Ultrasonic spinning rheometry test on the rheology of gelled food for making better tasting desserts

Cite as: Phys. Fluids 31, 113101 (2019); https://doi.org/10.1063/1.5122874

Submitted: 01 August 2019 . Accepted: 16 October 2019 . Published Online: 01 November 2019 Taiki Yoshida

, Yuji Tasaka

Phys. Fluids 31, 113101 (2019); https://doi.org/10.1063/1.5122874 31, 113101

© 2019 Author(s).

Ultrasonic spinning rheometry test

on the rheology of gelled food for making better tasting desserts

Cite as: Phys. Fluids 31, 113101 (2019); doi: 10.1063/1.5122874

Submitted: 1 August 2019 • Accepted: 16 October 2019 •

Published Online: 1 November 2019

Taiki Yoshida,1,a)

Yuji Tasaka,1

and Peter Fischer2

AFFILIATIONS

‌1 Laboratory for Flow Control, Faculty of Engineering, Hokkaido University, N13W8, Sapporo 060-8628, Japan 2Laboratory of Food Process Engineering, Institute of Food, Nutrition and Health, ETH Zürich, Schmelzbergstrasse 7, 8092 Zürich, Switzerland‌

‌Note: This paper is part of the Special Topic on Food and Fluids.

a) Author to whom correspondence should be addressed: yoshida@ring-me.eng.hokudai.ac.jp

ABSTRACT

Rheological properties of gelled foods that may relate to the physics of the fluids in the swallowing process of complex food components are determined by ultrasonic spinning rheometry (USR) [T. Yoshida et al., “Efficacy assessments in ultrasonic spinning rheometry: Linear viscoelastic analysis on non-Newtonian fluids,” J. Rheol. 63, 503–517 (2019)]. Through rheological evaluations of thixotropic gelled food, the inaccuracies in standard rheometer data to capture the true-rheological property are discussed first with steady rotational and oscillatory tests; the inaccuracies arise from commonly existing problems that cannot be directly observed in standard rheometers (wall-slip, shear banding, shear localization, elastic instability, etc.). The results evaluated by standard rheometers would be related to the measurements being specific response, depending on the geometry of the measurement device. The USR test discussed here shows the potential to overcome these problems in the rheological evaluation of gelled foods and reflects the advantages offered by USR such as spatial, local, and oscillation cycle measurements; the results with the transient flow curve that has not previously been discussed can be usefully interpreted, and the stability of the food materials in the unsteady shear displayed is of great importance in understanding which rheology indicates the better texture.

