# A Novel Approach for Modeling the Non-Newtonian Behavior of Simple Liquids: Application to Liquid Water Viscosity from Low to High Shear Rates

^{*}

## Abstract

**:**

## 1. Introduction

“we believe that all liquids exhibit similar non-Newtonian behaviour […] and therefore it has a fundamental physical and theoretical significance”.

## 2. Influence of External Actions in the Elastic Mode Theory

#### 2.1. Generalized Expression of the Elastic Energy Functional

_{B}molecules or atoms) property due to the non-zero temperature of the system at equilibrium. The random variable $\overrightarrow{a}$ can optionally represent the instantaneous position $\overrightarrow{u}$ or the orientation $\overrightarrow{\mathsf{\Omega}}$ of the local director of a basic unit. The starting point of this model is the assumption that $\overrightarrow{a}\left(\overrightarrow{r}\right)$ can be developed into a Fourier series (whose coefficients are referred to as elastic modes) on the lattice. Therefore, for component a

_{x}of $\overrightarrow{a}$, one has:

_{a}is an elastic constant.

_{eq}+ E

_{ext}

_{eq}is given by Equation (2) and ${E}_{\mathrm{ext}}=\frac{1}{2}{\displaystyle \int {f}_{E}dV}$, ${f}_{E}$ being the energy per unit volume, which describes the coupling between random microscopic motions and deterministic macroscopic motions.

_{eff}tends to 1. In other words, when the particles have been made independent of each other by a sufficiently intense external action, the size of the thermal cloud is zero and the particle behaves similar to a classical object of the material point mechanics.

#### 2.2. Application to Translational Macroscopic Motion

_{t}= T. In this case, it is assumed that $\overline{\lambda}$ is written in the following form:

_{eff}~ 1. Therefore, by increasing the velocity gradient, the fluid system goes from a “coherent” state where all fluid particles are indistinguishable in a large volume, to an “incoherent” state where all fluid particles are independent from each other. This result suggests to identify this dynamical transition as some kind of Bose–Einstein condensation, where it is the velocity gradient, thus, the action temperature T

_{A}(see ref. [8]) is the relevant parameter, and not the thermodynamic temperature T as modeled in the framework of standard statistical mechanics. This transition will be illustrated in the case of water vapor in Section 3.2. This feature will also be discussed in more detail in a forthcoming paper. Another way to understand this result regarding the reduction of thermal fluctuations $<\left|{u}^{2}\right|>$ caused by velocity gradients is presented in Appendix A in the form of an expression that is reminiscent of the uncertainty principle of Quantum Mechanics.

_{pm}. Let us remember that ${c}_{0}$ characterizes the velocity at which the shear information propagates in the medium. Strictly speaking, in a real experiment, the relation ${c}_{0}=\sqrt{K/\rho}$ is only valid in an ideal case where the no-slip condition is rigorously satisfied on all surfaces in contact with the liquid. In a less perfect case (e.g., presence of bubbles, more or less hydrophobic surface), the information will take longer to cross the system, so it is as if ${c}_{0}$ had to be replaced by an effective celerity ${c}_{eff}=\sqrt{{K}_{eff}/\rho}$ where K

_{eff}represents an effective shear elastic constant such that ${K}_{eff}={C}_{K}K$ with ${C}_{K}<1$. If the non-perfection surface can be pictured as region where the fluid is freely slipping, the coefficient ${C}_{K}$ is a parameter that can be interpreted as the ratio of the detached surface to the total surface.

_{N}that quantifies the disorder which governs all properties, and since the disorders are additive, it is possible to define a new mean value $\u2329\lambda \u232a$ of λ by an average using the function H

_{N}such that: ${H}_{N}\left(v,\u2329\lambda \u232a\right)=\frac{1}{V}{\displaystyle \underset{V}{\int}{H}_{N}\left(v,\lambda \left(\overrightarrow{r}\right)\right)dV}$. Since ${H}_{N}\left(v,\lambda \right)$ is large for small λ, the value of $\u2329\lambda \u232a$ is determined by low gradient values rather than high ones. Therefore, it is the quietest regions of the sample that control the reduction of fluctuations in $<\left|{u}^{2}\right|>$; the larger the corresponding volume, the greater the contribution of these quietest regions.

_{t}(T, ρ). These state equations have been determined for water in ref. [7] and for potassium and thallium in ref. [9]. We will explore in the following section the consequences of the present modeling for various experiments done to determine the dynamic viscosity of water. We recall that the calculation program corresponding to ref. [7] can be freely downloaded from ref. [10].

## 3. Application to Different Experiments of Water Viscosity Measurements

#### 3.1. Exploration of the Liquid-Like Phase

^{−1}, the difference between ${H}_{N}\left(v,\lambda \right)$ and ${H}_{N}\left(v\right)$ is decades smaller than the smallest error bars associated with water viscosity measurements (i.e., there is no difference for calculations to consider ${H}_{N}\left(v\right)$ instead of ${H}_{N}\left(v,\lambda \right)$).

^{−1}and 1706 s

^{−1}. According to Figure 4, it can be seen that these values are distributed around the “separation curve” of Equations (8) and (10). However, these values represent a maximum value of the gradient in the sample. Now, it was mentioned earlier that ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}$ is essentially determined by the quietest flow regions, i.e., here, by what happens in the water tank, which has a radius about 100 times larger than the capillary tube. It is deduced that the required average ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}$ is such that ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}\approx \left|{\left(\nabla {v}_{f}\right)}_{\begin{array}{l}\mathrm{tube}\\ \mathrm{wall}\end{array}}\right|/{100}^{3}$, i.e., the value to be considered for this experiment is of the order 10

^{−3}s

^{−1}(or a value of λ ≈ 1.8 × 10

^{−16}). Figure 4 shows that, for such a value, there is no calculable difference between ${H}_{N}\left(v\right)$ and ${H}_{N}\left(v,\lambda \right)$.

