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Polyethylene oxide solutions have a behavioral flexibility that provides researchers with the opportunity to use constitutive law models in a variety of ways for their MRI characterization. Our results obtained in numerical simulation carried out in 2D and 3D for speed profiles of a solution of PEO deployed by the simple method of the cylindrical Couette geometry considering the fluid Newtonian defect, allowed to identify the behavior of fluid complex (rheo-fluidifying threshold fluid). The relevance and the interest of the method are examined by analyzing these results generated by the numerical data obtained, since these profiles depend on the non-Newtonian properties of the fluid which one does not know a priori and which one seeks to measure by postulating first to the power law of Ostwald, then to the truncated power law.

It is well known that the use of an industrial product of any kind always requires the use of materials in generally liquid, pasty or solid states with a choice based on the physical, chemical, mechanical and rheological properties obtained by experimental means. Natural polymers or synthetic polymers, with very maneuverable properties, are no longer a scarce commodity and their uses are widespread and varied in industrial sectors, basic academic research and in the commercial sector; the case of polyethylene oxide has better advantages because of its low cost and its behavioral flexibility. For example, PEO is used as an additive in polymerization reactions to prevent foaming and promote polymerization (in the production of vinyl chloride and acrylonitrile butadiene styrene) [

The rheological properties of polyethylene oxide solutions are related to the concentration, molecular weight, nature of the solvent, external parameters (temperature, pressure) and microstructural interactions (polymer-solvent or polymer-polymer) [

In our study, we used Couette geometry to simulate and evaluate the non-uniform flow curves of the PEO solution at an entanglement concentration. We then study the rheological behavior of this solution in order to convert the imposed or measured experimental quantities of torque and rotational speed into Couette geometry in the form of a shear rate constraint relationship. The process initiated from the Newtonian model used by default, continues with the Oswald law model (which has remarkable shortcomings in not taking into account the localization observed when the solution is flowed at low velocities of rotation since the existence of a flow threshold is not predicted by this model) and then to the model of the truncated power law. In particular, we would like to show that the use of the Reynolds number R e which is the main parameter governing the different types of fluid flows in the Couette cylindrical geometry is possible.

The sample of the material used in this study is a solution based on polyethylene oxide (PEO) with a molar mass of 106 (g・mol^{−}^{1}) and a concentration of 1.8% wt. PEO is a linear chain nonionic polymer [CH_{2}-CH_{2}-O]n. The choice is made on the geometry of Couette with imposed shear (i.e. we measure the torque on the inner cylinder after having imposed its rotational speed ω = 2πN (rd./s), of the coaxial cylinders type with wide gap in front of the microstructure size to avoid the effect of size which makes the smooth surface of cylinder R 1 sensitive to slippage, hence its coating sometimes with the Emeri canvas. The shear in the geometry is considered homogeneous, which can ensure that the material has the same structure throughout the air gap. The two coaxial cylinders are of inner radius R 1 = 13.375 ( mm ) and rotates at an angular velocity of 0.002 to 0.9 (rd./s), resulting in low rotational speeds of 0.0026 to 1.20 (cm/s) and shear ratio of 0.2 to 96 (s^{−1}). The cylinder R 1 with height h = 40.12 (mm) ensures sufficient contact to increase the torque. The outer radius R 2 = 13.55 ( mm ) is fixed, the curvature R_{1}/R_{2} = 0.99. The numerical simulations of the velocity profiles are carried out with the software MATLABR2008b. The conversion of the macroscopic data into a behavior law requires an adapted analysis.

The first work is based on the determination of the shear rate on which the constituent laws predominantly lead to the equations of the velocities. In cylindrical coordinates, this requires knowledge of the speed gradient from which they derive [

γ ˙ = ∂ V ( θ , r ) ( r ) ∂ r − V ( θ , r ) ( r ) r = r ∂ ∂ r ( V ( θ , r ) r ) (1)

The search for the velocity profile under the multiple flow hypotheses is done as follows:

1) Laminate which imposes a low speed for a displacement in layer of lamellae with respect to each other causing friction forces;

2) Isothermal or permanent to express the independence of variables over time;

3) Incompressible;

4) In the cylindrical coordinate system (r, θ, z) where the components of the axial, tangential and radial velocities are respectively V z , V ( θ , r ) , V r , taking into account the symmetry of the problem with respect to z and at θ and in the absence of an axial pressure gradient, V z = 0 and V ( θ , r ) is independent of θ;

5) The forces of inertia are at all points and at all moment’s negligible vis-à-vis the forces of viscosity, which is expressed by: R e ≪ 1 .

