Experimental and simulative determination and correction of the effective gap extension in structured coaxial measuring systems

The use of structured measuring systems to prevent wall slip is a common approach to obtain absolute rheological values. Typically, only the minimum distance between the measuring surfaces is used for further calculation, implying that no flow occurs between the structural elements. But this assumption is misleading, and a gap correction is necessary. To determine the radius correction Δr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta r$$\end{document} for specific geometries, we conducted investigations on three Newtonian fluids (two silicon oils and one suspension considered to be Newtonian in the relevant shear rate range). The results show that Δr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta r$$\end{document} is not only shear- and material-independent, but geometry-dependent, providing a Newtonian flow behaviour in a similar viscosity range. Therefore, a correction value can be determined with only minute deviations in different Newtonian fluids. As the conducted laboratory measurements are very time-consuming and expensive, a CFD-approach with only very small deviations was additionally developed and compared for validation purposes. Therefore, simulation is an effective and resource-efficient alternative to the presented laboratory measurements to determine Δr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta r$$\end{document} for the correction of structured coaxial geometries even for non-Newtonian fluids in the future.


Introduction
One fundamental issue while measuring rheological properties of complex materials is wall slip (e.g. for suspensions). In general, the presence of wall slip leads to an underestimation of the actual viscosity, which can, for example, not only lead to an incorrect design of production plants, but also create considerable problems in the field of simulation, where the viscosity often serves as an input parameter. Wall slip is characterized by a jump in velocity directly at the wall, i.e. true wall slip or strongly increasing velocity gradient in close proximity to the wall, i.e. apparent wall slip (Pahl et al. 1991). However, true wall slip is rarely observed in reality (Haase et al. 2017), since all surfaces show a certain roughness at the atomic level (Zhu and Granick 2002;Granick et al. 2003). Therefore, the term wall slip will be used in the following as a synonym for apparent wall slip.
The reason for wall slip is the development of a thin slipping layer directly at the wall, with a much lower viscosity compared to the bulk material (see Fig. 1a). This is especially prominent for suspension flows, since a particle depleted area with a strongly reduced viscosity occurs at the wall (Banfill 1991;Macosko 1994;Barnes 1995;Jesinghausen et al. 2016). Several phenomena lead to these low particle concentrations, with geometric constraint, i.e. the particles cannot penetrate the wall, being probably the most important. For monodisperse, spherical particles the typical size range of the slipping layer is in the range of the particle radius (Yilmazer and Kalyon 1989;Kalyon 2005;Ballesta et al. 2012;Jesinghausen et al. 2016). Other relevant mechanisms include particle migration due to gradients in shear rate, repulsive particle-wall-interactions and particle layering on surfaces caused by the flow (Besseling et al. 2010;Korhonen et al. 2015;Cloitre and Bonnecaze 2017).
Since the correction of wall slip always requires different sets of coaxial measurement geometries and in general a high measuring effort (Mooney 1931;Yoshimura et al. 1988;Kiljański 1989), the prevention of wall slip by roughened or rather structured geometries is usually practiced (Pawelczyk et al. 2020). In this case, the tip-to-tip distance H between the surfaces defines the measuring gap for further calculation (see Fig. 1b). The use of such geometries does not prevent the particle-depleted layer, but it shifts it to a region that is not relevant to the essential flow field. However, the assumption of a distinct flow-interface at the geometry tips is void, as various investigations revealed, and leads to calculation errors (Pawelczyk et al. 2020). Fincke and Heinz (1961) conducted research on the determination of yield stress of coarse suspensions by using grooved coaxial measuring geometries for wall slip prevention. In order to determine, if the material located in the grooves participates in the shear flow or not, they measured Newtonian fluids with these geometries. The results have shown that the effective radius differs from the true radius by a few tenths of a millimetre. After applying these correction values to the calculation of a Newtonian oil, no deviations between smooth and grooved measuring system could be observed. Since the effective radius and the actual radius do not differ that much, the authors state that the flow penetration depth in the grooves is only small. However, according to Fincke and Heinz, this is dependent of the fluid behaviour and the applied shear stress. Bauer et al. (Bauer et al. 1995), who investigated concentrated diblock copolymer solutions in grooved coaxial systems, determined the correction values in advance by measuring several Newtonian fluids. The correction value ΔR was found to be proportional to the radius R but independent of viscosity and shear rate. With these corrected radii, they were able to obtain the same results for smooth and grooved measuring geometries, even for non-Newtonian fluids. However, they state that while the correction factors can be used to include the influence of the grooved surface, they do not consider the disturbance of the flow field. This is particularly important when measuring materials that exhibit a shear-dependent build-up of structures that leads to shear-thickening behaviour. Since a structured geometry can cause this behaviour at a lower shear rate than a smooth geometry, a wrong interpretation of the flow behaviour could be possible. Similar results were obtained by Ahuja et al. (Ahuja and Singh 2009), who compared simulated viscosity values of smooth and serrated geometries for low concentrated suspensions. When using the tip-totip distance in the simulation, deviations between smooth and serrated walls occurred. However, when simulating the serrated walls with the actual gap, i.e. tip-to-tip distance H plus two times the serration-depth, the deviations vanished, indicating a significant flow in the serration.
Besides the mentioned investigations concerning coaxial systems, various research on parallel-plate systems were conducted. However, as this paper focuses on coaxial systems, other measuring geometries will not be discussed in detail. Information on the investigations concerning these systems and the underlying theoretical backgrounds can be found in (Beavers and Joseph 1967;Neale and Nader 1974;Nickerson and Kornfield 2005;Carotenuto and Minale 2013;Minale 2014Minale , 2016Carotenuto et al. 2015;Paduano et al. 2018Paduano et al. , 2019Pawelczyk et al. 2020).
The presented studies clearly show that the application of structured geometries in coaxial systems leads to an increase in the effective radius and to the need for radius corrections to determine correct rheological values. However, for materials prone to wall slip, such as suspensions, those geometries are widely used and partially irreplaceable, but often without the necessary correction. In order to develop a simple method for determining the radius correction of any geometry, we conducted investigations on two homogeneous Newtonian fluids and a medium dense suspension comparing laboratory measurements and CFD for various kinds of coaxial measuring geometries, primarily differing in groove widths and depths. In a first step, correction values for the radii, which can be seen as theoretical penetration depths, were derived via experimental comparison of structured geometries to smooth geometries for the Newtonian fluids. Second, as this would surely simplify future investigations, a CFD simulation was built to determine the correction values, which shows a strong agreement to the experiments. At last, the results were transferred to a medium dense suspension and the applicability is demonstrated. Fig. 1 a Schematic illustration of the velocity profile in the two-plate-model with the velocity of the upper plate u x , the slip velocity u s , the gap width H , the structure depth d and the slip-layer or particle depleted area ; b prevention of wall slip through profiled measuring surfaces with new defined gap height H 1 3

