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Poiseuille flow of nonlocal microstructured fluid

In this paper, an Eringen's type nonlocal flow law is developed to account for the microstructured effects in modelling the behaviour of heterogeneous fluids. The axisymmetrical Poiseuille flow of Newtonian and non-Newtonian fluids is studied using the nonlocal mechanics, and some original analytical results are obtained for this geometrical configuration. It is shown that the nonlocal Newtonian fluid exhibits a kind of pseudo plastic behaviour because a plug flow zone exists within the concept of nonlocal mechanics. Furthermore, the shear rate profile shows some nonlinearity for the nonlocal Newtonian fluid, a property which cannot be accounted for with the so-called local theory. Viscoplastic fluids, modelled by Herschel–Bulkley or Bingham laws, are also investigated using this nonlocal generalization. Hypergeometric functions are used to compute velocity profiles. A general parametric study illustrates the nonlocal specificities of the viscoplastic flow for such heterogeneous materials. Finally, the Newtonian nonlocal model is calibrated using a discrete layered axisymmetrical Poiseuille flow.


Introduction
Recently, flow of concentrated suspensions or emulsions have shown finite size effects such as spatial cooperativity or localized plastic shear banding. According to Goyon et al. [1], there is a strong need to develop a theoretical framework to describe the non-local flow behaviour of glassy and jammed systems, such as soft glasses and granular materials. These authors suggested that nonlocal theory can be used to model these finite size effects. Nonlocal mechanics is a continuum mechanics theory that may be able to capture some detailed phenomena ranging from small to large scale . Nonlocal mechanics was largely developed in the 60's notably for solid mechanics applications (see the historical perspective of Maugin [2]; see also the book of Eringen) [2,3]. Among the different nonlocal mechanics theories, integral-based nonlocal models have found a large diversity of applications, especially in the fields of heterogeneous, discrete solids and microcracking media where they are used to eliminate the stress singularities in the vicinity of a crack. The application of integral-based nonlocal mechanics to fluid mechanics was developed in the 70's, with the Eringen seminal paper on integral-based nonlocal Newtonian fluids [4]. Recently, with the interest in micro fluids and atomic fluids mechanics, nonlocal concepts have found new applications (see for instance Todd and Hansen [5] or Puscasu et al [6]). The nonlocal kernels or nonlocal viscosity can be calibrated using simple atomic fluids (see for instance Alley and Alder [7] or more recently Puscasu et al [6]). It is worth mentioning that some spatial related nonlocal concepts have also been introduced to explain the flow cooperativity phenomenon [1,8,9]. The nonlocal aspect of material parameters has been introduced by the previously referenced authors through a cooperativity length scale, which is linked to the microstructure geometry and used to compute a nonlocal viscosity parameter. A theoretical investigation of nonlocal fluids is also reported in the recent paper of Zhu et al [10]. In this study, a nonlocal Newtonian and non-Newtonian model is presented in a differential form. This model can be viewed as being analogous to the Eringen's model for solid mechanics (see the differential format of the elasticity stress-strain law of Eringen [11]) which has now been generalized to fluid materials. Axisymmetrical Poiseuille flow is studied using this nonlocal model, and some original analytical results are obtained for this geometrical configuration. The effect of the nonlocality on the modification of the viscoplastic fluid behaviour is also studied. Regarding Newtonian behaviour, the proposed model is compared to a discrete layered axisymmetrical Poiseuille flow and the nonlocal characteristic length is calibrated with the discrete layers thickness.

Behaviour and flow geometry
The uniaxial Herschel-Bulkley law is studied in a nonlocal form in order that some nonlocal effects is the positive operator.  is the shear stress and  is the shear rate. The length scale parameter c l controls the nonlocal aspect of the constitutive law. Eq. (1) can be also written as: The Herschel-Bulkley law covers many of the viscoplastic rheological laws including the particular case for given values of 0 and n. For instance, the Bingham law is obtained for n = 1; the power law is obtained for 0=0 and the Newtonian law for n = 1 and 0 = 0. Eq. (1) can be considered as the Eringen's type nonlocal generalization of the Herschel-Bulkley law. This kind of Laplace operator affecting the stress variable was already suggested by Eringen [11] for elastic solids materials. In this paper axisymmetric Poiseuille flow is considered only in the direction of the x-axis and associated with a constant pressure gradient, denoted by A>0. With this geometrical configuration, if we assume that gravitational forces are negligible, the x-axis term of the stress equilibrium can be written as Eq (3): The flow conditions dictates that x(x,r) =p(x). Substituting  for xr the x-axis term of the divergence of the stress tensor leads to Eq. (4).
A dx dp dr Integration of equation (4) results in the relationship: where B is a constant. The only possible value for B is zero to have a finite value of the shear stress along the radial axis.