Published under license by AIP Publishing. https://doi.org/10.1063/1.5122874.,

1. ‌INTRODUCTIONRheology of foods related to the chewing and swallowing pro- cesses has been extensively investigated as a safety issue and recently also to improve the quality of life of consumers; for example, usual oatmeal is good for people who have weak chewing and swallow- ing abilities, especially among older people and infants, but due to very soft texture, chewing is not enjoyable when good taste and good texture are valued. Differently, gelled foods have prob- lems with aspiration even with the better impression arising from eating. Both questionnaires and rheological tests have been used in rheological research, but questionnaire research is affected by personal preferences and rheological tests are not always suitable despite attempts to provide quantitative evaluations; this is because of both the complexity of the chewing and swallowing processes andthe rheological properties of foods. Rheological evaluations, there- fore, require an understanding of complex properties together with the development of a rheometry that captures both the complex properties and individual and preferential processes of swallowing involved.Developments in research have tried to evaluate complex- rheological properties, such as shear thinning, yield stress, viscoelas- ticity, effective viscosity with multiphase dispersion, and other qual- ities, by creating novel rheometers with different geometries. The precision expressed in the results with recent rheometers (different from measurement accuracy) is ensured with sensitive torque sen- sors and with the improvements in the geometry of the rotational part, but little attention has been paid to the “physics of fluids” in the measurements by rheometers. This is the reason why there are “com- mon problems of the rheometers” arising from, for example, theCouette inverse problem,1 and no fully acceptable solution for these problems is known, though extensive trials to overcome specific problems have been made.2–4It is thought that non-Newtonian characteristics could be one source of the common problems (not directly observable), which include shear history effects,5 shear banding,6,7 shear localization,8 wall-slip,9,10 elastic instability,11–14 and other phenomena. When rheological evaluations are attempted without considering these influences, the obtained results will reflect the specific response as it is dealt with in the measurement device, not exposing or clarify- ing the true-rheological properties. These problems are caused by the fluid characteristics, and efforts to solve the problems have to be approached from the perspective of fluid mechanics.To improve the quality of rheometry, utilizing spatiotempo- ral velocity distributions of the fluid motion is required because the velocity information reflects all the rheological information of a complex fluid. Approaches to achieve this from fluid mechanics have mainly focused on standard rheometers coupled with inner visualization techniques (e.g., Refs. 12–14) and have made it pos- sible to directly identify the common problems (usually invisible). Then, the influence exerted by those problems has been investigated by integrating standard rheometer readings with ultrasonic imaging. Gallot et al.12 proposed a technique that explains the unstable shear- banding flow of non-Newtonian fluids, and on this basis, Fardin et al.13 found that that the potential impact of inner flow patterns could be observed for both complex as well as Newtonian fluids in the large amplitude oscillatory shear (LAOS).Research into velocity-profiling rheometry15–17 has dealt with time-averaged velocity profiles limited to steady flow states; as one of the examples of velocity-profiling rheometry, Derakhshandeh et al.15 performed measurements of transient behaviors of thixotropic fluids using a Couette rheometer with a wide gap and ultrasonic velocity profiling (UVP).18 This was able to detect yield- ing regions of the fluid from quasisteady velocity profiles, also when the torque measurements may be influenced by existence of wall- slip that could lead to critical errors in evaluations of the rheological characteristics. This leaves problems, as one of the non-Newtonian characteristics involves both shear-rate-dependence in the rheo- logical properties and time-dependence (structural recovery, relax- ation, and other aspects), and techniques and measurements must be able to detail time-dependent properties for evaluations of transient rheological properties.We have developed a novel velocity-profiling rheometry that supplements the rheological evaluations beyond the capabilities of standard rheometers and have termed it ultrasonic spinning rheom- etry (USR).19–23 Based on the ultrasonic measurement of instanta- neous velocity profiles, the technique has advantages, such as ease of handling and offers the option of employing it with opaque flu- ids, and it has been applied to the measurement of foods in gen- eral.24 The basic concept of this rheometry is that velocity profiles are substituted into the equation of motion to estimate the rheolog- ical properties, and the potential value of USR with various complex fluids has been established in previous reports.20–23In this paper, the rheological properties of gelled foods that may relate to flows in the swallowing process of complex food materials are investigated by means of USR. We examine both test mate- rials prepared with a recipe suggested by a food company devel- oped to enable better eating and also intentionally modified recipes.A comparison of results of rheological evaluations of both mate- rials could lead to the establishment of a methodology necessary to understand what is involved in optimum swallowing sensations. Here, as a test material, a milk dessert gelled by the chemical attrac- tion between low-methoxyl (LM) pectin (mainly used as a thickener for foods, such as fruit jam and pastelike sweets) and including cal- cium ions was chosen, and three test fluids prepared with different recipes were examined by comparative experiments involving the standard rheometer and the USR. In Sec. III, the inaccuracies in the capture of the true-rheological property with standard rheome- ters are discussed by examining the rheological evaluations of steady rotational and oscillatory tests. After that, a rheology for better texture of complex food materials is discussed by considering the findings obtained by USR (Sec. IV).
2. MATERIALS AND METHODOLOGY
   1. ‌Recipe for test materialsThe test material, “Fruiche” is a popular dessert in Japan and is available as a basic source material from House Foods Group, Inc., Japan. The Fruiche source includes much fruit pulp, O(1–10 mm) in mean diameter, irregularly shaped and deforming easily under shear stress. Mixed with whole milk, the completed Fruiche dessert changes drastically into a gel due to the aggregating reaction between low-methoxyl (LM) pectin25 (see the detailed chemical features in Ref. 26) dispersed in the Fruiche source and calcium ions dissolved in the milk as depicted in Fig. 1. Here, the whole milk is provided by Yotsuba Milk Products Co., Ltd., Japan; the nutrient composition is protein 34 g/l, lipid 40.5 g/l, carbohydrate 48.5 g/l, Na 0.39 g/l, and Ca 1.14 g/l. The chained molecules of the LM pectin structures the network linked by the calcium ions, and the structure is commonly called an “egg carton model.” From the perspective of rheology, the complete Fruiche displays highly complex-rheological properties (shear-thinning viscosity, viscoelasticity, yield stress, thixotropy, and modifications of the effective viscosity by multiphase dispersion). These would give rise to the common problems with the standard rheometer mentioned in Sec. I.The basic source of Fruiche is designed by the company for good texture in chewing and the following swallowing, and to do that, the best weight ratio of the milk to the source is set as 1:1. Here, to evaluate how differences from the recipe affects the rheology, we‌FIG. 1. Schematic outlines of LM-pectin cluster and the egg-carton model.