Theoretical background
Concentric cylinder measuring systems consist of two cylinders with the same symmetry axis. The material is sheared in a gap between the inner cylinder (bob, radius r i and length L ) and the outer cylinder (cup, radius r o and length L ). Since the surface area of the cylindrical plane increases from r = r i to r = r o , the shear stress occurring in the measuring gap decreases with increasing radius r (see Eq. (1)). If dealing with a Newtonian fluid, the shear rate can be expressed through Eq. (2), where i and o are the angular velocity at the inner or outer cylinder, respectively, and = (r i ∕r o ) 2 (Pahl et al. 1991;Mezger 2012).
For a Searle type cylinder system, which was used in this paper, only the inner cylinder is rotating. Thus, o = 0 and the viscosity for a Newtonian fluid can be expressed as shown in Eq. (3) (Pahl et al. 1991;Mezger 2012).
Usually, the flow behaviour of the sample is not known or rather non-Newtonian and, therefore, the calculation according to Eqs. (1) to (3) is not applicable anymore. To this end, an approximation method using the so-called representative shear rate ̇r ep and representative shear stress rep is commonly applied, as it is fast and simple even though better, but more complicated correction methods exist (Krieger and Elrod 1953;Giesekus and Langer 1977). Schümmer (Schümmer 1970) showed that for any fluid a representative radius r rep exists, where the shear rate and the shear stress are equal to those of a Newtonian fluid. Based on that, Giesekus and Langer (Giesekus and Langer 1977) developed equations for the shear rate ̇r ep and shear stress rep , which are independent of the flow behaviour (see Eqs. (4) and (5)). It is to mention that the factor originates from a series expansion, with usually only the first series element to be used for further calculation, as for narrow gaps ( 1 − ≪ 1 ) the deviation is negligibly small. (1) Since this approximation is implemented in the evaluation method for the concentric cylinder measuring systems of the rheometer used in this paper, the expressions and ̇ will generally refer to the above-mentioned representative values. However, the viscosity remains independent of whether the representative values were used for the shear stress and the shear rate because the factor is truncated in the viscosity calculation.