^{−1}and 217 s

^{−1}. However, as in the experiment of Korosi et al., one must consider what happens in the water tank, which has a radius 200 times larger than that of the capillary tube. It is deduced that the average ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}$ is such that ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}\approx \left|{\left(\nabla {v}_{f}\right)}_{\begin{array}{l}\mathrm{tube}\\ \mathrm{wall}\end{array}}\right|/{200}^{3}$, of the order 10

^{−5}s

^{−1}(or a value of λ ≈ 1.8 × 10

^{−18}). Here, again, it can be considered that there is no difference between ${H}_{N}\left(v\right)$ and ${H}_{N}\left(v,\lambda \right)$.

_{S}= 25.4 µm and a density ρ

_{S}= 2.42 g/cm

^{3}. The analysis of these data in ref. [7] showed that the dissipative distance d = R

_{S}should be considered here. It is assumed that a maximum value of the velocity gradient on the ball surface can be determined from the following formula: $\left|{\left(\nabla {v}_{f}\right)}_{\begin{array}{l}\mathrm{ball}\\ \mathrm{surface}\end{array}}\right|=\frac{2{R}_{\mathrm{S}}\left({\rho}_{\mathrm{S}}-\rho \right)g}{9\eta}$. In the studied pressure range, $\left|{\left(\nabla {v}_{f}\right)}_{\begin{array}{l}\mathrm{ball}\\ \mathrm{surface}\end{array}}\right|$ varies between 51 s

^{−1}and 77 s

^{−1}along the isotherm at 293.15 K. If we simply assimilate these values to ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}$, it is found that the relative deviation between ${H}_{N}\left(v\right)$ and ${H}_{N}\left(v,\lambda \right)$ varies from 5 × 10

^{−6}% to 3.5 × 10

^{−7}%, which is totally negligible in the calculation of viscosity, given that the uncertainty of the data is ±5%. Now, the diameter of the tube is 27.6 times larger than the diameter of the ball, therefore, if we take it as the diameter of an effective water tank then we obtain ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}\approx 3\times {10}^{-3}{\mathrm{s}}^{-1}$ (or a value of λ ≈ 5 × 10

^{−16}). This leads to the same conclusion as in the previous examples.

^{−3}s

^{−1}and 2.1 × 10

^{−2}s

^{−1}. It is then deduced that the relative deviation between ${H}_{N}\left(v\right)$ and ${H}_{N}\left(v,\lambda \right)$ varies from 5 × 10

^{−14}% to 4 × 10

^{−12}%, which is totally negligible in the calculation of viscosity.

#### 3.2. The Dilute-Gas Limit

**(i)**if $\overline{\lambda}$ is sufficiently small, i.e., satisfies condition (11), then one has $\underset{\rho \to 0}{\mathrm{lim}}v\to 2$ and $\underset{\rho \to 0}{\mathrm{lim}}{H}_{N}\left(v,\lambda \right)\cong {H}_{N}\left(2,0\right)=N-1\propto {\rho}^{2}$. In total, in the weak perturbation limit, $<\left|{u}^{2}\right|>$ diverges as $1/{\rho}^{2}$. In practice, this divergence of $<\left|{u}^{2}\right|>$ is limited by the finite sample size, comparable to the fluctuative distance ${d}_{N}$;

**(ii)**If, instead, $\overline{\lambda}$ is sufficiently large, according to Equation (9), ${H}_{N}\left(v,\lambda \right)\propto \left(1/{\lambda}^{3}\right)\left(1-1/{N}^{3}\right)$. Since, according to Equation (17), $\underset{\rho \to 0}{\mathrm{lim}}\lambda \propto {\rho}^{-\frac{4}{3}}$, and since $\underset{\rho \to 0}{\mathrm{lim}}\left(N-1\right)\propto {\rho}^{2}$, then, in first approximation, $\underset{\rho \to 0}{\mathrm{lim}}{H}_{N}\left(v,\lambda \right)\propto {\rho}^{6}$. In total, for $\rho \to 0$, in the strong perturbation limit, $<\left|{u}^{2}\right|>$ is proportional to ${\rho}^{2}$.

^{−1}. It is then less and less visible as the density increases.

_{N}function. However, the liquid term ${\eta}_{l}$ represents about half of the total viscosity value and therefore, in order to validate this analysis, it is important to show that ${H}_{N}\left(v,\lambda \right)$ does not differ significantly from ${H}_{N}\left(v\right)$ in the corresponding experimental conditions.

^{4}s

^{−1}and 1.476 × 10

^{5}s

^{−1}is obtained. Figure 6 shows that if we assign these values to ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}$ then we are in the region where Equation (10) varies strongly with the velocity gradient parameter (on the other hand, it can be observed that the function H

_{N}does not vary with temperature). As was done in the previous section, a ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}$ value is determined here using the ratio of the lengths of the two capillary tubes. It is then deduced that ${\left(\nabla {v}_{f}\right)}_{\mathrm{av}}$ varies between 3136 s

^{−1}and 10 286 s

^{−1}. The relative deviation between ${H}_{N}\left(v\right)$ and ${H}_{N}\left(v,\lambda \right)$ varies from 2 × 10

^{−3}% to 7 × 10

^{−2}%, which is totally negligible in the calculation of viscosity given that the uncertainty of the viscosity data is ±1%.

_{N}function. It is observed that the deviation only starts to become significant in relation to the experimental uncertainties in the region near the critical point. It can be concluded that it is possible to use Equation (8) instead of Equation (10) for all usual viscometry experiments with a vapor density lower than 0.05 g/cm

^{3}.