‌FIG. 2. Photographs of qualitative rhe- ology for test materials prepared by different recipes and placed on 30○ inclined glass plates. (a) Milk:source

= 2:1 mix, (b) milk:source = 1:1 mix, and

(c) milk:source = 1:2 mix.

intentionally prepared the Fruiche in different ratios (milk:source

= 2:1 and 1:2) from the recipe. After the mixed materials were placed on a 30○-inclined glass plate for 10 min, different behav- iors were observed, as shown in Fig. 2. In Fig. 2(a) (milk:source

= 2:1), the mix has fluidity and adheres on the surface of the plate. In Fig. 2(b) (1:1), the material had firmly adhered to the glass sur- face and its shape deformed only a little toward the lower part of the slope within 10 min. In Fig. 2(c) (1:2), the material had slipped on the surface while keeping its shape the same with the feature in Fig. 2(b). These observations represent the simplest rheological test and reflect important rheological characteristics of the test materi- als qualitatively, such as adhesion, deformability, fluidity, and oth- ers qualities, and these features can be used to verify the evaluated properties.

1. ‌Steady rotational and oscillatory shear tests with the standard rheometerA rheometer with geometry of Taylor-Couette system (Anton Paar MCR-502, parallel-plate geometry PP25) was used in the rhe- ological tests of the completed Fruiche prepared by the different recipes (Sec. II A). As is obvious from Fig. 2, all the three different test materials show large difference in the rheological characteristics. Although a variety of geometries depending on the characteristics of the test materials should be chosen, the same parallel plate was used for all the test materials to evaluate the test materials under the same condition. Otherwise, the geometric difference will affect the rheo- logical evaluations. The test materials are maintained at a constant temperature (T0 = 15 ○C).For the steady rotational and oscillatory tests, the following conditions and considerations are maintained to ensure accuracy:(a) one-directional flow (strain), (b) the shear rate is a linear functionin the axial direction, (c) the walls are subject to nonslip condi- tions, (d) the fluid is homogeneous, (e) inertial effects from fluid motion are disregarded, and (f) there are no secondary flows. A thin layer O(0.1–1 mm) of the test material is required to satisfy the assumptions; however, this thickness is inadequate to prevent the appearance of non-Newtonian behaviors,9,10,23 and the shear rate γ˙ and shear stress τ are regarded as an apparent shear rate γ˙app and an apparent shear stress τapp.In the ideal condition of oscillatory tests, the fluid response should be determined depending on its rheological characteristics (e.g., viscous, viscoelastic, and elastic) as shown in Fig. 3(a), and Lis- sajous curves can be derived by considering the shear strain/shear rate and shear stress as horizontal and vertical axes [Fig. 3(b)]. Clearly seen, the rheological characteristics can be distinguished by the feature of the Lissajous curves. Lissajous curves consisting of shear strain and shear stress draw circles for viscous, ellipse for vis- coelastic, and diagonal lines for elastic materials. In the nonlinear viscoelastic regime, the fluid response may not show clear sinu- soidal characteristics due to occurrences of unexpected phenom- ena. To understand the invisible phenomena of the fluid with the nonlinearity in the gap of the rheometer, large amplitude oscilla- tory shear (LAOS) measurements (e.g., Refs. 27 and 28) are per- formed for the rheological evaluations of complex fluids. The basic concept behind the LAOS measurement is that the shear stress in response to sinusoidal shear deformations are evaluated with the approximations to ensure the accuracy mentioned above. Nonsinu- soidal responses depending on the rheological characteristics with complexities (e.g., viscoelasticity, yield stress, and multiphase disper- sions) will be obtained using the LAOS. So, it is possible to qualita- tively understand the nonlinear rheological response caused by the complex fluid characteristics. The purpose of LAOS measurement in‌FIG. 3. (a) Ideal fluid response repre- senting shear stress against the applied shear strain, and (b) ideal features of Lis- sajous curve in the case of shear strain vs shear stress.