Wall slip correction
The fundamental issue with wall slip is that the wall slip velocity u s leads to an apparent shear rate ̇a , which is always higher than the true shear rate ̇ applied to the material. Thus, the obtained viscosity is lower than the true viscosity. To overcome this, Yoshimura and Prud'homme developed a correction method for coaxial systems, which allows for the determination of true shear rate and true viscosity (Yoshimura et al. 1988).
As the correction method is based on the use of two gaps and coaxial systems generally have a fixed gap width, the use of two systems with different radii is necessary. According to Yoshimura and Prud'homme (Yoshimura et al. 1988), the ratio between outer and inner radii has to be the same for both systems (see Eq. (6)).
Adjusting equal shear stresses for both systems, the angular velocity can then be expressed by: Since the fluid angular velocity term f is equal for both systems, and the slip velocities are a function of stress only, which results in equal bracketed terms, Eqs. (7) and (8) can be simplified and combined to calculate f ,according to Eq. (9).
With regard to the representative shear stress rep shown in Eq. (5), it can also be expressed by: The true viscosity can then be calculated by:

Geometries
The reference system used in this investigation was a smooth concentric cylinder (CC) system (Anton Paar CC27) with an inner radius r i = 13.333mm , an outer radius r o = 14.462mm and a measuring length L = 40.026mm . For wall slip correction, two other coaxial systems called CC14 and CC25 were manufactured (see Table 1). Both systems fulfil the requirements of the ISO 3219 for standardized systems. In order to determine the influence of structural parameters on the effective gap size, further structured coaxial systems were developed based on the CC25-system (see Fig. 2a). Three different groove depths ( d 1 = 0.3mm , d 2 = 0.65mm , d 3 = 1mm ) and two different groove widths ( w 1 = 1.1mm (narrow), w 2 = 2.1mm (wide)) were analysed, while the groove shape is always rectangular. The measuring length of the inner cylinder of all systems is L = 36.84mm . Concerning the groove depths, it is to mention that due to the rectangular shape combined with the curved surface of the cylinders, the depth of a groove is always the largest depth possible when referring to the outer or inner radius, respectively. This leads to the inner cylinder having the maximum groove depth in the middle of a groove, while the outer cylinder having the maximum groove depth at the edge of a groove (see Fig. 2b). The systems were 3D-printed out of stainless steel with a small oversize and then reworked on a lathe to ensure the dimensional accuracy. The individual dimensions can be found in Table 1, where the corresponding geometries and dimensions are given in the form "rdddw" (r = rectangular, ddd = groove depth in 10μm , w = groove width in rounded mm).

Laboratory measurements
All measurements were conducted using a Physica MCR 501 rheometer from Anton Paar. Since it is possible to temper coaxial systems quite efficiently, the mean temperature of all measurements was within T = 20.0 ± 0.01 • C . The rheometer has a maximum torque accuracy of 0.5%, which equals 0.2μNm . For the fluids measured in this paper with AK 1000 being the one with the lowest viscosity, a shear rate of 0.1s 1 results in a torque error of about 5% , which reduces to only 0.4% at 1s 1 . For rheometre controlling and data acquisition, the Anton Paar software "RheoPlus" (RHEOPLUS/32 V3.31) was used. The materials analysed in this investigation were Wacker AK 1000 and AK 5000 silicon oils, which have a kinematic viscosity of = 1000mm 2 ∕s and = 5000mm 2 ∕s, respectively (manufacturer specification) and show a nice Newtonian behaviour in the examined shear rate range (see Fig. 3). The AK 5000 was also used to prepare a suspension together with 21.23vol% Spheromers® CA40 particles ( x 50,3 = 48.5μm measured with Sympatec QICPIC).  As silicon oils are a very homogeneous material, no premeasurement treatment was necessary. The suspensions, however, were prepared individually before every measurement procedure, which consists of three single measurements, by mixing the right amount of particles with the silicon oil. This was done to ensure the reproducibility by preventing variations in concentration due to inhomogeneities, which could have developed over time, even though that would have been very unlikely. After mixing the suspensions, they were degassed in an ultrasonic bath at 19 • C for 1 h and then mixed again. As the suspensions heated up to 30 • C due to internal friction, they were cooled to a target temperature of 20 • C with constant stirring in a water bath before pouring them into the measuring cup.
In order to prevent the formation of air bubbles, the materials were carefully poured along the wall of the geometries. Subsequently, the measuring bob was lowered very slowly ( 100μm∕s ) into the cup and the temperature was set to 20 • C . After excess material was removed or additional material was filled into the measuring gap, the measuring program was started (see Table 2).
For an easier comparison, the measured (apparent) viscosity values of the structured geometries a are normalized at each shear rate according to the reference viscosity CC27 and are labelled n , as depicted in Eq. (12).

