number, ρ, while f1, responsible for large forcing scales, does not. 3
(6 )
(g , r ) =
[X] means: units of X. 4
a´
E (g ) ´ k 0-5
2 3
3
=
f1 (g) ´ f2 (g , r ) . g5 3
(8 )
Phys. Scr. 91 (2016) 074003
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This simple ansatz for the dissipation region is originally due to Kraichnan (1959) (loc. cit. Pope 2000) and might be considered as universal, although here are alternatives (Pope 2000). However, the specific model choice for f2 does not really matter here. But in any case our Reynolds number proxy, ρ, is a strong control of f2.
Spectral ranges
In the case of developed viscous turbulence, three wavenumber ranges may be distuingished. Their borders are labeled by limiting wavenumbers or length scales. Figure 2 shows two selected such spectra in dimensionless form against a continuous dimensionless wavenumber, g, wherein the following values may be distuingished:
The ideal inviscid case, ρ ¼ ∞
(i) The lower bound of the so-called inertial (or Kolmogorov) subrange, k 0 = 2p l 0 . (ii) The often much larger Kolmogorov wavenumber k = 2p l and its related microscale, l , with the property l l 0 . It is related with the kinematic viscosity ν and the dissipation rate ò of TKE via (1). The latter equation shows that if n 0 , also l 0 . (iii) The so-called energy-containing length-scale, , is located between l 0 and l . It is, together with its associated wavenumber k = 2p representative for the whole inertial range.
This ideal case is characterized by (l, t ) (0, 0), or, equivalently, by (k, w ) (¥ , ¥). Consequently the integrals in (4) read as follows (Baumert 2013), =
Eˆ (w ) dw = p 2 W2 .
(11)
0
k = g = k0
2 p 3 = 7.874,
(12) (13)
These relations hold for r = ¥ only. 4. Stirred vessels, ρ < ∞ The dynamic balances of and Ω in the interior of a stirred vessel are given in general terms as follows (Baumert 2013): ¶ ¶t ¶W ¶t
¶ ⎛ ¶ ⎞ ⎟ = nt S 2 - , ⎜nt x ¶ ⎝ j j⎠
(14)
⎞ 1 ⎛ S2 ¶ ⎛ ¶W ⎞ ⎟= ⎜ ⎜nt - W2 ⎟ . ⎠ p⎝ 2 j ⎝ ¶x j ⎠
(15)
3
å ¶x j=1
3
å ¶x j=1
Here nt = (p W) is the isotropic eddy viscosity; S = áSe2ñ1 2 is the effective (rms) shear rate of the mechanical stirring process, and is the total dissipation rate of smallscale mechanical energy within the vessel.
(9 )
It describes slow, quasi-periodic but chaotic, non-regular quasi-inertial motions and is not essentially influenced by small-scale friction but governed almost exclusively by the forcing mechanism. Therefore it does not essentially depend on our Reynolds number proxy, r = l 0 l .
Box model
While (14) and (15) represent a complex nonlinear system of partial differential equations, here we prefer without too much loss of generality a simple quasi-homogeneous box concept. In particular we write
Sub-spectrum f 2 ðg; ρÞ: Kraichnan’s 1959 model
f2 (g, r ) in (8) describes the so-called dissipation spectral region wherein the smallest eddy motions are concentrated. These motions have quasi-laminar character (for details see pp 232, 233 in Pope 2000) where here (like in von-Karman’s range) the specific mixing mechanisms of the inertial range are missing, too: ⎞⎤ - c2⎟ ⎥ . ⎟⎥ ⎠⎦
0
¥
òw
W p = = 1.571. 2 w0
f1 (g) in (8) is von Karman’s (1948) spectral model (see p 747 Pope 2000) valid in the low-wavenumber spectral range:
4
E (k ) d k =
g =
Sub-spectrum f 1 ðg Þ: von Karman’s 1948 model
⎡ ⎛⎡ ⎛ ⎞4 ⎤1 g ⎢ 4⎥ ⎜ ⎢ f2 (g , r ) = exp - g ⎜ ⎟ + c2 ⎢ ⎜⎢⎣ ⎝ r ⎠ ⎦⎥ ⎝ ⎣
¥
They can analytically be carried out and give—after some algebra—the following results:
The range 0 k < k 0 covers chaotic, quasi-periodic motions situated still outside the ‘inertial subrange’. Here turbulent mixing is weak because specific mechanisms of the proper inertial range (k 0 k k ) are missing. Sometimes these mechanisms are termed space-filling bearings (Herrmann et al 1990) or devil’s gear (Getriebe des Teufels, see Pöppe 2004). This is meant when we wrote above that we consider here turbulence in a narrow sense. Further, this spectral interval is substantially influenced by the specific forcing mechanism of turbulence, e.g. form of propellors, convective processes, pumps etc.