Shear rate profile and evolution of the plug flow zone
For "local" viscoplastic materials, two cases can be distinguished. The first case occurs at low values of the pressure gradient if the shear stress does not exceed the yield stress in the gap; in this case, no flow occurs. In the second case, the material is sheared at a radius greater than a critical value rc where the shear stress is higher than the yield stress 0. Thereafter for nonlocal materials, if R ≥ rc ≥ 0 the flow is described by the nonlocal stress law: The following dimensionless parameters can be considered: It is noted that the material is sheared if  * < 1. According to Eq. (5), the dimensionless shear stress can be written as: Consequently, we can write Eq. (6) in terms of a dimensionless radius: the solution of which is given by: For Herschel-Bulkley and Bingham fluids, the plastic zone is defined by the position of the critical radius rc which is defined as being at the beginning of the sheared zone:  (12) It should be noted that this plastic zone equation has a unique solution of rc * within the interval ]0 ; 1[: The local case can be determined asymptotically as the length scale lc * tends to zero, this being associated with the local critical radius Consequently, for nonlocal viscous fluid, Eq. (13) leads to rc * = lc * . It follows that the nonlocal viscous fluid has a plug flow zone of radius lc * within a Poiseuille geometry. This means that the material is not sheared inside a zone controlled by the characteristic length scale which is linked to the material's microstructure. From a physical point of view, this can be attributed to the fact that inside the nonlocal critical radius, interactions between inclusions induce a "pseudo-plastic" material behaviour leading to unsheared zones. For Herschel-Bulkley fluids, the shear rate is given by Eq. (10): For a Bingham fluid, the shear rate profile is obtained from Eq. (14) by taking n = 1. Shear rate profiles of the nonlocal Herschel-Bulkley fluids (0 * = 0.25) for a shear-thinning behaviour (n = 0.5), for a shear-thickening behaviour (n = 1.5) and for Bingham fluids are plotted in Figure 1.   [8]. Moreover, it can be noted that the linear shear rate profile obtained for the local Bingham behaviour disappears when nonlocal effects are introduced into the constitutive law.
In the case of a power law fluid, the shear rate can be obtained from Eq. (14) by taking 0 * = 0 and for Newtonian fluid, n = 1 and 0 * = 0. Shear rate profiles for nonlocal viscous fluids are plotted in Figure   2 for a power law shear-thinning behaviour (n = 0.5), a power law shear-thickening behaviour (n = 1.5) and a Newtonian fluid. The same apparent viscosity increase (of the nonlocal fluid), obtained for viscoplastic fluids can be seen in figure 2 for viscous fluids. Similarly, in the case of Bingham materials, the linearity of the shear rate profile obtained for the local Newtonian behaviour is lost when nonlocal effects are introduced in the constitutive law.

Velocity profile
Dimensionless velocity can be defined as For Herschel-Bulkley fluids and for r * > rc * the non-dimensional velocity can be obtained by integrating the shear rate profile Eq. (14). To perform the spatial integration, Eq. (14) can be factorized as follows: which can be expanded in Taylor  One can recognize that the double series is equal to a hypergeometric function of two variables F1 as defined by Appell [12]: where the rising Pochhammer symbol (q)k is used in the double series, and is defined by Then, according to Appell, Eq. (16) can be written as follow: In the case of r * < rc * , velocity continuity can be used to obtain the entire velocity profile: Velocity profiles for nonlocal viscoplastic fluids (0 * = 0.25) are plotted in Figure 3 for a shearthinning Herschel-Bulkley behaviour (n = 0.5), a shear-thickening Herschel-Bulkley behaviour (n = 1.5) and Bingham behaviour. It appears that the velocity values decrease with respect to the nonlocal length scale for all dimensionless radii. where V2 is a constant defined with V * (1) = 0 and 2F1 the Gauss hypergeometric function defined as follow [13] for This integration result can be proven using the basic properties of hypergeometric functions. Firstly, the shear rate profile equation is factorized by r * as follows: It is interesting to note that the Taylor series of the function (1-z²) b is equal to the following hypergeometric function: The velocity profile can be obtained by integrating the shear rate profile between x = lc * and x =1. the solution of which is given by the following integration rules of hypergeometric functions [13,14]: The resolution of equation (29) is then: Finally, for a Newtonian nonlocal fluid, the velocity profile is simply reduced to: Velocity profiles for nonlocal viscous fluids are plotted in Figure 4 for a shear-thinning behaviour (n = 0.5), a shear-thickening behaviour (n = 1.5) and a Newtonian behaviour. It is interesting to note that when r * is less than lc * , the velocity is constant and equal to V * (lc * ); the velocity profile then exhibits a plug flow zone exactly as found in the case of viscoplastic materials.