Simulation
CFD simulations were conducted using the open source software OpenFOAM (version v2012). The STL-files for the geometry were compiled using the freeware Blender and meshed via the blockMesh-and snappyHexMesh-tool of OpenFOAM. Since the flow field in the measuring gap can be described two-dimensional within a reasonable error margin, the virtual measuring gap was designed with a length of 0.3mm for easier meshing and further simulated in 2D using the solver simpleFoam. For further analysis of the flow field, a customized solver was developed, whose functionality is explained in "Simulation results -investigation of the penetration depth and the flow field via simulation".
Even though this is basically a dynamic simulation, there is only interest in the steady-state flow field. Therefore, the so-called MRF (multiple reference frame) method was used, which works with a stationary mesh, but still considers the rotation by adjusting the momentum equation. Additional geometries were constructed describing the rotating zone, whose outer radius is equal to (r i + r o )∕2 of the geometry in question.
The post-processing and extracting of individual cell values was done via the freeware ParaView. Combined with a self-written MATLAB algorithm, a profound evaluation of the flow field and the penetration depths was possible. Fig. 3 Individual measurements #1-#3 and mean value for silicon oil AK 5000 measured with CC27 system at T = 20.0 ± 0.01 • C ; a shear stress over shear rate; b viscosity over shear rate

Determination of correction values
The idea behind the correction is that a general radius correction value Δr can be derived for individual measuring systems and applied to a huge variety of materials. Based on the correction method presented for parallel plate systems (Nickerson and Kornfield 2005), we developed a method to determine the radius correction in coaxial systems. However, some assumptions need to be made for applying this correction.
If there was no influence of the structured measuring surfaces, meaning the interface of the flow field is indeed the tip-to-tip-distance, the same viscosity must be measured using the structured geometries and the CC27 reference system. As this is clearly not the case, a correction is necessary. The approach for the correction is that the true viscosity true , which has to be equal to the reference viscosity CC27 , can be calculated from the apparent viscosity a , which is measured with the structured systems, by implementing a correction radius Δr . This correction value needs to be included both for the inner radius Δr i and the outer radius Δr o . However, to guarantee a stable numerical solution method, the assumption Δr i = Δr o = Δr has to be made. Therefore, based on Eq. (3) by truncating all equal parameters except = (r i ∕r a ) 2 and r i , which differ because of the Δr , the following relation between true and a can be derived (see Eq. (13)). As true and a as well as r i and r o are known, the radius correction Δr can be easily calculated. To apply the radius correction, true and a need to be corrected for their flow behaviour a priori. Therefore, it can be assumed that Eq. (13) is not only valid for Newtonian fluids. Corresponding application to shear-thinning fluids can be found in (Pawelczyk et al. 2020).

Results and discussion
The following chapters address separately the results obtained during the laboratory measurements and the simulation. The main focus of the laboratory results is to determine the radius corrections by comparing the values measured with the structured geometries with the reference values, applying these corrections and transferring them to another homogeneous Newtonian fluid.
The simulation results focus firstly on the evaluation of the simulation method and then a comparison between simulation and laboratory results will be conducted, followed by the transfer to suspensions.