⎛ ⎞17 3 g ⎜ ⎟ . f1 (g) = ⎜ 2 + c ⎟ g ⎝ 1⎠
òk
˙ = nt S 2 - - Y ,
(16)
⎛ 2 ⎞ ˙ = 1 ⎜ S - W2 ⎟ , W ⎠ p⎝ 2
(17)
where Ψ will be discussed further below. Steady state
Equation (17) has the simple steady-state solution
(10)
W=S 5
2.
(18)
Phys. Scr. 91 (2016) 074003
H Z Baumert and B Wessling
Not so equation (16) which knows only growing solutions for a stationary Ω. This can easily be seen when we insert the results of Baumert (2013), nt = (p W) ,
(19)
= W p,
(20)
together with (18) in (16). Obviously the resulting differential equation has a stationary solution = if and only if Y=
S· . p· 2
(21)
Note that at the same time holds = Y . The interpretation of this result goes as follows: Wihin our box model concept, the TKE loss term Ψ describes the wall friction outside the core region of the vessel.
Figure 3. Valid steady-state combinations of parameters l 0 , l . Note that always two states are possible, the upper branch without turbulence in the narrow sense, and the lower branch with a developed Kolmogorov spectrum with the property l l 0 . The forcing length scales are here l 0 = 10 cm, 2 cm, 1 cm, 5 mm, respectively, from outside to inside.
5. Microscale bifurcation Spectral integration
For finite ρ we integrate (5) in the sense of (4) and get
with
⎛ 1 3a 1 ⎞ ´ 2 3 ⎜⎜ 2 3 - 2 3 ⎟⎟ , =2 k0 ⎠ ⎝ k
(22)
=
⎛ l2 3 l2 3 ⎞ 3a ⎟, ´ 2 3⎜ 0 2 3 2 (2 p )2 3 ⎠ ⎝ (2 p )
(23)
=
3 a ´ (2 p )2 2
(24)
3
´ 2 3 (l 20
3
- l2 3) .
a = 25
3
- l2 3 )]3 ,
6
(27)
• The shear frequency, S, is controlled by the rotation speed of (e.g.) the propellor(s) used to stir the vessel. I.e. for the solution of (30), S is thus a given, prescribed input quantity. • The longest turbulent length scale l 0 is controlled by geometry (e.g. radius, length of stirring propellor etc) and represents also a given, prescribed input quantity. • The microscale l however is not given. It is the dynamic relaxation result of the forcing and thus an output quantity.
(28)
As we see in figure 3, for a given length scale l 0 (i.e. along a chosen curve), each admissible shear value S allows for two values of the microscale, l :
(25)
p )-1.
(26)
Now (25) can be inserted into (1) to give the following implicit relation for the Kolmogorov scale, l : ⎛ n 3 ⎞1 l = ⎜ ⎟ ⎝ ⎠
4
⎧ ⎫1 n ⎬ =⎨ 2 3 2 3 ⎩ [a¢ S (l 0 - l )]3 ⎭ ⎪
⎪
⎡ ⎤3 4 n ⎥ , =⎢ ⎢⎣ a¢ S (l 20 3 - l2 3 ) ⎥⎦
4
(a) The upper value is associated with the strangling of turbulence due to the close proximity of l and l 0 . This proximity does not leave enough space for a developed turbulence spectrum (in the narrow sense). Note that l 0 represents a turbulence characteristic which does not directly affect the colloidal structure which was present under laminar or under no-flow conditions and reflects the non-equilibrium character of dispersions/emulsions. (b) The (mostly much) lower value of l represents a turbulence which strongly affects (and will seriously influence) the original colloidal structure and reflects the non-equilibrium character of dispersions/emulsions.
such that finally ⎡ ⎤3 4 n S ⎥ . l = ⎢ ⎢⎣ a¢ (l 20 3 - l2 3 ) ⎥⎦
(29)
For the numerical treatment of (29) we multiply both 2 3 3 4 and get after some algebra the sides with [l 20 3 - l ] following working equation (l2 l 0)2
3
= l2 + a ´
n (S ) S
(31)
The meaning of (30) is most easily understood by highlighting input and output variables and assuming mechanical stirring:
where a¢ = (25
´ p = 5.598.
Physical interpretation
This result we insert together with (18) in (20) and get after some algebra
= [a¢ ´ S ´ (l 20
6
(30)
6
Phys. Scr. 91 (2016) 074003
H Z Baumert and B Wessling
Thus finally we have
Unfortunately nature gives no stability guarantee for this state because there exists the alternative of a strangeled spectrum as discussed above under (a).
