Flow Classification and Its Application to Fluid Processing

Mathematically, the problem of flow field classification can be analyzed by the eigenanalysis of the deformation-rate tensor; however, such analysis technique have not been commonly applied in fluid processing. We derive a simplified objective flow classification scheme based on the invariants of the strain-rate tensor and the vorticity tensor. Multiaxiality of flow, which is related to the type of elongation, and converging/bifurcating flow, is characterized by the strain-rate tensor, while rotation contribution that protects fluid element from stretching is characterized by the relative intensity of an objective vorticity to the strain-rate. The spatial distributions of flow classification quantities offer an essential tool in understanding the flow pattern structure, and therefore can be useful to get insights into the connection between the geometry and the process performance.


Introduction
In fluid processes, such as transport, mixing, reaction, and heat transfer, flow pattern is of essential importance to the process performance. Flow pattern or topology is largely controlled by the channel geometry and/or driving elements as well as the operation conditions such as flow rate. Analyzing the flow pattern should be useful for understanding fluid processes in connection with the different geometric structures of channels. In real flow of interest, local flow pattern changes from one spatial point to another, and is not well-defined rheometic flows such as simple shear or simple elongation used in rheological characterization of fluid. In such situations, flow classification scheme of non-rheometric flow is required.
Local flow pattern or topology is determined by the deformation-rate tensor; therefore, the problem of flow field classification can be analyzed by eigenanalysis of the deformation-rate tensor (Tanner and Huilgol, 1975;Chong et al., 1990). However, such analysis technique have not been commonly applied in fluid processing, partly because since this analysis involves the flow topology, intensity, and directions, physical interpretation is not straightforward in real flow fields. It would be useful to be able to calculate quantities that identify the flow topology irrespective of the flow intensity and directions for any flow pattern.
In this paper, we formulate a simplified flow classification scheme based on the invariants of the strain-rate tensor and the vorticity tensor. Application of our flow field classification to flow of a non-Newtonian fluid in melt-mixing in twin-screw extrusion demonstrates that the difference in the channel geometry highly affects the distribution of elongational flow (diverging and bifurcating flows).