Laboratory results
In order to obtain reference values, the silicon oils were measured with the standard geometry CC27. Since a wall slip correction needs to be conducted for the suspension (see "Wall slip correction"), it was measured with the CC25 and CC14 systems. All measurements were repeated three times at T = 20.0 ± 0.01 • C following the measurement procedure and calculations presented earlier. For further investigation, the three individual measurements were averaged. Figure 3 shows the exemplary results for the silicon oil AK 5000, which shows a nearly ideal Newtonian behaviour in the observed shear rate range. This is also valid for the other silicon oil and the suspension. The linear regressions with the corresponding values can be seen in Table 3.

Laboratory results -determination of radius corrections using silicon oil AK 5000
In order to determine the radius corrections, the normalized viscosities were measured for AK 5000 with all systems (see Fig. 4), with the error bars representing the gaussian error. The silicone oil AK 5000 was preferred to the AK 1000 for this investigation, since, due to the higher viscosity, a higher measuring accuracy can be expected. However, as the shear rates up to ̇= 2s 1 show some fluctuations, only shear rates larger than ̇= 2s 1 will be used for deriving the correction. All measurements show viscosity values which are lower than the reference viscosity and, as expected, with a greater deviation for the wide grooves. Furthermore, a clear groove depth dependency can be observed for all systems, as the normalized viscosity decreases with increasing groove depth. However, the Newtonian flow behaviour stays unharmed. One can conclude that there is no shear rate dependency of the penetration depth, which is supported by the results obtained by Bauer et al. (Bauer et al. 1995). Therefore, a radius correction was derived for the individual groove depths and widths by averaging over all shear rates larger than ̇= 2s 1 (see Fig. 5). The error bars represent the gaussian error.
As to be suspected, Δr increases with increasing groove depth, meaning that the flow penetrates the grooves deeper. This is different from grooves in parallel-plate geometries where a maximum penetration depth could be observed (Pawelczyk et al. 2020). However, the same behaviour might appear for greater groove depths. Nevertheless, some differences can be pointed out between the gap widths. The first thing becoming obvious is the range of the values for Δr . For the narrow grooves, the correction radius is in the range of approximately 37 − 43μm , whereas this range is 83 − 93μm for the wide grooves. Furthermore, for the narrow grooves, Δr seems to show a linear dependency of the groove depth ( R 2 = 0.9944 ). For the wide grooves, this conclusion can only be drawn with certain limitations (R 2 = 0.945).
The increase of Δr with increasing groove depth is not surprising. An explanation for the increasing correction factor could be that as the groove depth increases, the fluid experiences less resistance when entering the groove. It is intended to clarify this trend with further investigations in the future but up to now no further prediction is possible. However, whether the penetration depth really increases with increasing groove depth cannot be investigated easily via laboratory measurements and must be considered in more detail via simulations (see "Simulation results"). Figure 6 shows the corrected values for all measurement systems according to Eq. (13) for AK 5000.

Application of the radius correction to silicon oils
As can be seen, the corrected values match the reference viscosity (measured with CC27) with nearly no deviation (see also Table 4). This is not surprising, as the values for Δr were calculated also using Eq. (13). However, the general purpose of radius corrections is to have a material-independent, but geometrydependent characteristic value, with which the measured fluids can be corrected. Therefore, it is desirable that the value determined above can also be used for other fluids, which we have already shown for a parallel plate setup (Pawelczyk et al. 2020). In order to prove that correlation for coaxial systems, the obtained correction values for the AK 5000 were also used to correct the AK 1000 measurements (see Fig. 7). The correction values were also applied to suspension measurements (see "Application of the radius correction to silicon oils").
Although the corrected values for AK 1000 do not match the reference viscosity as good as they do for the AK 5000, the results still look very satisfying, which again is supported by the results obtained by Bauer et al. (Bauer et al. 1995). When taking a closer look at the mean deviations depicted in Table 5, it can be assumed that the correction values Δr are indeed transferable to other materials, provided that they are also Newtonian. As the difference in viscosity between AK 1000 and AK 5000 is not that great, this assumption will be verified in further investigations by using Newtonian fluids with much higher viscosities in the future. Whether this also applies to suspensions with nearly Newtonian flow behaviour is investigated in "Application of the radius correction to suspensions".