_{max}(and therefore λ) can become infinite in the limit $\rho \to 0$. In other words, N

_{eff}tends to 1 in this limit. This still implies that the particles can only be considered as isolated in the limit $\rho \to 0$. However, it should be noted that the divergence of λ is theoretical since, from an experimental point of view, one cannot reach $\rho =0$. Thus, in practice, the divergences do not exist but only result in large numbers. Finally, it should be noted that in the limit $\rho \to 0$, the cutoff pulsation ${\omega}_{c}$ of the inertial mode theory tends to zero as ${c}_{0}$. In other words, there exists in this limit only the transient regime, which persists for an infinitely long time (i.e., the inertial modes are irrelevant).

#### 3.3. Experiments from Low to High Shear Rates at Atmospheric Pressure and Room Temperature

#### 3.3.1. Presentation of Devices and Experimental Results

^{−1}. This qualitative behavior is not specific to the double cone experiment, but is also observed in the simple cone experiment as well as with the second experimental device.

^{3}and the 2008 IAPWS formulation (Ref. [17]) gives a liquid water viscosity η = 1.0016 mPa·s. The elastic mode theory (Ref. [7]) gives a liquid water viscosity η = 1.0018 mPa·s with a Knudsen term ${\eta}_{Knu}=0.0379\mathrm{mPa}\xb7\mathrm{s}$. However, the apparatus constants in these experimental devices were set to find η = 1.002 mPa·s. This said, the absolute values given by the experimental devices vary quite strongly, while the variations of viscosity as a function of the velocity gradient can be always superimposed for a given type of experiment. Changing the rise time by a factor of 2 to 4 leads to the same results to experimental precision.

#### 3.3.2. Experimental Results Analysis

^{−1}and indicates a positive curvature of the viscosity variation for lower velocity gradients, while the curvature is negative for high values of the velocity gradient. One has the feeling that the curve tends asymptotically to zero when the velocity gradient becomes very large. From a theoretical point of view, given Equation (9), Equation (19) is written in the limit:

^{11}s

^{−1}, using the parameters of Figure 10. In practice, for this velocity gradient value, the turbulence has already appeared and the present modeling is no longer valid (see Appendix B).

_{t}, are not impacted by the coefficient C

_{K}. Specifically, the value of exponent v, which appears in all calculations, is not affected by this replacement.

_{eff}, the data in Figure 9 corresponding to the simple cone experiment can be analyzed similarly to those from the double cone experiments. First of all, it can be seen in Figure 12a that the plateau corresponding to subregion 2 is strongly shifted in absolute value. Moreover, this plateau is rather narrow. The theoretical curve determined with Equation (19) shows that the beginning and the end of subregion 3 can be reproduced correctly, while the middle appears as a bump. Figure 12b shows that the evolution of the viscosity is only a consequence of the stress variation. In this same figure, it can be seen that the stress has a significant change of evolution around 400 s

^{−1}and then suddenly returns to a “normal” behavior beyond 970 s

^{−1}. If the bump is ignored for the moment, the theoretical curve requires that the parameter d

_{N}be equal to the cone radius. This is perfectly consistent with what was found in ref. [8] with the case of the plate-plate rheometer, which also contained a free surface. The presence of this free surface also has the effect of inducing wall slip, which is translated in the elastic mode theory by a decrease in the static shear elastic constant K value. Indeed, the theoretical curve requires the decrease of K by a coefficient C

_{K}= 0.7655. This is also in perfect agreement with what has been shown in ref. [8].

_{K}, which must decrease until it reaches a value 1.49 times lower. The relatively sharp decrease of the viscosity during the stalls observed at high shear rate in the simple cone experiments can be interpreted as a reduction of the slip for certain values of the rate, phenomena probably related to the existence of a free surface for the liquid combined with the unavoidable vibrations associated with relatively high velocities of rotating mechanical systems.

_{N}corresponding to the height of water in the cell. In addition, a lower value of the static shear elastic constant K must be introduced. This is in accordance with the fact that there is the presence of a free surface as in the simple cone experiment, which induces a slip, but it appears as an intensification of this slip because of the more hydrophobic material of the Couette cell compared to the stainless steel of the cone. The value of C

_{K}is perfectly compatible with the results of Badmaev et al. (Ref. [18]) to reproduce the values of the liquid water shear elastic modulus G′ for low wettability of the contact surface (see the corresponding discussion in ref. [8]).

^{−1}and 1130 s

^{−1}. Again, this bump represents an “abnormally” strong slip. Figure 13c shows that this bump can be reproduced by only decreasing the value of the coefficient C

_{K}until it reaches a value 1.76 times lower. This analysis is therefore consistent with that of the simple cone experiment.

_{N}and C

_{K}in each experiment. The analysis was done in relative form using an empirical value of the viscosity ${\eta}_{0}$. To determine an absolute value of ${\eta}_{0}$, the value of the dissipative distance d must be fixed. To be consistent, the analysis of subregion 1 should be described using the inertial mode theory presented in ref. [8]. This theory introduces the notions of action temperature T

_{A}, of viscous mass ${K}_{\mathrm{A}0}^{t}$ (further details on this notion are given in Appendix C) and of threshold stress ${\sigma}_{T}=\left({K}_{N}+{K}_{gas}\right)\frac{\xi}{e}$, where ${K}_{N}=\frac{K}{{H}_{N}\left(v,\lambda =0\right)}$ represents the macroscopic static shear elastic modulus, ${K}_{gas}$ is the shear elastic modulus of the released gas and ξ is the correlation length between the fluid basic units. The parameter e represents a characteristic distance of the experimental set-up. In the case of the plane-plane rheometer with a small air gap e

_{g}, it was shown in ref. [8] that, for these conditions, e = d = e

_{g}. It has been shown that ξ is compatible with the value ξ = e

_{g}at zero strain (corresponding to a zero-action temperature) and then decreases when the strain increases until reaching the value ξ

_{0}corresponding to an action temperature consistent with the establishment of a Newtonian regime. Let us recall that ξ

_{0}is an intrinsic property of the fluid, which represents the distance over which the fluctuations of the unit cells are correlated in the bulk phase at thermodynamic equilibrium (see ref. [7]).