The constitutive equation which connects the stress to the shear rates is given by Equation (2):

τ ( ω , r ) = η γ ˙ ( r ) (2)

This shear rate is derived from the expression of the velocity given by Equation (3):

V ( ω , r ) ( r ) = ω R 1 2 R 2 2 − R 1 2 ( − r + R 2 2 r ) (3)

or:

{ V ( ω , r ) ( R e , r ) = R e K N G A ( − r + R 2 2 r ) with: K N G A = μ R 1 ( R 2 + R 1 ) e 2 (4)

K N G A is the period of fluid flow in the air gap in (s^{−}^{1}).

The Equation (4) reformulates the Equation (3) in consideration of hypotheses (5), so that the flow velocity takes into account the property of the solution related to the Reynold number R e = ( ω R 1 e ) / μ with μ = η / ρ , the angular velocity ω (rd./s) and the geometry of the duvet outlet device having internal radii R 1 and outer R 2 . The resulting shear rate is given by Equation (5):

γ ˙ ( r ) = 2 ω R 1 2 R 2 2 R 2 2 − R 1 2 1 r 2 (5)

Thus the stress at the inner cylinder R 1 or at any radial position r c mm is given by the Equation (6):

τ ( ω , r ) ( R 1 ) = M 2 π h R 1 2 (6)

A fluid is Newtonian if its dynamic viscosity is independent of stress and shear duration. There are a large number of fluids very commonly used which have a more complex flow behavior. In the case of polymer solutions when the concentration of the polymer is greater than the overlap concentration (c > c*), the interactions are essentially attractive and the polymers attract each other and attach to each other [

The expressions of the shear rate and the stress in the air gap depend strongly on the radius of the virtual cylinder (r). Because of this dependence, these two quantities are measured at the same place and for a Newtonian fluid we can write: τ ( r ) = η γ ˙ ( r ) . In practice, rheometers indicate average values of stress and shear rate local in the air gap for this Newtonian fluid.

≺ σ ≻ = R 1 2 + R 2 2 4 π H R 1 2 R 2 2 M and ≺ γ ˙ ≻ = R 1 2 + R 2 2 R 2 2 − R 1 2 ω (7)

If the fluid is not Newtonian, man can expect a difference between the law of actual behavior of the material σ = f ( γ ˙ ) and that given by the rheometer ≺ σ ≻ = f ( ≺ γ ˙ ≻ ) . However, when the cell used has a small difference that is to say when R 2 − R 1 ≺ ≺ R 1 , it is possible to confuse the macroscopic law of the rheological behavior given by the rheometer with most non-Newtonian local rheological behaviors. In this work, it is essential to use a wide gap rheometer to identify the actual varying behavior of the sheared fluid.

1) Numerical Simulation in 2D

The second work of our rheological study after the first which allowed to establish the laws of the profiles of speed will lead to their simulation in the centered Couette geometry with a wide gap: the inner cylinder is moving at an angular velocity ω (rd./s) or controlled by R e (Reynolds number), the outer cylinder is fixed, for a polyethylene oxide solution with a concentration of 1.8 wt%. According to Equation (3) and Equation (4), two profiles of dimensionless velocities are used: The first profile of _{ }at the level of the inner cylinder R 1 ). The second triangle profile is also obtained from the ratio of Equation (4), with respect to its same effective value V 0 (mm/s). The two velocity profiles are obtained following numerical simulations with MATLABR2008b at an angular velocity ω = 0.9 (rd./s) and at R e = 0.0075 .

2) 3D numerical simulation and analysis

The main interest of 3D modeling is to see the velocity profiles in spatial mode in order to graph the different zones of the velocity profile and the conditions of inhomogeneous flows in steady state

Works of [^{−1}) → 0.

3) Observations

For the two 2D curves of ^{−1}) over a significant distance, and they fall to γ ˙ 1 (s^{−1}) as in and then tend to zero and remain around this value for larger distances of the inner cylinder. They thus correspond to the velocity profiles obtained for a shear flow of giant micelles [

This rheological study not only leads to the obtaining of rheological quantities of fluids, but also provides a window through which other complementary characteristics are deduced, particularly the behavior that binds the stresses to the flow of the fluids. Fluids according to theoretical, linear and non-linear mathematical models existing in the current literature [

{ r c ≺ r ≺ R 2 ⇒ V ( ω , r ) = 0 and R 1 ≺ r ≤ r c ⇒ V ( ω , r ) = ω R 1 2 R 2 2 − R 1 2 ( − r + R 2 2 r ) o u V ( R e , r ) = R e K ( − r + R 2 2 r ) (8)