Simulation results
For validation purposes, the CC27 reference system was first simulated in 2D at three representative shear rates, ̇= 5.72s 1 (low shear), ̇= 37.6s 1 (middle shear) and ̇= 82.7s 1 (high shear) with a virtual length of L = 0.3mm . To compensate even minor measurement variations, an averaged viscosity over three shear rates in the range of the above-mentioned values was used as input parameter. Given the shear rate and the fluid viscosity, it is possible to calculate the torque on the cup and the bob. As convergence criterion, a constant torque for at least 10,000 iteration steps was defined. Following the equations given in "Theoretical background", it is then Fig. 6 Corrected and uncorrected normalised viscosities n over the shear rate ̇ for silicon oil AK 5000 calculated for narrow groves (a) and wide grooves (b)  possible to calculate the shear stress from the torque and the corresponding viscosity which should match the input viscosity if everything is set up correctly. The results are shown in Table 6. When looking at the deviations, it becomes clear that this approach, including the MRF method, the 2D simulation and the calculation of viscosity, seems appropriate and can be used in the following investigation.

Simulation results -comparison between laboratory and simulation results
The simulations performed for the reference system were also performed for all structured systems with the same initial conditions and convergence criteria. The results are presented in Table 7.
Even though the deviations in viscosity are slightly larger compared to the reference system, they are below 1.5% for all simulations. Therefore, it is possible to calculate the correction factors for similar grooved geometries only via simulation. As the calculation of Δr is viscosity based with a minute deviation, the results are not shown here, since no further insight would be achieved.