_{A}becomes sufficiently large in front of the reaction temperature T

_{A0}so that we can consider the flow regime as Newtonian. Thus, by definition, we write:

_{A}>> T

_{A0}and $t>>{\omega}_{c}^{-1}={\tau}_{c}$, where ${\omega}_{c}$ represents the cut-off pulsation of the inertial modes. In the case of the experiment with the HAAKE viscometer or with the Couette cell, E

_{A}(X) is identified with the experimentally determined or imposed stress σ and ${v}_{f}\left(X,t\right)$ with the maximum radial velocity. The parameter ${C}_{cal}$ represents the calibration constant that must be applied to ${\eta}_{0}$ to compensate for the offset of the experimental data corresponding to the plateau of subregion 2.

_{1}, σ

_{2}, ε

_{1}and ε

_{2}are four empirical constants. It is immediately seen that Equation (24) has the correct boundary properties when σ = 0 and σ >> σ

_{1}. We preferred here a description in terms of the shear stress σ rather than in terms of strain rate, which allows us to have a single input parameter for the whole modeling. The evolution of the parameter ξ is shown in Figure 14c and we note that the variation is limited to subregion 1 at low strain rates. Therefore, as expected, the value of ξ is equal to ξ

_{0}in subregion 2.

^{−1}). Thus, the measurements made essentially correspond to a regime in which the medium behaves as a liquid.

_{0}. This is consistent with the fact that ${T}_{\mathrm{A}}^{*}$ takes a higher value for the same velocity gradient in the simple cone experiments than with the double cone. It is observed that the medium reaches the same liquid-like regime in subregion 2 with ${T}_{\mathrm{A}}^{*}$ values between 25 and 40.

_{gas}was negligible in the experiment with the double cone and about three times lower in the experiment with the simple cone. This shows the importance of taking into account the gas released during the shear action.

_{0}. This very fast variation of ξ is consistent with the fact that very large values of ${T}_{\mathrm{A}}^{*}$ are quickly obtained, as can be seen in Figure 16c. This also explains that subregion 2 is reached for lower values of the velocity gradient than in the previous experiments. The decrease of the effective viscosity in subregion 1 when the strain rate increases is no longer related to the evolution of ξ, but corresponds to the transient regime. Indeed, in the experiments with the HAAKE viscometer, the characteristic time ${\tau}_{c}$ of this transient regime is of the order of 0.1 s, i.e., the first experimental point recorded already corresponds to at least 10${\tau}_{c}$. On the other hand, in the experiments with the Couette cell, the characteristic time ${\tau}_{c}$ is of the order of 10 s. Subregion 2 is reached after a time of about 4${\tau}_{c}$. Although the medium transits very quickly to a Newtonian liquid-like regime, there is still a time needed for the steady state to set in and, thus, for subregion 2 to be established. This transient regime was “hidden” by its very short duration in the HAAKE experiments.

“It was discovered here that simple liquids (e.g., argon, chlorine and water) behave rheologically [i.e., they should shear thin and shear thicken] in much the same way as these more chemically complicated fluid mixtures [e.g., mineral oil, polysaccharide xanthan gum]”.

**Figure 17.**Liquid n-octane experiment at atmospheric pressure and T = 293.15 K with the HAAKE simple cone by imposing a linear strain ramp. (

**a**) Viscosity variation as a function of velocity gradient. The black horizontal line represents the expected viscosity value according to ref. [19]. (

**b**) Stress variation as a function of velocity gradient.

## 4. Synopsis and Conclusions

**(i)**The intrinsic parameters are “necessary”, they are of a physico-chemical nature and concern the system at thermodynamic equilibrium: molecular composition, thermodynamic quantities such as the critical parameters, the phase diagram as well as all thermodynamic functions, to which are added, in the framework of the model, the elastic constant K

_{0}, the shear celerity c

_{0}, the cutoff wave-vector q

_{c}of the elastic modes, the correlation length ξ

_{0}and the number of atoms/molecules in the basic unit n

_{B}, all related to the thermodynamic parameters.

**(ii)**The extrinsic parameters are “contingent”: they depend on the type of experiment (here, a “flow” in the broad sense of the term) that we want to carry out: the size, shape and volume of the samples through the fluctuative distance d

_{N}and dissipative distance d, the nature of the walls of the container through the parameter K

_{N}, the parameter λ associated with the average velocity gradient caused by the imposed mechanical stress and possibly other additional external fields such as an electric or magnetic field for more complex experimental situations.

_{0}defined by Equation (10) of ref. [7]. This behavior is in perfect agreement with the fact that, sufficiently close to thermodynamic equilibrium (not net flow), any finite volume of fluid must be considered as a solid, as abundantly demonstrated by numerous rheology experiments with sub-millimeter size samples at very low shear stress or strain amplitude and frequency (see ref. [8] for more details).

_{eff}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. “Uncertainty Relationship”

^{14}s

^{−1}at 200 K and 10

^{15}s

^{−1}in the normal liquid phase. These gradient values are very large and far exceed the values of the experiments we have analyzed in this paper. They are also large in front of the values of λ for which turbulence appears in the usual experiments, as described in Appendix B. Therefore, the coefficient in front of ${\hslash}^{2}/4$ is greater than 1 in the usual experiments with liquid water.