4) The critical values of rotation speed and Reynolds number

In order to show all the influence of the angular velocity or of the Reynolds number on the modeling of the velocity profiles and consequently on the flow of our fluid in the Couette cylinder, we will again carry out two representations of the same profiles sized with the variation of these two parameters

and

5) Location of the shear

When the stress field in geometry is heterogeneous, the shear rate is zero in the zones where the shear stress τ is less than the flow threshold τ c (Pa). This is the case in cylindrical pipe flows where the shear stress is proportional to the radius: the material is sheared only near the walls and the central part of the material (where τ < τ c ) is transported at constant speed, then the shear is localized. This problem is found in the rheometry of threshold fluids where the velocity field measured locally by MRI during the flow of a concentrated emulsion in the Couette geometry is represented. In the case of our study by a numerical simulation in Figures 1-3, we note that the material is not sheared in the whole of the air gap: there is a zone near the outer cylinder where the speed is zero. Moreover, the size of the sheared zone decreases when the rotation speed ω of the inner cylinder or the Reynolds number R_{e} is decreased as shown by the work [

Thus, we can say that at low rotational speeds which correspond to very low numbers of Reynolds

6) Shear band

The fact that we obtain two different zones at different shear rates γ ˙ < γ ˙ 2 and γ ˙ > γ ˙ 1 translates a mechanical instability (existence of an elastic limit) [^{−1}) and γ ˙ 2 (s^{−1})

force (centrifugal force) is the driving force of Taylor-Couette instability in a simple fluid, elastic in stability is generated by the (centripetal) forces exerted along curved current lines [

The velocity profile at an almost constant slope at a distance and sharply decreases to near zero and maintains this value for greater distances. We still have the coexistence of a solid region and a liquid region in such a flow. However, there is here a discontinuity of the shear rate at the interface between the two regions: the shear rate is critical at a point of shear rate γ ˙ c (s^{−1}) in the liquid region and equal to zero in the solid region. This behavior is closer to what is usually called the shear band [

The velocity profile at an almost constant slope at a distance and sharply decreases to near zero and maintains this value for greater distances. We still have the coexistence of a solid region and a liquid region in such a flow. However, there is here a discontinuity of the shear rate at the interface between the two regions: the shear rate is critical at a point of shear rate γ ˙ c (s^{−1}) in the liquid region and equal to zero in the solid region. This behavior is closer to what is usually called the shear band [

7) Position of location and link between microscopic and macroscopic measurements

Since the paragraph 3.2, the explanatory approach of the behavior of our material in the geometry to start and will continue for the determination of the quantities that will allow to establish the law of complete behavior of the material. It is necessary to determine the threshold of the material which will be added to a power law. In fact, the position of location shows in the Couette geometry the area or shear fluid in fraction and the zone or the fluid shears totally according to thicknesses r c (mm)

τ ( R ) = M / 2 π h R 2 (9)

M is the torque applied to the inner cylinder of radius R 1 and of height h given by Equation (9) above. The measurement of the critical radius r c (mm). To which the material ceases to flow imposes knowledge of the critical stress characterizing the flow threshold by other means and the knowledge of the torque applied to the inner cylinder will make it possible to deduce the location r c (mm) as in Equation (9) (i.e. τ c ( r c ) = M / 2 π h r c 2 ).

We obtain a threshold stress τ c (Pa), a constraint which is sometimes too low to be detected by a simple experiment such as an inclined plane test [

γ ˙ ( r ) = ∂ V ( r ) ∂ r − V ( r ) r (10)

At the interface of the sheared and non-sheared zones characterizing the stoppage of flow, the associated shear rate is given by the slope of the velocity profiles. This rate corresponds to γ ˙ c (s^{−1}), and below this threshold γ ˙ c (s^{−1}), value, there is no stable flow. The existence of a threshold shear rate associated with a stress threshold is a general property of threshold materials discovered recently [^{−1}), to lower values, the whole sample is sheared. Also for higher values γ ˙ c (s^{−1}), r c (mm) increases and the fluid is sheared throughout the air gap. γ ˙ c (s^{−1}) is the same for two different air gaps [

σ ( r ) = σ i R i 2 r 2 (11)

Since the local shear rate is the spatial derivative of the velocity profile (slope of the profile), the local viscosity can be calculated as compared to the overall viscosity. The localization of shear is due to the fact that the shear stress τ ( ω , R ) measured at low rotational velocity ω (rd./s) of the inner cylinder passes below the threshold at the interface and of Equation (12), the interface of the two zones of flow and of non-flow (dead zone) by the equation.

R c ( ω ) = R 1 τ ( ω , r ) τ c (12)

When ω (rd./s) tends to 0, R c ( ω ) then tends to R 1 , i.e. the thickness of flowing material tends to zero and the shear stress τ ( R 1 , ω ) on the inner cylinder tends towards τ c (Pa) [

σ = k γ ˙ n (13)

8) Medialization

This situation is presented as the behavior of stress fluids in simple shear which has a viscosity plateau. It is modeled according to Equation (14).