⎫ ⎛ S [t ]) ⎞m d n (t ) 1⎧ = ⎨n 0 + g ´ ⎜ ⎟ - n (t )⎬ . ⎝ S ⎠ dt ti ⎩ ⎭ *
This is our first attempt to describe the effects of changes in nanoscopic structures of colloidal systems with respect to viscosity. It is not clear whether the linear ansatz is the correct one among the many other relaxation models. The value of ti can be measured in principle but is not yet available. Its value is important because its relation to intrinsic turbulent time scales matter. Possibly it is related with the internal spatial scale, li , characterizing the colloidal structures. In contrast to the number of open questions, the results achieved so far not only qualitatively explain the previously only phenomenologically observed turbulent viscosities under non-Newtonian conditions.
6. Discussion Smart fluids, smart control?
We have shown in section 5 that a shear-thickening fluid in a stirred vessel may attain two different steady states with respect to the turbulent status. One of them allows colloidal structures to survive, the other one not. Clearly the system cannot stay in both states at the same time. So dynamic transitions between them are physically possible and of practical relevance. The transitions can be understood as ‘bifurcations’ (in the sense discussed in chemical contexts e.g. by Glansdorff and Prigogine 1971). Questions remain. E.g. how does the system choose, for a given set of initial conditions {0 , W0} and forcing parameters {l 0, S}, between the two admissible steady states, and under which conditions (including initialization) transitions between them occur in which specific dynamic form. Possibly smart fluids deserve smart means to keep them under control?
Acknowledgments HZB thanks Snezhana Abarzhi in Pittsburgh and the other organizers of the workshop Turbulent Mixing and Beyond at the Abdus-Sallam International Center for Theoretical Physics (ICTP) in Trieste, for considering our contribution for oral presentation. He further thanks Alexander J Babchin in Tel Aviv for hints on early efforts by Yakov Frenkel (1894–1952) towards a kinetic theory of liquids and certain generalized phase transitions. He thanks further Michael Eckert in Munich for discussions on historical aspects. BW would like to thank Wolfgang C Petersen (1921–1984) enabling him to continue his basic research on colloidal systems.
Relaxation of the ‘inner fluid’
Due to the inner structure of shear-thickening colloidal systems we have to expect that a viscosity according to (3) is never instantaneously realized. Dynamic adjustments to a new micro-mechanical state take certain relaxation steps. They are governed by the laws of irreversible thermodynamics (Wessling 1991, 1993, 1995) and their time scales may become comparable to turbulent adjustment times. Therefore, to get a complete dynamic picture of the stirred fluid with our focus on learning how one of the two steady states is selected and how potential transitions between them are controllable, we actually have at least to augment the equation system (16) and (17) by an additional equation for the dynamic relaxation of the kinematic viscosity (the ‘inner fluid’) against instantateous changes in the shear rate, S = S (t ). At least in the present moment we have no detailed knowledge of the inner relaxation behavior of our test fluid. Therefore we have to apply here the most simple ansatz— linear or ‘first order’ relaxation (‘nudging’). We augment (16) and (17) by the following (linear) relation, d n (t ) 1 = {˜n (S [t ]) - n (t )}. dt ti
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(32)
This equation describes the inner microscopic-dynamic relaxation of n = n (t ) wherein n˜ is the steady-state value of the kinematic viscosity and known from above as the Ostwald–de-Waele relation ⎛ S [t ] ⎞ m n˜ (S [t ]) = n 0 + g ´ ⎜ ⎟ . ⎝ S ⎠ *
(34)
(33)
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Phys. Scr. 91 (2016) 074003
H Z Baumert and B Wessling
conductive polymers. A key to understanding polymer blends or other multiphase polymer systems Synth. Met. 45 119–49 Wessling B 1993 Dissipative structure formation in collodial systems Adv. Mater. 5 300–5 Wessling B 1995 Critical shear rate-the instability reason for the creation of dissipative structures in polymers Z. Phys. Chem. 191 119–35 Wessling B 2000 Conductive polymers as organic nanometals Handbook of Nanostructured Materials and Nanotechnology vol 5 ed H S Nalwa (New York: Academic) ch 10, pp 501–575 namely Fig. 29/p 545 Wessling B 2007 Conductive polymers as organic nanometals Handbook of Conjugated Polymers Processing and Applications 3rd edn ed T A Skotheim and J R Reynolds (Boca Raton, FL: CRC Press) ch 1, pp 1–3 to 1–75, here namely fig 1.10, p 1–11
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