**Figure A1.**Representation of Equation (A3) in the normal liquid and supercooled phases of water between 200 K and the critical temperature. The thick red curve represents the liquid coexistence curve in the normal liquid phase and then extends into the supercooled phase with the atmospheric isobar.

^{15}s

^{−1}and 10

^{16}s

^{−1}. These values are thus comparable to those obtained for the liquid phase. These values can therefore be reached more easily with gases than with liquids.

**Figure A2.**Representation of Equation (A3) in the gaseous phase of water between the triple point and critical temperature, and from the triple point gas density to the critical one. The thick red curve represents the vapor coexistence curve.

## Appendix B. Criterion for Transition to Turbulence

^{−8}s, it is deduced that the condition will be fulfilled when the velocity gradient will be of the order of ten megahertz. In Section 2.2, the following notation has been introduced:

**Figure A3.**Evolution of the critical function ${H}_{N}\left(v,\lambda ={l}_{pm}/d\right)$ (with d = 0.01 cm) versus the temperature along the atmospheric isobar of liquid water, between the triple point temperature T

_{tr}and the saturated vapor pressure curve temperature T

_{σ}.

- In the case of tubes, L corresponds to the diameter of the tube D, and, if the capillary tube is small enough, then d = D/2 and the critical Reynolds number is such that ${\mathrm{Re}}_{\mathrm{crit}}\cong 2{H}_{N}\left(v,\lambda =\frac{{l}_{pm}}{d}\right)$. At room temperature, we deduce from Figure A3 that ${\mathrm{Re}}_{\mathrm{crit}}\cong 2242$. This value typically corresponds to the minimum value allowed for this type of experimental device.
- In the case of channels, L corresponds to the distance between the two parallel planes, and, if the thickness of the channel is small enough, then d = L and the critical Reynolds number is such that ${\mathrm{Re}}_{\mathrm{crit}}\cong {H}_{N}\left(v,\lambda =\frac{{l}_{pm}}{d}\right)$. At room temperature, we deduce from Figure A3 that ${\mathrm{Re}}_{\mathrm{crit}}\cong 1121$. This value typically corresponds to the minimum value allowed for this type of experimental device.
- In the case of flow over a flat plate, L corresponds to the critical distance x
_{c}to the upstream edge of the plate. Typical values of x_{c}are of the order of magnitude of a few centimeters, therefore, at room temperature, the minimum value of the critical Reynolds number is ${\mathrm{Re}}_{\mathrm{crit}}\cong \mathrm{112,100}$. This value typically corresponds once again to the minimum value allowed for this type of experimental device.

## Appendix C. Notion of Viscous Mass

**viscous mass**. The expression for ${K}_{\mathrm{A}0}^{t}$ defined by Equation (20) in ref. [8] involves the function ${H}_{N}\left(v\right)$, which means that ${K}_{\mathrm{A}0}^{t}$ was assumed to be independent of the dynamic state of the fluid. We have seen that in practice this approximation is sufficient in the Newtonian regime, but, beyond this regime, the dependence on the parameter λ can no longer be ignored and must be considered by replacing ${H}_{N}\left(v\right)$ with ${H}_{N}\left(v,\lambda \right)$. In the framework of the generalized formalism introduced in this article, it is then interesting to discuss more deeply this notion of viscous mass. Its expression is defined by Equation (20) of ref. [8] as the sum of two terms such that:

_{0}, the mean free path by the dissipative distance d and the mass of an atom/molecule by the viscous mass. In other words, the microscopic quantities of the kinetic theory of gases are replaced by macroscopic quantities at the sample scale. Despite a form of dissimilarity in the expression of the viscous masses, Equations (A12) and (A13) show a uniformity in the description of the viscous terms.

_{f}is that of the particle with respect to the laboratory reference frame and the reference velocity is the speed of light c, whereas, in the present modeling, the former is the average relative velocity between two neighboring particles in the fluid ${\overline{v}}_{m}={\left(\nabla {v}_{f}\right)}_{\mathrm{av}}{l}_{pm}$, and the latter is the shear velocity ${c}_{0}$. In a normal fluid flow, the relativistic aspect is totally negligible as far as the transported energy is concerned, and, therefore, the total mass can be assimilated to the rest (or proper) mass, while the viscous “relativistic” aspect, although negligible in the Newtonian regime, becomes fundamental beyond, since it governs not only the variation of dissipation associated with the increase of viscosity (see Figure 8), but also the appearance of instabilities such as turbulence (see Appendix B).

^{−11}Pa, which is much lower than the value of the triple point pressure of water, a situation which is not reached in the usual experiments.

**subshearic**) to the regime (which we call

**supershearic**) where it increases according to a power law in λ satisfying Equations (9) and (13), is obtained when λ = 1, i.e., when

**Figure A4.**Evolution of Equation (A16) versus v and N. This surface separates the subshearic regime to the supershearic one.

_{N}is a self-similar function (e.g., at the transition v = 1, all derivatives of the function H

_{N}(v) are such that $\underset{v\to 1}{\mathrm{lim}}\frac{{d}^{n}}{d{v}^{n}}{H}_{N}\left(v\right)=\frac{{\left(\mathrm{ln}N\right)}^{n+1}}{n+1}$; therefore, all derivatives are infinite for N infinite and one could expect to find the Feigenbaum number. The small difference can be explained by the fact that the cascade considered, for example, to describe the appearance of turbulence in Appendix B is never infinite, but always finite. Under these conditions, we end up with a value that is always lower than the limit value for a quadratic logistic function (Ref. [21]).