{ τ ≺ τ c ⇒ γ ˙ = 0 but: τ ≥ τ c ⇒ τ = f ( γ ˙ ) = k γ ˙ n (14)

and in this case, the final law of the behavior is:

τ = τ c + K γ ˙ n (15)

where τ c (Pa) corresponds to the threshold constraint, it is deduced from the equation rheological equation. The rheograms of the experimental data are plotted in the log-log coordinates. The number n represents the slope of the line obtained and is also the index of structure. For 0 < n <1 the fluid is rheo-fluidifier, then n = 1 is the Bingham case and n > 1 the rheo-thickening case. K is the consistency in [N sec^{2}/m^{2}], is given by the point of intersection of the line with the axis corresponding to γ ˙ = 1 (s^{−1}). Since γ ˙ in cylindrical coordinates is written:

γ ˙ = r ∂ ∂ r ( V ( θ , r ) r ) (16)

We have:

V ( θ , r ) = r ω ( ( R 2 / r ) 2 / n − 1 ( R 2 / R 1 ) 2 / n − 1 ) (17)

The system of Equation (8) is repeated for a complex fluid in the form:

{ r c ≤ r ≤ R 2 ⇒ V ( θ , r ) = 0 but: R 1 ≤ r ≤ r c ⇒ V ( θ , r ) = r ω ( R 2 / r ) 2 / n − 1 ( R 2 / R 1 ) 2 / n − 1 (18)

This model of velocity profile fairly well represents the flow of PEO solution

{ τ ≺ τ c ⇒ γ ˙ = 0 τ ≥ τ c ⇒ τ τ c = ( γ ˙ γ ˙ c ) n (19)

τ c and γ ˙ c correspond respectively to the stress and the shear rate at the interface between the sheared zone and the non-sheared zone of the Equation (13). When r c = R 2 (mm), radius of the outer cylinder, there is no localization and the velocity profile remains analogous to that obtained by the Ostwald model.

In this case the Ostwald law is not sufficient to analyze the non-localized velocity profiles. On the other hand, in the case of low rotational speeds where the flow is localized, the characterization of the flow is performed by the truncated power law. In this case, the speed profile is given by:

{ r c ≤ r ≤ R 2 ⇒ V ( θ , r ) = 0 R 1 ≤ r ≤ r c ⇒ V ( θ , r ) = n 2 r γ ˙ c ( ( r c 2 ) 2 / n − 1 ) (20)

A second fundamental reason for the inadequacy of Ostwald Waele’s power law is that it has a linear range of viscosity at low shear rate on the log-log diagram when this law is expressed in terms of viscosity. However, the fluid is rheo-fluidifying, and therefore characterized by the decrease of the apparent viscosity when the shear rate increases. This assumes that this Ostwald power law is valid only within a certain range of shear rates. The Tile model satisfactorily represents the evolution of the viscosity as a function of the shear rate. Using the

values of n, r c , γ ˙ c in the equation of the truncated model, we obtain the linear curves of

In this paper, we have presented the basic method for identifying the flow curve of a shear fluid in a Coutte cell and then the searching for the most appropriate procedure for the determination of the rheological quantities that relate to this fluid. This method is simple and efficient through the results obtained numerically, but one can get rid of certain limitations related to the size of the air gap [

Ngarmoundou, N., Ousman, R.G., Mahamat, B. and Beye, A.C. (2017) Flow Curves in the Centered Cylindrical Couette Geometry of a Polyethylene Oxide Solution. Open Journal of Fluid Dynamics, 7, 673-695. https://doi.org/10.4236/ojfd.2017.74044

r: virtual cylinder position (mm)

r c : flow and rest area interface (mm)

R i : radius of the virtual cylinder (mm)

R 1 : radius of the inner cylinder (mm)

R 2 : radius of the outer cylinder (mm)

K: consistency (N sec^{2}/m^{2})

n: flow index

V z : axial component of the speed (mm/s)

V r : radial component of velocity (mm/s)

V ( θ , r ) : tangential component of velocity (mm/s)

V 0 : maximum tangential velocity expressed as a function of angular velocity (mm/s)

M: couple (Nm)

R e : Reynolds number

γ ˙ : shear rate (s−1)

γ ˙ c : critical shear rate (s−1)

τ: shear stress Pa

τ c : Critical shear stress (Pa)

ω: rotation speed (rd./s)

ω c : critical rotation speed (rd./s)

σ i : shear stress at the virtual cylinder (Pa)

η: viscosity (Pa・s)

μ: kinematic viscosity (m^{2}/s)

ρ: density (g/cm^{−}^{3})

PEO: polyethylene oxide

C: critical point

1-2: index at point 1, respectively 2

i: virtual point index