Flow rate
In the Poiseuille geometry, the flow rate is the key macroscopic parameter that is measured during the flow. A dimensionless mass flow rate is defined as follow: The dimensionless flow rate can be linked to the dimensionless velocity by the following integral: The evolution of the dimensionless flow rate with respect to the characteristic length for viscoplastic materials is plotted on Figure 5. It clearly appears that the flow rate decreases with an increase in the characteristic length. For Newtonian fluids, Eq. (35) provides the following dimensionless flow rate: The evolution of the dimensionless flow rate with respect to the characteristic length for viscous materials is plotted on Figure 5. As for nonlocal viscoplastic materials, it is evident that the flow rate decreases with increasing characteristic length.  signifies Herschel-Bulkley).

Discrete system: definition and velocity profile
In order to give a physical basis to the nonlocal model, the nonlocal characteristic length is calibrated on the dimensionless layer thickness a * of a discrete concentric material flowing in a Poiseuille geometry. The calibration consists of finding the value of lc * that gives the best fit between the nonlocal and the discrete velocity profiles Layer n r * =n.a * =1 Layer n-1 Layer n-2

Layer i+1
Layer i The stress equilibrium on an elemental layer provides the following discrete equation: which can be simplified in a dimensionless format: . For a Newtonian fluid, the dimensionless shear stress is equal to the r-derivate of the velocity; it is then possible to write: where C0 is a constant computed with the boundary condition Vn * = 0. It is then possible to obtain the following expression: Due to the axisymmetry of the problem, one can note that such a microstructured flow contains a plug flow-like zone of radius a * located around the centre. A parallel can be drawn with the nonlocal model that also possesses a plug flow zone of radius lc * located at the centre of the Poiseuille geometry. The plug flow zone can be attributed to the finite-sized microstructure of the Newtonian fluid. Eq. (43) can be written substituting a * i by the dimensionless radius r*: It is interesting to note that this solution is also the solution of the second order discrete equations resulting from the combination of Eq.
This equation can be integrated to provide the following velocity profile: where C1 is a constant computed with the boundary condition V * (1) = 0. Finally, the computation of C1 shows that the continualised velocity profile is equivalent to the discrete velocity profile given by Eq. (44).

Calibration of the nonlocal model with the discrete system
A mean square error method is used pointing this context to find the value of lc * , with respect to a * , that gives an optimal fit to the nonlocal velocity profile with respect to the discrete (or continualised) velocity profile. This method has been successfully applied to calibrate the characteristic length lc * value for the application of Eringen nonlocal elasticity for wave propagation [15]. The calibration of lc * consists of minimizing the mean square of the difference between the discrete and the nonlocal velocity profile. This difference can be written as follows: Figure 7: Comparison of velocity profiles obtained with the discrete system, the continualisation of the discrete system and the calibrated nonlocal modelling for n = 4 and n = 16.
It appears that the calibrated Eringen's nonlocal model is able to provide a good approximation of the discrete system and can be considered as an efficient engineering model with respect to the flow profile of the discrete system. It is also worth noting that the continualization of the discrete system provides a better model as it resulted in the same velocity profile.

Conclusions
A nonlocal model has been applied to the rheology of viscoplastic fluids. This model can be classified as an Eringen's type integral-based nonlocal model, which can be also expressed in a differential form. This nonlocal model accounts for some specific microstructural effects characteristic the flow analysis of heterogeneous fluids (such as cement paste, foam, emulsions, granular materials, suspensions, fibred materials…).
Analytical solutions are presented for the Poiseuille axisymmetrical stationary flow. By considering a nonlocal viscous material as a particular case of a nonlocal viscoplastic fluid, it has been shown that a nonlocal viscous fluid may exhibit a pseudo plastic behaviour, this is because a plug flow zone of dimensionless radius lc * appear at the centre of the cylindrical section. In a certain sense, this paper suggests, within a nonlocal mechanics perspective, that it is not necessary to introduce yield stress behaviour in a theoretical model to observe a plug flow zone. This idea was differently presented by Barnes and Walters [16], or more recently by Barnes [17], when discussing the existence of yield stress in viscous fluids.
The comparison of the nonlocal material with a discrete layered material for a Newtonian behaviour also shows the physical basis and potential of the proposed modelling. The introduction of the length scale lc into the nonlocal model, which is related to the nonlocal shear stress, has been calibrated in terms of the microstructure size of the heterogeneous fluid, as recently reported by Challamel et al [18] for non viscous elastic solids. This phenomenological parameter can be inversely adjusted from rheological data.
These results which apply to a simple geometrical configuration need to be generalized for other configurations including some more complex rotational flows.