Flow classification
Flow pattern at a point in real flow fields is classified into converging, bifurcating, and planar flows. Rheometric elongational flows are special cases of converging and bifurcating flows, while simple shear flow is classified as planar flow. Elongational and shear flows in real flow fields are different from the ones in rheometic flows. For example, whereas the elongation rate is fixed to twice as compression rate in simple uniaxial elongational flow, the ratio between the elongation and compression rates in general elongational flows does not fixed. In other words, elongational flows in general are uneven multiaxial flow. The question is how to characterize flow topology of such nonrheometric flow in real flow fields.
Since the flow pattern or topology is a kinematic property, criterion for flow classification should be based on the kinematic quantities of flow, not on the rheological properties of fluid, such as the parameters in constitutive models of fluids. Such flow classification criterion can be applied to any real flow of complex fluids irrespective of the rheological properties of the fluid. Astarita (1979) discussed the properties that the flow classification criterion should have. First, it should be a local quantity in the sense it is defined from the local flow field rather than the whole flow field. Since the flow topology varies with spatial position (for instance, see Figure 1), local criterion is used to characterize the spatial variation of the flow topology.
Next, the criterion should be objective; it should be invariant under change of reference frame since the flow topology is independent of the frame of reference. Furthermore, when constructing a constitutive equation representing the rheology dependent on the flow topology, the flow classification criterion can be directly included in the constitutive equation. For this application, the criterion must fulfil the objectivity property that is required for the constitutive equation. We derive scalars representing flow topology that satisfies these properties.
Local flow around a point x is expressed as from which we identify ( ) v x is translation and ∇v determines local flow pattern. The velocity gradient is further decomposed into the strain-rate tensor observe that local flow pattern is superposition of multiaxial strain-rate and rotational motion and therefore consider separate characterization of the flow multiaxiality and rotational contribution.
We rederive a scalar of D which characterizes flow multiaxiality, called strain-rate state (Nakayama et al., 2016;Lund and Rogers, 1994;den Toonder et al., 1996), from the geometric viewpoint of the strain-rate tensor. Consider following three invariants of D : trD , : D D , and det D . In this paper, we discuss incompressible flow which implies tr and represents the intensity of D . For incompressible flow, the third-order invariant is reduced to The flow multiaxiality is quantified with det D as explained below.
The strain-rate tensor consists of three vectors , which represents the strain-rate vector on the plane normal to , , . This geometric interpretation enables us to relate the multiaxiality of flow and the value of det D . In planar flow, one of the strain-rate vectors vanishes, and thus the parallelepiped collapses, leading to det 0 = D . By contrast, when all three strain-rate vectors are nonvanishing, det D has a finite value, which indicates a triaxial flow and represents a general kind of elongational flow including rheometric and non-rheometric ones. Uniaxial elongation consists of compression in two directions, , 0 represents a converging flow. On the other hand, biaxial elongational flow consists of , 0 , which represents bifurcating flow. In short, the sign of det D discriminates flow multiaxiality, which is a toplogical property of flow independent of the flow strength.
The value of det D is determined by not only the type of multiaxial flow but the intensity of the strain-rate  where the numerical factor is determined to be Since s is based on the invariants of D , s is objective. The multiaxiality of flow at a spatial point is quantified by the strain-rate state. The strain-rate state has been used to analyze the relation between the channel geometry and non-Newtonian flow (Nakayama et al., 2016(Nakayama et al., , 2018, and discuss the elongational flow distribution in turbulence (Lund and Rogers, 1994;den Toonder et al., 1996).
We illustrate the relation between the flow multiaxiality and the value of s. For this purpose, we only need to see the three principal strain-rates calculated by diagonalizing D , where T is orthonormal matrix. In biaxial elongational flow, 1 2 0 , We add a comment on the complete characterization of the multiaxial flow for better understanding what s-value implies. In general, multiaxial flow is described by three properties: the type, the strength which is commonly comcharacterize by elongational/shear rate, and the directions. These three properties are defined with the eigenanalysis of D in the first place. From this viewpoint, s only determines the type of multiaxial flow. The advantage is that spatial variation of the multiaxiality can be directly visualized and analyzed with s since s is a scalar.
Next, we consider the contribution of the vorticity tensor on the multiaxial flow. The vorticity tensor or equivalently the vorticity vector has three invariants: intensity of vorticity, and direction relative to the principal axis of D which is measured with two angles.
The vorticity represents the rigid-body rotational motion of fluid elements. This rotational motion itself does not strain the fluid element, but reduce the stretching effect of D by shifting the direction of the fluid element away from the highest stretching direction of D . From the flow pattern viewpoint, we concern the relative contribution of Ω to the multiaxial flow.
We note that Ω itself is not objective, but the relative rotation rate to that of the principal axes of D is objective (Astarita, 1979). Here, we introduce relative rate-of-rotation tensor according to Astarita (1979)   In contrast that The irrotationality E is a measure of the strain-dominance of the flow. Especially, the flow is strain-dominated or irrotational when 1 E = , half rotational when 1/ 2 E = , and fully rotational when 0 E = . Simpler definition of irrotationality, where Ω instead of rel Ω is used, have been often used in the literatures to study flow-type distribution (Ottino, 1989;Yang and Manas-Zloczower, 1992;Yao and Manas-Zloczower, 1998;Jongen, 2000;Yao et al., 2001;Connelly and Kokini, 2006).
Two scalars, s and E , characterize different aspects of the topology of flow, and are combinedly used to classify flow as depicted in Figure 3. The type of elongational flow is determined by the strain-rate state s while the relative effect of vorticity on the multiaxial flow is determined by the irrotationality E . Figure 3 shows that rheometric flows (such as simple shear, simple elongation, equi-biaxial elongation etc.) are special cases in general flow classification.
In the past, other methods of flow classification have been discussed. A method based on the invariants of ∇v that appear in its eigenpolynomial 3 2 0 P Q R λ λ λ − + − = has been proposed (Chong et al., 1990), though ∇v is not objective. ∇ v v , has been used (Einarsson et al., 2018). Positive ∆ indicates the eigenvalues of ∇v are all real, namely Ω is vanishing, while negative ∆ indicates the complex eigenvalues due to finite vorticity. In this case, the sign of ∆ is used to discriminate the fully strain-dominant (fully irrotational) or not; however, the interpretation of the value of ∆ is not clear. Difficulty in interpretation of Q , R , and ∆ is because they reflect not only the flow type but also the intensity of D and Ω . By contrast, with s and E , only flow topology can be determined regardless of the intensity of D and Ω .