^{−10}, ${m}_{vi}^{l}/{m}_{B}=1/{H}_{N}\left(v,\lambda \right)$ tends to a constant smaller than 1. This region of λ is the one typically corresponding to the Newtonian plateau. The subshearic regime where the viscous mass is smaller than the basic unit rest mass corresponds to the fact that, in a laminar flow, the dissipation is weak and, therefore, the equivalent in mass converted into heat is also small compared to the energy transported by the flow. According to Equation (9), it can be observed with Figure A5 that the supershearic regime no longer depends on N as soon as N is greater than 10.

**Figure A5.**Logarithmic plot of the function $1/{H}_{N}\left(v,\lambda \right)$ and Equation (A16) for v = 1.71492. The states below the semi-transparent light red surface correspond to the subshearic regime, and above to the supershearic one.

**Figure A6.**Logarithmic plot of the function $1/{H}_{N}\left(v,\lambda \right)$ (black curve) and the inverse of Equation (A17) in the limiting case where v = 2 and N → ∞. A zoom corresponding to the dotted black rectangle shows the transition zone between the two power laws.

## References

- Panchawagh, H.V.; Sounart, T.L.; Mahajan, R.L. A model for electrostatic actuation in conducting liquids. J. Microelectromech. Syst.
**2009**, 18, 1105. [Google Scholar] [CrossRef] - Samaali, H.; Perrin, Y.; Galisultanov, A.; Fanet, H.; Pillonnet, G.; Basset, P. MEMS four-terminal variable capacitor for low power capacitive adiabatic logic with high logic state differentiation. Nano Energy
**2021**, 55, 277. [Google Scholar] [CrossRef] - Perrin, Y.; Galisultanov, A.; Hutin, L.; Basset, P.; Fanet, H.; Pillonnet, G. Contact-Free MEMS Devices for Reliable and Low-Power Logic Operations. IEEE Trans. Electron. Devices
**2021**, 68, 2938. [Google Scholar] [CrossRef] - Barnes, H.A.; Hutton, J.F.; Walters, K. An Introduction to Rheology; Elsevier: Amsterdam, The Netherlands, 1989; Volume 3. [Google Scholar]
- Heyes, D.M. Non-Newtonian behaviour of simple liquids. J. Nonnewton. Fluid Mech.
**1986**, 21, 137. [Google Scholar] [CrossRef] - Woodcock, L.V. Origins of shear dilatancy and shear thickening phenomena. Chem. Phys. Lett.
**1984**, 111, 455. [Google Scholar] [CrossRef] - Aitken, F.; Volino, F. A new single equation of state to describe the dynamic viscosity and self-diffusion coefficient for all fluid phases of water from 200 to 1800 K based on a new original microscopic model. Phys. Fluids
**2021**, 33, 117112. [Google Scholar] [CrossRef] - Aitken, F.; Volino, F. A novel general modeling of the viscoelastic properties of fluids: Application to mechanical relaxation and low frequency oscillation measurements of liquid water. Phys. Fluids
**2022**, 34, 043109. [Google Scholar] [CrossRef] - Aitken, F.; Volino, F. New equations of state describing both the dynamic viscosity and self-diffusion coefficient for potassium and thallium in their fluid phases. Phys. Fluids
**2022**, 34, 017112. [Google Scholar] [CrossRef] - Aitken, F.; Volino, F. “EOSViscoDiffH2O” from the Notebook Archive. 2022. Available online: https://notebookarchive.org/2022-04-9od0wto (accessed on 13 December 2022).
- Wagner, W.; Pruß, A. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data
**2002**, 31, 387. [Google Scholar] [CrossRef] - Korosi, A.; Fabuss, B.M. Viscosity of liquid water from 25 to 150 °C. Measurements in pressurized glass capillary viscometer. Anal. Chem.
**1968**, 40, 157. [Google Scholar] [CrossRef] - Hallett, J. The Temperature Dependence of the Viscosity of Supercooled Water. Proc. Phys. Soc.
**1963**, 82, 1046. [Google Scholar] [CrossRef] - Först, P.; Werner, F.; Delgado, A. The viscosity of water at high pressures—Especially at subzero degrees centigrade. Rheol. Acta
**2000**, 39, 566. [Google Scholar] [CrossRef] - Mariens, P.; Van Paemel, O. Theory and experimental verification of the oscillating disk method for viscosity measurements in fluids. Appl. Sci. Res.
**1956**, 5, 411. [Google Scholar] [CrossRef] - Yasumoto, I. Viscosity of Water Vapor in the Temperature Range from 6 °C to 29° C. Bull. Chem. Soc. Jpn.
**1970**, 43, 3917. [Google Scholar] [CrossRef] - Huber, M.L.; Perkins, R.A.; Laesecke, A.; Friend, D.G.; Sengers, J.V.; Assael, M.J.; Metaxa, I.N.; Vogel, E.; Mareš, R.; Miyagawa, K. New international formulation for the viscosity of H
_{2}O. J. Phys. Chem. Ref. Data**2009**, 38, 101. [Google Scholar] [CrossRef] - Badmaev, B.; Dembelova, T.; Damdinov, B.; Makarova, D.; Budaev, O. Influence of surface wettability on the accuracy of measurement of fluid shear modulus” Colloids Surfaces A Physicochem. Eng. Asp.
**2011**, 383, 90. [Google Scholar] [CrossRef] - NIST Chemistry Webbook. NIST Standard Reference Database Number 69. Available online: https://webbook.nist.gov/chemistry/ (accessed on 13 December 2022).
- Kolmogorov, A.N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. C.R. Acad. Sci. URSS
**1941**, 30, 301–305. [Google Scholar] - Feigenbaum, M.J. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys.
**1978**, 19, 25–52. [Google Scholar] [CrossRef] - Bergé, P.; Pomeau, Y.; Vidal, C. L’ordre Dans le Chaos; Hermann: Paris, France, 1988. [Google Scholar]

**Figure 1.**Semi-logarithmic plot of the variations of Equations (8) and (10) when $\overline{\lambda}$ is a constant independent of v for v = 1.5. The thick red curve represents the limit of validity of ${H}_{N}\left(v\right)$ when ${\left(\overline{\lambda}\right)}_{\mathrm{threshold}}={N}^{-\left(2+v\right)}$ as given by Equation (11).

