On turbulence in dilatant dispersions Helmut Z. Baumert
[email protected]
Bernhard Wessling
[email protected]
Abstract. This paper presents a new theory on the behaviour of shearthickening (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, λ0 , labeling the begin of the inertial part of the wavenumber spectrum; (ii) the energy-containing scale, L; (iii) Kolmogorov’s micro-scale, λK , related with the size of the smallest vortices existing for a given kinematic viscosity and forcing; (iv) the inner (‘colloidal’) micro-scale, λi , typically representing a major stable material property of the colloidal fluid. In particular, for small ratios r = λi /λK ∼ O(1), various interactions between colloidal structures and smallest turbulent eddies can be expected. In the present paper we discuss particularly that for ρ = λ0 /λK → O(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 [Baumert, √ 2005, 2013]. It predicts e.g. von Karman’s constant correctly as 1/ 2 π = 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 λK , potentially approaching λi . The second branch characterizes a state family with ρ → O(1) and thus strangled turbulence, ρ ≈ O(1). Stability properties and a potential dynamic commuting between the two solution branches had to be left for future research. Keywords: Turbulence, kinematic viscosity, eddy viscosity, colloidal systems, emulsions, dispersions, shear-thickening, mixing, stirring, reactor, non-Newtonian fluid, Kolmogorov scale, bifurcation Submitted: Physica Scripta (IOP) 29 March, 2015 Revised: January 12, 2016
1. Introduction
latant) fluids exhibit increased viscosity upon increased shear stress. Their manufacturing is often strongly controlled by turbulence characteristics like the energycontaining length scale, L, 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
General While the behavior of Eulerian fluids under turbulent conditions is well understood in the meantime [Baumert , 2013], this is not the case for Newtonian and still less for non-Newtonian fluids. Shear-thickening (di-
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Baumert & Wessling: Mixing in dispersions rate of the equipment preparing or using such fluids. 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. 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, λK . 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, λK , 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 λK matters when ρ → O(1) and r → O(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 physicochemical functions. For a colloidal fluid under turbulent conditions, Kolmogorov’s microscale λK is a most important characteristic because if λK 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 superstructure1 . 1 This
can easily be understood, because for manufacturing
According to Kolmogorov, λK follows 3 1/4 ν . λK =
(1)
Here the flux [energy per time and volume, m2 ·s−3 ] is dissipated by the smallest feasible eddies at scale λK [m] into heat; ν [m2 ·s−1 ] is the kinematic viscosity. It holds generally (but not always) that λK λ0 . The higher , the smaller λK 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 material. In nonNewtonian, colloidal dispersions the situation is different. The internal interactions between different fluidmechanical and colloidal variables are subtle2 . In addition to the (until recently) very limited understanding of turbulence they represent a significant extra challenge. 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 Andrej N. 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 Pjotr L. Kapitza and Lev D. Landau participated and the mathematician Alexander M. Obukhov as dispersions, one is already intuitively using stronger turbulence the smaller the final particle size should be. 2 The latter actually represent fine-scale solid-state structures with features like elasticity, load limitations, inner friction etc.
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Baumert & Wessling: Mixing in dispersions well. The summary report was written by Kolmogorov [1942], accompanied by relevant remarks 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 3 ) were present in the room. The documented oral comment by Landau should be read with attention4 : ”. . . 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: Baumert , 2005, 2013] for the vortex-vortex interactions was still missing.
2. Smart fluids / colloidal systems Today so-called smart fluids become increasingly relevant in various industries, from medicine over smallscale 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. As their kinematic viscosity is not a material constant, smart fluids have always non-Newtonian character. They all are at least 2-phase systems, whereas 3 In the mathematical literature, Fokker-Planck equations are sometimes called Kolmogorov-forward equations because Kolmogorov gave them an axiomatic foundation [Kolmogoroff , 1931]. 4 It is documented by many authors, here cited from p. 219 in Falkovich [2011], emphases by the authors.