Simulation results -investigation of the penetration depth and the flow field via simulation
One of the basic assumptions to derive the radius correction is the even flow penetration of the grooves at the inner and outer cylinders, meaning Δr i = Δr o = Δr . However, because of the small penetration depth and optical accessibility, it is very complicated to access this experimentally, but a lot easier via simulation.
In this respect, it should be emphasized again that the radius correction is only a correction value and cannot be directly compared with the actual penetration depth of the fluid. While it can be considered an approximation to some extent, it serves more as a relative comparison between the different structures and, what is the main task, as a correction value.
In order to evaluate the relations between Δr i and Δr o , some post-processing of the simulation results or the individual cell values, respectively, is necessary. The main reason for this is that the actual penetration depth of the flow field could only be observed at the outer cylinder, as the flow velocity at the inner cylinder is too high to detect the fluid penetration into the structure, which is much slower. Therefore, a post processing was designed and integrated in a modified solver, which recalculates the velocity field in the measuring gap as follows. All elements in the region 0 < r < r i are seen as a solid state vortex having the radius dependent overlaying velocity u(r) = r ⋅ i with the angular velocity of the inner cylinder i . The overlaying velocity field u(r) in the region r i < r < r o is approximated linear with the boundaries u r i = i ⋅ r i and u r o = 0 . These overlaying velocity values are subtracted from the calculated velocity values of the cells at the corresponding radii. As all elements laying in the region r > r o can also be seen theoretically as a solid state vortex, but with an angular velocity of o = 0 , no recalculations for these elements need to be conducted. This manipulation emphasizes the areas of interest and enables a direct comparison, that is not dominated by the high velocities of the inner cylinder. Figure 8 illustrates the areas just mentioned, and Fig. 9 shows the comparison between an unmodified and a modified flow field to be analysed further. A vortex-like flow field appears, which indicates how deep the fluid penetrates the structures, since the  velocities differ maximally from the assumed ideal velocities with no penetration at all. Even though there was no correction in the outer gap, the results differ from the original measurement since the maximum velocity is lower and no longer masks the flow in this area. An algorithm in MATLAB was written, which calculates the penetration depth over the groove width on the basis of a velocity plot in radial direction (see Fig. 10a). Starting from the inner or outer radius, respectively and going into the groove, a circumferential speed of v c = 0m∕s in the new calculated velocity field was defined as a reference point for calculating the penetration depth, as this point describes the reversal of the flow direction (see Fig. 10a and b). The penetration depth has a maximum in the middle of the groove (angular direction), as it should be expected, and is nearly the same for all inner/outer groove-overlapping situations (see Fig. 11). It is to mention that only a quarter of the measuring system was considered to reduce simulation time making use of the symmetric shape. The values for two directly opposing grooves are given as an example in Fig. 12.   Fig. 8 Schematic visualisation of the regions for recalculating the flow field in the measuring gap Fig. 9 High shear velocity field with the corresponding vectors in the measuring gap of two opposing grooves of the system r0651 before recalculation (a) and after recalculation (b) with the new post processing. It must be noted that the scale is different in both depictions, even though the colours and the vector lengths appear to be similar. This has been done for better clarity, since the overall velocity is much lower in figure (b) Fig. 10 a Schematic visualization for calculating the penetration depth, presented for the r0651 grooves for one position (groove middle) with the outer radius (dotted line), the line for the velocity plot (solid line) from r o into the groove and the calculated penetration depth (dashed line, schematic illustration); b circumferential speed v c over radius ( 0 = r o to end of groove) with marked penetration depth at v c = 0 Table 8 shows the individual averaged penetrations depths for the inner and outer grooves of all systems. To obtain these values, the penetration depths for each individual groove, as presented in Fig. 12, were first averaged over the groove width. This was done for all different overlapping cases (Fig. 11) at the inner and outer cylinders. Then, all these "mean penetration depths" for each individual groove were averaged, to generate only one characteristic parameter for the inner and the outer grooves, respectively. In order to be able to compare the results of the algorithm for the different shear rates more precisely, the first two decimal places are given at this point.
As can be seen in Table 8, the penetration depths do not depend on the shear rate, as only minute deviations occur, which might have their origin in computational accuracy. Therefore, all further discussion will focus on the results of the low shear rate region, but is valid for all regions.
Analysing the penetration depth for the outer and inner grooves, it is obvious that no general conclusion can be drawn, whether is greater at the inner cylinder or vice versa. For the narrow grooves the maximum difference ( 2.08% ) is smaller than for the wide grooves ( 6.57% ). Generally, no trend can be seen for both systems. For example, is greater at the inner grooves for one system (r0302), but for Fig. 11 Different overlapping situations for the modified flow field, exemplary presented for the r0651 geometry another system (r1002), this behaves the other way round. Therefore, the assumption Δr i ≈ Δr o seems to be a valid assumption for the small and the wide grooves, meaning the approach presented in Eq. (13) is not only justified, but is also related to the real flow behaviour.
However, looking at the averaged penetration depths (Table 8) reveals no clear trend or correlation. This might be a hint that the state of groove overlapping is important. Therefore, additional investigations analysing the maximum penetration depth, i.e. in the middle of the groove (see Fig. 12), at certain points were carried out, in fact at two opposite grooves (GG), at inner groove opposite outer wall (IGO) and at outer groove opposite inner wall (OGI) (see Table 9). Additionally, since the groove depths differ slightly in the groove middle (see Fig. 2), the ratio of penetration depth to groove depth is listed to obtain more significant information.
The results clearly show that for many systems, the fluid penetrates the whole groove depth. It can also be seen that there is a direct relation between the maximum penetration depth and groove depth, as with increasing groove depth the maximum penetration depth increases, too. This also applies to the groove width, since the same relationship exists. However, this matches the assumptions made in "Simulation results -comparison between laboratory and simulation results" of higher penetration depths with increasing groove depths and widths due to the lower resistance of the cylinder walls. Taking a closer look, it stands out that no great deviations exist between inner and outer grooves, except for d GG,o ∕d o for the system r1002, while the reason for this exception remains unknown. When comparing the penetration depths for inner and outer grooves for those systems, where the fluid does not penetrate the entire groove (r0651 and r1001), it can also be seen that here, too, the penetration depths for inner and outer grooves are in a similar range with only small deviations, which again supports the assumption of Δr i = Δr o .