**Figure 2.**Semi-logarithmic plot of the variations of Equation (12) when $\overline{\lambda}$ is a constant independent of v for v = 1.5.

**Figure 3.**Semi-logarithmic plot of the variations of Equations (8) and (10) when $\overline{\lambda}$ is given by Equation (13) for v = 1.5. The thick red curve represents the limit of validity of ${H}_{N}\left(v\right)$ when ${\left(\lambda \right)}_{\mathrm{threshold}}={N}^{-\frac{2+v}{1+v}}$ as given by Equation (14).

**Figure 4.**Semi-logarithmic plot of the variations of Equations (8) and (10) along the isotherm 295 K when $\overline{\lambda}$ is equal to zero or is given by Equations (13) and (17) with water equations of state. The variation of the density scale corresponds to liquid water from its density on the saturation vapor pressure curve ${\rho}_{\mathsf{\sigma},\mathrm{Liq}}\left(295\mathrm{K}\right)\cong 0.99775{\mathrm{g}/\mathrm{cm}}^{3}$ to a high density of 1.3 g/cm

^{3}. The dissipative distance is d = 100 µm.

**Figure 5.**Semi-logarithmic plot of the variations of Equation (6) along the isotherm 295 K when $\lambda $ is equal to zero or is given by Equations (13) and (17) with water equations of state. The variation of the density scale corresponds to the steam from its density on the saturation vapor pressure curve ${\rho}_{\mathsf{\sigma},\mathrm{Gas}}\left(295\mathrm{K}\right)\cong 1.927\times {10}^{-5}{\mathrm{g}/\mathrm{cm}}^{3}$ to ${\rho}_{\mathrm{tr},\mathrm{Gas}}/100$, where ${\rho}_{\mathrm{tr},\mathrm{Gas}}$ represents the triple point gas density of water. The dissipative distance is d = 100 µm.

**Figure 6.**Semi-logarithmic plot of the variations of Equations (8) and (10) along the isochor 6.1036 × 10

^{−6}g/cm

^{3}when $\lambda $ is equal to zero or is given by Equations (13) and (17) with water equations of state. The temperature range corresponds to that explored by Yasumoto’s experiment (Ref. [16]). The dissipative distance is d = 100 µm.

**Figure 7.**Deviation plot $100\left({H}_{N}\left(v\right)-{H}_{N}\left(v,\lambda \right)\right)/{H}_{N}\left(v\right)$ for steam. The water vapor density range is from ${\rho}_{\mathrm{tr},\mathrm{Gas}}$ to the critical density ${\rho}_{c}$ and the temperature range is from the triple point temperature T

_{t}to the critical temperature T

_{c}. The thick red curve represents the vapor coexistence curve of water. The dissipative distance is d = 100 µm.

**Figure 8.**Experimental results of liquid water viscosity as a function of the velocity gradient obtained with the double cone by realizing either a linear stress ramp or a linear strain ramp. T = 293.15 K.

**Figure 9.**Experimental results of liquid water viscosity as a function of the velocity gradient obtained with the different experimental devices used. T = 293.15 K.

**Figure 10.**(

**a**) Comparison of theoretical model (red curve, Equation (19) with ${d}_{N}=1.3144\mathrm{cm}$) with the experimental results (blue points) for liquid water viscosity as a function of the velocity gradient. (

**b**) Deviation of the experimental data with Equation (19). Only the experimental points corresponding to subregions 2 and 3 are represented. Double cone experiment at T = 293.15 K.

**Figure 11.**Second derivative of Equation (19) as a function of the velocity gradient using the parameters corresponding to Figure 10.

**Figure 12.**Liquid water experiment at T = 293.15 K with the HAAKE simple cone by imposing a linear strain ramp. (

**a**) Experimental results (orange triangles) versus theoretical model (black curve with ${d}_{N}=3.1781\mathrm{cm}$ and C

_{K}= 0.7655) for the viscosity as a function of the velocity gradient. (

**b**) The experimental stress versus the velocity gradient. (

**c**) Experimental results (orange triangles) versus theoretical model (black curve with ${d}_{N}=3.1781\mathrm{cm}$ and C

_{K}= 0.5137) for the viscosity as a function of the velocity gradient.

**Figure 13.**Liquid water experiment at T = 293.15 K with the Couette cell by imposing a linear strain ramp. (

**a**) Experimental results (green diamonds) versus theoretical model (black curve with ${d}_{N}=1.6787\mathrm{cm}$ and C

_{K}= 0.01086) for the viscosity as a function of the velocity gradient. (

**b**) The experimental stress versus the velocity gradient. (

**c**) Experimental results (green diamonds) versus theoretical model (black curve with ${d}_{N}=1.6787\mathrm{cm}$ and C

_{K}= 0.00617) for the viscosity as a function of the velocity gradient.

**Figure 14.**Liquid water experiment at T = 293.15 K with the HAAKE double cone by imposing a linear strain ramp. The model parameters specific to this run are C

_{cal}= 1.03789, d = 88.5 µm and e = 0.0908 cm. (

**a**) Deviation of the experimental results with Equation (23). (

**b**) Stress variation as a function of time t for the first 100 s. (

**c**) Representation of the reduced correlation length ${\xi}^{*}$ (left coordinate axis) and the reduced action temperature ${T}_{\mathrm{A}}^{*}$ (right coordinate axis) as a function of velocity gradient using the experimental stress σ as an input parameter (σ

_{1}= 0.099478 Pa, σ

_{2}= 1.17158 × 10

^{−4}Pa, ε

_{1}= 10.358 and ε

_{2}= 2.402). (

**d**) Experimental results (blue points) versus theoretical model (red curve) for the viscosity as a function of the velocity gradient for all the subregions.