this paper was to clarify the rheological complexities in the test mate- rial, and it will also help to reveal the vagueness in the rheological evaluations of the standard rheometer.

1. ‌Ultrasonic spinning rheometry (USR)The experimental apparatus is an open cylindrical container made of acrylic resin; the container has 2-mm-thick side walls, 145-mm inner diameter, and is 60-mm high. The container was mounted at the center of a water bath to control the temperature, T0, of the test fluids and to avoid any influence from coreflected ultrasonic waves. Oscillations of the cylinder were controlled by a stepping motor to a set the oscillation angle Θ and frequency f o, where the motor was attached at the bottom of the container. The oscillating motion was controlled as a sinusoidal angular velocity, Uwall sin(2πf ot), where Uwall = 2πf oRΘ (see Ref. 23 for details). UVP-Model Duo (Met-Flow S.A., Switzerland) was used to mea- sure instantaneous velocity distributions. To obtain the azimuthal velocity component, an ultrasonic transducer (resonance frequency 4 MHz and 5-mm active element diameter) was mounted with a gap offset Δy from the center coordinates of the cylindrical con- tainer. With axisymmetric flow and the radial velocity component negligible, the azimuthal velocity uθ is calculated from the geomet- ric relation uθ = uξr/Δy. Empirically, Δy = 15 mm was selected, and further details of the setup for the transducer were detailed in Ref. 19.–Important theoretical considerations for the linear viscoelastic analysis in USR are as follows: assuming that the fluid flows are one- directional and axisymmetric, Cauchy’s equation of motion is given as ρ∂uθ/∂t = ∂τ/∂r + 2τ/r, where ρ is the density of the test fluids and τ is the shear stress. To describe the relation of uθ and τ, the Maxwell model, τ + (μ/E) ∂τ/∂t = μ (∂uθ/∂r uθ/r), is selected as the simplest expression to represent the linear viscoelastic character- istics, where μ and E indicate the viscosity and elasticity of the fluid. From the Fourier transform with respect to t, Cauchy’s equation and the Maxwell model can be modified as‌iωρuˆ = (  ∂ + 2 )τˆ, (1)can be used to distinguish the rheological characteristics of test flu- ids: Constant phase lag regimes correspond to rigid rotation, and such regimes may be regarded to occur with fluids having elastic properties or very high viscosity values. Regimes with a changing phase lag indicate fluidization areas and thereby can be regarded as fluids with viscous or viscoelastic properties. For example, radial profiles showing discontinuous variations indicate the existence of boundaries between different regimes of a rheological property.Simultaneous consideration of both linear viscoelastic analy- sis and phase lag in the USR offers the possibility to elucidate the rheological properties with the transient behaviors. Such transient- rheological properties cannot be measured by standard rheome- ters because of the limitations of methodology, and several valuable findings by the USR will be discussed next.
1. ‌RHEOLOGICAL EVALUATIONS BY THE STANDARD RHEOMETERIn this section, rheological evaluations of the test materials pre- pared with different recipes are examined by the standard rheometer with the parallel plate geometry described in Sec. II B. The steady rotational and oscillatory tests using the standard rheometer mea- sured the rheological properties of the test materials (Secs. III A and III B), and then, inaccuracies with the standard rheometer in measuring the true-rheological properties, arising from the com- mon problems that may occur between the plates, are discussed in Sec. III C.