Application of the radius correction to a suspension
Our investigation clearly shows that the determination of the radius correction values can be greatly simplified by simulations. However, the use of structured systems is only necessary for wall slip materials such as suspensions. Therefore, it is of high interest, if the radius corrections determined for the silicon oil AK 5000 are also applicable to suspensions within an acceptable deviation. To this end, a 21.23vol% suspension of silicon oil AK 5000 and Spheromers® CA40 was prepared and corrected for slip using the described methods (see "Wall slip correction") Fig. 12 Penetration depth over normalised groove width w norm for the inner and outer grooves calculated for the measuring system r0651 (a) and r0652 (b) at high shear for two directly opposing grooves while measured with smooth surface systems. The true viscosity true is illustrated in Fig. 13, with the error bars representing the gaussian error in case of true and the standard deviation in case of a . As expected, the uncorrected viscosity values are slightly lower than the corrected viscosity, indicating wall slip. However, it is quite significant, that the uncorrected values show an apparent slightly shear thinning behaviour, whereas the corrected values indicate a Newtonian flow behaviour with a mean viscosity of = 11.369 ± 0.017Pa ⋅ s . The shear thinning effect could be attributed to a higher slip-velocity at high shear rates, showing that wall slip may also lead to a false conclusion regarding the flow behaviour. However, a general shear-thinning behaviour of the suspension could be excluded, since not only the presented values, but also previous measurements conducted with a parallel-plate system and according wall slip correction showed a Newtonian flow behaviour (Pawelczyk et al. 2020).
After the true viscosity was determined as a reference, the suspensions were measured with the structured systems, and the radius correction values determined for pure silicon oil AK 5000 were applied. Figure 14 shows the uncorrected and the corrected normalised viscosities together with the reference viscosity. Again, due to measuring errors in the very low shear rate range, only the values above ̇= 2s 1 are considered. The slight tendency to apparent shear-thickening, especially for the system r1001 can be attributed to measurement errors due to minimal dimensional inaccuracies of the measuring systems. However, as this behaviour is not strongly pronounced, it will be neglected in the further course of investigating the transferability of the radius corrections.
The good agreement of the corrected values with the reference curve again strengthens the fact that the Δr is transferable and measurements with structured geometries are possible for suspensions. This is due to the fact that the corrected normalised viscosites differ from the reference values by a maximum deviation of only 3.5% (measuring system r1001) (see Table 10).
When looking at these deviations, it should always be kept in mind that the wall slip corrections were performed using two self-manufactured systems. Therefore, slight deviations may be explained by inaccuracies of the measuring systems and the flow behaviour. Nevertheless, these deviations are within a normal measurement uncertainty, so that, including the results for the silicon oil AK 1000, the conclusion can be drawn that the radius corrections can be generally applied to Newtonian fluids. However, further investigations with Newtonian fluids would be useful to verify this assumption. In this context, it is to mention that the same research was conducted with shear thinning fluids, but no general solution could be found, as the correction values seem to be dependent not only on the shear rate but also on the extent of the shear thinning flow behaviour. Therefore, these investigations are important and will be pursued in the future.

Conclusions
The general approach for measuring suspensions, which are prone to wall slip, is to use structured geometries without recalculating the effective radius for the determination of rheological properties. However, our investigation clearly shows that structured geometries lead to false values with much lower viscosities, which need to be corrected in any case. We are able to show that the effective radius can be corrected with a geometry-dependent, but material-and shear rate-independent correction value Δr , provided Newtonian fluids measured and the assumption of an equally penetrating fluid at the inner and outer cylinders is made, meaning Δr i = Δr o = Δr . This assumption could be backed up via simulative results. The generality of the correction factor was shown by determining the mean radius corrections using a reference fluid, in this case the silicon oil AK 5000, and using these values to correct other Newtonian fluids, namely silicon oil AK 1000 and a suspension consisting of silicon oil AK 5000 and 21.23vol% Spheromers® CA40 particles. While the application of the correction values to AK 1000 lead to very small deviations of only 0.05 − 0.37% , the deviations for the suspension were slightly higher with 0.327 − 2.511% . For some systems, especially for the system r1001, the values improved by only a small percentage compared to the uncorrected values, so that a correction does not seem necessary. However, this is still a quite promising result, especially considering that the investigation of the suspension was based on self-manufactured measurement systems and other measurements could be enhanced significantly. It is very likely that this purely geometrical correction works for all Newtonian fluids. However, it has to be proven for much higher viscosities in the future.
Since both, the fabrication of structured systems and the measurement of the material in the laboratory, are very timeconsuming and costly, a simulative approach to determine the radius corrections was investigated. The CFD results lead to two conclusions. On the one hand, the simulation is able to reproduce the laboratory results with high precision. On the other hand, it could be shown that the assumption of Δr i = Δr o = Δr to correct the apparent viscosity measured with structured systems is justified, since the averaged penetration depths between inner and outer cylinders only differed little, if at all. Therefore, the use of a simulative approach for determining the radius correction should be preferred in any case.
Funding Open Access funding enabled and organized by Projekt DEAL.
Data availability If desired the raw data will be given to any researcher on a reasonable request.
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