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 wide-spread 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, λ0 , λK , λi ) get statistical chances to interact. The non-Newtonian behavior of colloidal dispersions/emulsions and suspensions, especially the shearthickening (dilatant) behaviour which can very often be observed, is the object of broad research work [Brown and Jaeger , 2012, 2014; Wagner and Brady, 2009, cf.]. However, several aspects are widely overlooked or ignored: first the fact, that colloidal dispersions and emulsions exhibit a complex 3-dimensional inner structure [Wessling, 1993, cf.] 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, U ×L Re = (2) ν 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 r.m.s. 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 shearthinning (pseudoplastic) fluids. In the former case viscosity increases with increasing shear rate so that the fluid becomes ‘thicker’. The most prominent example
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Baumert & Wessling: Mixing in dispersions 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, 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 3D5 network of 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].
erence shear S∗ = 0.19 s−1 (Hz), ν0 = 0.43 m2 ·s−1 ; n = {−0.5, +0.65, +1.5}. In the following we exclusively deal with shearthickening fluids (n > 0) and use the notion of turbulence in a narrow sense defined later. However, the theoretical foundations go beyond.
3. Turbulent Kinetic Energy (TKE) Turbulent Kinetic Energy (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 sense6 , the spectrally integrated turbulent kinetic energy is given as follows: Z ωK Z kK ˆ E(ω) dω (4) E(k) dk = K = k0
ω0
Here the values k0 and kK denote the integration limits for the inertial (‘Kolmogorov’) spectral range. They will be discussed further below. The values ω0 and ωK are corresponding frequencies. Turbulent wavenumber spectrum: E(k, ρ)
Figure 1. The effective kinematic viscosity in shearthickening (green), shear-thinning (red) and Newtonian fluids (blue) following an Ostwald-de-Waele relation (3) where for illustrative purposes ν0 = 0. 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 ν0 = ν0 (T ) and γ = γ(T ) are (generally temperature-dependent) empirical parameters: ν(S) = ν0 + γ × (S/S∗ )n .
(3)
Figure 1 shows three example fluids with (not completely fictive) parameters like γ = 10−6 m2 ·s−1 , ref5 To call it 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.
Within the inertial (‘Kolmogorov’) range the wavenumber spectrum is universal and follows the 5/3 law, i.e. E ∝ 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 properties [f1 ]=[f2 ]=17 ). To leave room for a generalized treatment we thus write the general wavenumber spectrum as follows: 2/3 × f2 (k, ρ) (5) k 5/3 Note that f2 , responsible for the smallest scales, essentially depends on the Reynolds number, ρ, while f1 , responsible for large forcing scales, does not. For the case ρ = ∞ we have f2 (k, ∞) = 1 (see relation (9) below) and it has been shown [Baumert , 2013] 2/3 that α = 13 (4π) = 1.80. E(k, ρ) = α × f1 (k) ×
ˆ Turbulent frequency spectrum: E(ω, ρ) In analogy to the above the generalized form of the frequency spectrum reads as follows: ˆ E(ω, ρ) = β × fˆ1 (ω) × 2 × fˆ2 (ω, ρ) . (6) ω 6 i.e. 7 [X]
covering only the intertial range means: units of X.
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Baumert & Wessling: Mixing in dispersions For the case ρ = ∞ we have fˆ2 = 1 and it has also been shown [Baumert , 2013] that β = 2. The frequency spectrum has been given here only for reasons of completness. Further below we concentrate exclusively on the wavenumber spectrum.
Further, this spectral interval is substantially influenced by the specific forcing mechanism of turbulence, e.g. form of propellors, convective processes, pumps etc.