Channel Geometry and Flow Pattern Distribution
We show how flow pattern is altered by the channel geometry. For flow in co-rotating twin-screw extrusion, we compare flow pattern distribution in different screw element geometries. In extrusion processes, flow is driven by screw rotation, which induces circumferential shear flow, and material-feeding flow rate, which induces pressure flow. Figure 4 shows a configuration of full-flight screw (FS) and neutral kneading block (NKB).
Although the shapes of FS and NKB are different, their cross-section is the same due to geometrical constraint in co-rotating twin screws so that they do not contact during the screw rotation.
We consider the incompressible flow of viscous shear-thinning fluid completely filled in the channel. The constitutive equation is was also calculated. Flow is assumed to be pseudo-steady to screw rotation and was solved with finite volume method with a software "R-FLOW"(R-flow Co., Saitama, Japan) under the screw rotation speed of 200 rpm and a flow rate of 60 cm 3 /s. The other detail of the numerical calculation is the same as described in Nakayama et al. (2016). Figure 5 shows shear-rate, irrotationality, and strain-rate state of the shear-thinning fluid flow in crosssections in FS and NKB. Note that a simpler definition of irrotationality ( )   Figure 5 for convenience. The strain-rate is largest in narrow-gap regions such as tip-barrel and inter-disc clearances, and is rather small in other areas. The shearrate distribution is similar in FS and NKB. Irrotationality distribution reveals that half-rotational flow is seen near wall while irrotational flow develops regions far from wall. This feature is common to FS and NKB. Note that the region of 1 ' E ≈ far from the wall is straindominated but the stretching should not be expected to be so strong since the shear-rate there is not large. By contrast, the strain-state distribution differs greatly between FS and NKB. In FS, planar shear flow prevails whereas in NKB bifurcating and converging flow develop at front and backside of the rotating disc, respectively. Figure 5 demonstrates that the strain-rate state distribution reflects the difference between the flow patterns of FS and NKB. Figure 6 shows the same plot as in Figure 5 but for the flow of a Newtonian fluid. Compared to the case of the shear-thinning fluid, Newtonian fluid flow exhibits a slightly thicker boundary layer near the wall;   Figure 4 that are in conveying fullflight screw and kneading block, respectively. Irrotationality is calculated with Ω for convenience.
nevertheless, there is no significant difference in the flow pattern structures between the shear-thinning and Newtonian fluid cases. This result demonstrates that flow pattern of viscous fluids in the channel shown in Figure 4 is mainly determined by the geometry of the channel.

Conclusions
We formulated a method of flow field classification using two scalars derived from the strain-rate tensor and the vorticity tensor. The two scalars are the strain-rate state, which measures multiaxiality of flow, and the irrotationality, which measures the strain-dominance of the flow. The method enable us to identify the local flow pattern in non-rheometric flow encountered in real flow and discuss the flow pattern distribution. Since the flow classification method proposed in this manuscript can be applied to flow in different fluid processes, it is expected to be an essential tool for understanding the fluid processes.