**Figure 15.**Liquid water experiment at T = 293.15 K with the HAAKE simple cone by imposing a linear strain ramp. The model parameters specific to this run are C

_{cal}= 1.42564, d = 110 µm and e = 0.0618 cm. (

**a**) Deviation of the experimental results with Equation (23). (

**b**) Stress variation as a function of time t for the first 100 s. (

**c**) Representation of the reduced correlation length ${\xi}^{*}$(left coordinate axis) and the reduced action temperature ${T}_{\mathrm{A}}^{*}$ (right coordinate axis) as a function of velocity gradient using the experimental stress σ as an input parameter (σ

_{1}= 0.12813 Pa, σ

_{2}= 8.3848 × 10

^{−5}Pa, ε

_{1}= 4.527 and ε

_{2}= 1.945).

**Figure 16.**Liquid water experiment at T = 293.15 K with the Couette cell by imposing a linear strain ramp. The model parameters specific to this run are C

_{cal}= 0.95707, d = 320 µm and e = 0.1 cm. (

**a**) Deviation of the experimental results with Equation (23). (

**b**) Stress variation as a function of time t for the first 100 s. (

**c**) Representation of the reduced correlation length ${\xi}^{*}$(left coordinate axis) and the reduced action temperature ${T}_{\mathrm{A}}^{*}$ (right coordinate axis) as a function of velocity gradient using the experimental stress σ as an input parameter.

Cone Diameter (mm) | Bottom Cone Angle (deg) | Upper Cone Angle (deg) | Bottom Cone Truncation (mm) | Height of the Cylindrical Shape Separating the Two Cones (mm) |
---|---|---|---|---|

63.562 | 1.023 | 5 | 0.054 | 2 |

Inner Cylinder Diameter (mm) | Outer Cylinder Diameter (mm) | Height of the Cell (mm) | Height of Water (mm) |
---|---|---|---|

20 | 21 | 40 | 16.787 |

Double Cone Water Volume (cm ^{3}) | Simple Cone Water Volume (cm ^{3}) | Couette Cell Water Volume (cm ^{3}) |
---|---|---|

~3.2 | ~2 | 3 |

**Table 4.**Characteristic values of the linear strain ramp slopes for the different experimental data shown in Figure 9.

Double Cone | Simple Cone | Couette Cell | |
---|---|---|---|

Slope of the linear strain ramp (s^{−2}) | 1.6148 | 0.9989 | 1.6715 |

Experiment duration (s) | 2500 | 1000 | 900 |

τ (s) | ${\mathit{\tau}}_{\mathit{\nu}}\approx \mathit{\tau}{\mathit{H}}_{\mathit{N}}\left(\mathit{v}\right)$ (s) | Stress Relaxation Time (s) | Interval Between Two Experimental Points (s) |
---|---|---|---|

10^{−8} | 10^{−2} | 10^{−1} | 10^{0} |

**Table 6.**Numerical values of the fundamental parameters for liquid water at atmospheric pressure and 293.15 K when using the modeling from ref. [7] and geometric characteristics of the double cone experiment.

Name (Unit) | Value |
---|---|

K (GPa) | 2.95364 |

${c}_{0}$ (m/s) | 1720.33 |

K_{N} (Pa) | 18,318.7 |

K_{gas} (Pa) | 652.8 |

d_{N} (cm) | 1.3144 |

ξ_{0} (Å) | 3.71768 |

v | 1.71492 |

**Table 7.**Numerical values of the fundamental parameters for liquid water at atmospheric pressure and 293.15 K when using the modeling from ref. [7] and geometric characteristics of the simple cone experiment.

Name (Unit) | Value |
---|---|

K_{eff} (GPa) | 2.2741 |

${c}_{eff}$ (m/s) | 1509.37 |

K_{N} (Pa) | 1728.6 |

K_{gas} (Pa) | 652.8 |

d_{N} (cm) | 3.1781 |

ξ_{0} (Å) | 2.7029 |

v | 1.71492 |

**Table 8.**Numerical values of the fundamental parameters for liquid water at atmospheric pressure and 293.15 K when using the modeling from ref. [7] and geometric characteristics of the Couette cell experiment.

Name (Unit) | Value |
---|---|

K_{eff} (GPa) | 0.06675 |

${c}_{eff}$ (m/s) | 258.59 |

K_{N} (Pa) | 4.9179 |

K_{gas} (Pa) | 652.8 |

d_{N} (cm) | 1.6787 |

ξ_{0} (Å) | 1.2857 |

v | 1.71492 |

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**MDPI and ACS Style**

Aitken, F.; Volino, F.
A Novel Approach for Modeling the Non-Newtonian Behavior of Simple Liquids: Application to Liquid Water Viscosity from Low to High Shear Rates. *Condens. Matter* **2023**, *8*, 22.
https://doi.org/10.3390/condmat8010022

**AMA Style**

Aitken F, Volino F.
A Novel Approach for Modeling the Non-Newtonian Behavior of Simple Liquids: Application to Liquid Water Viscosity from Low to High Shear Rates. *Condensed Matter*. 2023; 8(1):22.
https://doi.org/10.3390/condmat8010022

**Chicago/Turabian Style**

Aitken, Frédéric, and Ferdinand Volino.
2023. "A Novel Approach for Modeling the Non-Newtonian Behavior of Simple Liquids: Application to Liquid Water Viscosity from Low to High Shear Rates" *Condensed Matter* 8, no. 1: 22.
https://doi.org/10.3390/condmat8010022