Dimensionless wavenumber spectrum To introduce a dimensionless form of the wavenumber spectrum, we first remind the classical wavenumber as k = 2π/λ and define on this basis a dimensionless wavenumber, g, as follows: g = k/k0 = λ0 /λ,
(7)
where k0 and λ0 are defined further below. In the new terminology and with (5), the dimensionless wavenumber spectrum E is E(g, ρ) =
E(g) α×
2/3
×
−5/3 k0
=
f1 (g) × f2 (g, ρ) . (8) g5/3
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 E against a continuous dimensionless wavenumber, g, wherein the following values may be distuingished: (i) The lower bound of the so-called inertial (or Kolmogorov) subrange, k0 = 2π/λ0 . (ii) The often much larger Kolmogorov wavenumber kK = 2π/λK and its related microscale, λK , with the property λK ≤ λ0 . It is related with the kinematic viscosity ν and the dissipation rate of turbulent kinetic energy via (1). The latter equation shows that if ν → 0, also λK → 0. (iii) The so-called energy-containing length-scale, L, is located between λ0 and λK . It is, together with its associated wavenumber kL = 2π/L representative for the whole inertial range. The range 0 ≤ k < k0 covers chaotic, quasi-periodic motions situated still outside the ‘inertial subrange’. Here turbulent mixing is weak because specific mechanisms of the proper inertial range (k0 ≤ k ≤ kK ) are missing8. 8 Sometimes these mechanisms are termed space-filling bearings [Herrmann et al., 1990] or devil’s gear [Getriebe des Teufels, see P¨ oppe, 2004]. This is ment when we wrote above that we consider here turbulence in a narrow sense.
Figure 2. Kinetic-energy spectra for two cases of the Reynolds-number proxy ρ ∼ Re = λ0 /λK . Turbulence in the narrow sense is concentrated in the dimensionless wavenumber range g = 1 . . . gK . In the case of the red curve, this range extends until g → ∞. Sub-spectrum f1(g): von Karman’s 1948 model f1 (g) in (8) is von Karman’s (1948) spectral model [see Pope, 2000, p. 747] valid in the low-wavenumber spectral range: f1 (g) =
g p g 2 + c1
!17/3
.
(9)
It describes slow, quasi-periodic but chaotic, nonregular 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, ρ = λ0 /λK . Sub-spectrum f2(g, ρ): Kraichnan’s 1959 model f2 (g, ρ) 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 p. 232, 233 in Pope, 2000] where here (like in von-Karman’s range) the specific mixing mech-
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Baumert & Wessling: Mixing in dispersions anisms of the inertial range are missing, too: " #1/4 4 g f2 (g, ρ) = exp − γ + c42 − c2 . (10) ρ
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 .
too much loss of generality a simple quasi-homogeneous box concept. In particular we write K˙ =
νt S 2 − − Ψ, 1 S2 − Ω2 , π 2
˙ = Ω
(16) (17)
where Ψ will be discussed further below. Steady state Equation (17) has the simple steady-state solution √ Ω = S/ 2. (18)
The ideal inviscid case, ρ = ∞ This ideal case is characterized by (λK , τK ) → (0, 0), or, equivalently, by (kK , ωK ) → (∞, ∞). Consequently the integrals in (4) read as follows [Baumert , 2013], Z ∞ Z ∞ ˆ K= E(k) dk = E(ω) dω = π L2 Ω2 . (11) k0
ω0
They can analytically be carried out and give – after some algebra – the following results: gL
=
Ω ω0
=
√ kL = γL = 2 π 3 = 7.874 , k0 π = 1.571 . 2
(12) (13)
These relations hold for ρ = ∞ only.
4. Stirred vessels, ρ < ∞ The dynamic balances of K and Ω in the interior of a stirred vessel are given in general terms as follows [Baumert , 2013]: 3
∂K X ∂ − ∂t ∂xj
νt
∂K ∂xj
∂Ω X ∂ − ∂t ∂xj
∂Ω νt ∂xj
j=1 3
j=1
=
νt S 2 − ,
=
1 π
(14)
S2 2 − Ω . (15) 2
Here νt = K/(π Ω) is the isotropic eddy viscosity; S = hSe2 i1/2 is the effective (r.m.s.) shear rate of the mechanical stirring process, and is the total dissipation rate of small-scale mechanical energy within the vessel. Box model While (14, 15) represent a complex non-linear system of partial differential equations, here we prefer without
Not so equation (16) which knows only growing solutions for a is stationary Ω. This can easily be seen when we insert the results of Baumert [2013], νt
= K/(π Ω), = K Ω/π,
(19) (20)
together with (18) in (16). Obviously the resulting differential equation has a stationary solution K = K if and only if S·K √ . (21) Ψ= π· 2 Note that at the same time holds = Ψ. The interpretation of this result goes as follows: Wihin our box mocel concept, the TKE loss term Ψ describes the wall friction outside the core region of the vessel.
5. Microscale bifurcation Spectral integration For finite ρ we integrate (5) in the sense of (4) and get ! 3α 1 1 K = − × 2/3 − 2/3 (22) 2/3 2 kK k0 ! 2/3 2/3 λK 3α λ0 2/3 = × − (23) 2 (2π)2/3 (2π)2/3 3 α 2/3 2/3 2/3 = × × λ − λ . (24) 0 K 2 (2π)2/3 This result we insert together with (18) in (20) and get after some algebra h i3 2/3 2/3 = α0 × S × λ0 − λK
(25)
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Baumert & Wessling: Mixing in dispersions where
−1 α0 = 25/6 π .
(26)
Now (25) can be inserted into (1) to give the following implicit relation for the Kolmogorov scale, λK :
λK
=
=
3
ν
1/4
1/4 ν = h (27) i3 α0 S λ2/3 − λ2/3
0
ν
α0 S
such that finally
2/3 λ0
λK =
2/3
− λK
K
3/4
,
ν/S α0
2/3 λ0
2/3
− λK
(28)
3/4
.
(29)
For the numerical treatment of (29) we multiply both i3/4 h 2/3 2/3 sides with λ0 − λK and get after some algebra the following working equation, (λ2K λ0 )2/3 = λ2K + α00 ×
ν(S) S
(30)
with α00 = 25/6 × π = 5.598 .
(31)
Physical interpretation The meaning of (30) is most easily understood by highlighting input and output variables and assuming mechanical stirring: • 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 λ0 is controlled by geometry (e.g. radius, length of stirring propellor etc.) and represents also a given, prescribed input quantity. • The microscale λK however is not given. It is the dynamic relaxation result of the forcing and thus an output quantity. As we see in Figure 3, for a given length scale λ0 (i.e. along a chosen curve), each admissible shear value S allows for two values of the microscale, λK :
Figure 3. Valid steady-state combinations of parameters λ0 , λK . 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 λK λ0 . The forcing length scales are here λ0 = 10 cm, 2 cm, 1 cm, 5 mm, respectively, from outside to inside. (a) The upper value is associated with the strangling of turbulence due to the close proximity of λK and λ0 . This proximity does not leave enough space for a developed turbulence spectrum (in the narrow sense). Note that λ0 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 λK represents a turbulence which strongly affects (and will seriously influence) the original colloidal structure and reflects the non-equilibrium character of dispersions / emulsions. Unfortunately nature gives no stability guarantee for this state because there exists the alternative of a strangeled spectrum as discussed above under (a).
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
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Baumert & Wessling: Mixing in dispersions 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 {K0 , Ω0 } and forcing parameters {λ0 , 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? 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, 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, 17) by the following (linear) relation, dν(t) dt
=
1 {˜ ν (S[t]) − ν(t)} . τi
(32)
This equation describes the inner microscopic-dynamic relaxation of ν = ν(t) wherein ν˜ is the steady-state value of the kinematic viscosity and known from above as the Ostwald-de-Waele relation, m S[t] ν˜(S[t]) = ν0 + γ × . (33) S∗ Thus finally we have m dν(t) 1 S[t]) = ν0 + γ × − ν(t) . (34) dt τi S∗
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 τi 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, λi , 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. Acknowledgements. HZB thanks Snezhana Abarzhi in Pittsburgh and the other organizers of the workshop Turbulent Mixing and Beyond at the AbdusSallam 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.
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[email protected])
This preprint was prepared with AGU’s LATEX macros v5.01, with the extension package ‘AGU++ ’ by P. W. Daly, version 1.6b from 1999/08/19.