   1. ‌Results of the steady rotational testsThe original source of Fruiche includes numerous strawberry pulps as mentioned in Sec. II A [see Fig. 4(a)]. The sizes of the pulp component vary up to approximately 30 mm. The completed Fruiche with the dispersed ingredients critically influences the rheo- logical evaluations using the standard rheometer [Fig. 4(b)] as exem- plified by jamming of the gap between the parallel plates. Because the rheological evaluations in the standard rheometer should be doneθ ∂r rwithout very large ingredients in the test fluids, the pulp components were removed for all of the experiments in Sec. III.‌τˆ + iω μ τˆ = μ( ∂uˆθ − uˆθ ), (2)The steady rotational tests were conducted by both shear rateE ∂r rand shear stress sweeps. In the shear rate sweep, the angular speedwhere the Fourier-transformed velocity and shear stress are denotedrepresenting the shear strain rate was logarithmically ramped up1as û (r, ω) = [u (r, t)], τˆ(r, ω) = [τ(r, t)], with the angularfrom 10−1 s−1 to 5.0 × 101 s− [Figs. 5(a)–5(c)]. In the shear stressθ F θ Fsweep, the angular speed representing the shear stress was rampedfrequency ω. Considering the cost function2F(μ, E; r) = [iωρuˆ − (  ∂ + 2 )τˆ] s.t. τˆ + iω μ τˆ = μ(∂uˆθ − uˆθ ),up linearly from 0 to 100 Pa [Figs. 5(d)–5(f)]. The vertical and hor- izontal axes indicate the apparent shear strain rate and shear stressθ ∂r rω=ωoE ∂rr(3)defined in Sec. II B. The gray and black solid curves represent the results obtained with or without settling for 1000 s, after preshearing,μ and E can be determined by satisfying the optimization problem, determining the μ and E to minimize the cost function F. From Eqs. (1) and (2), the shear stress, the shear rate, and the flow curve can be obtained from the Fourier components obtained from the velocity (see Refs. 21 and 23 for detailed calculations).According to Yoshida et al.,20 the radial profile of the phase lags calculated from the velocity distributions,R[uˆθ(r, ωo)]‌φ(r) = tan−1{  I[uˆθ (r, ωo)] } − φwall, (4)respectively.<<<In the shear rate sweep results in Figs. 5(a)–5(c), the apparent shear stresses, τapp, at low shear rates showed large differences in the condition of time left at rest: for the 2:1 mix [Fig. 5(a)] in γ˙app 4 s−1, the τapp immediately after the preshearing is much lower than that after being left at rest, while the τapp agrees between the conditions in the 4 s−1 γ˙app; for the 1:1 mix [Fig. 5(b)], at γ˙app 1 s−1, the τapp immediately after the preshearing is much lower than that after being left at rest similar to 2:1 mix, however, the τapp immediately after the preshearing around γ˙app = 1 s−1 showed an‌FIG. 4. (a) Photographs of ingredients dispersed in the original source and (b) apparent shear stress vs apparent shear rate obtained by the standard rheometer with the ingredients dispersed in the test material milk:source = 1:1 mix.

<

< <

abrupt but slight drop. Then, the curves obtained in the different conditions behave with similar manner at the 1 s−1 γ˙app value. For the 1:2 mix [Fig. 5(c)] and for all of the γ˙app range, there are clearly differences between the conditions in the leaving time; after leav- ing at rest, the τapp shows a plateau at 0.2 s−1 γ˙app 1 s−1. These results show significant differences that have to be understood for the different rheological characteristic, for example, at the 1:2 mix [Fig. 5(c)], as a typical flow curve,9 the measured plateau τapp indi- cates the occurrence of wall-slip (or shear banding) that would be caused by a weak thixotropic behavior.

In the measurements with a shear stress sweep conducted to evaluate the critical yield stresses at 2:1 mix [Fig. 5(d)], τapp after being left at rest for 1000 s is larger than that immediately after the preshearing. This feature is also observed for 1:2 mix [Fig. 5(f)], while the τapp for 1:1 mix [Fig. 5(e)] after being left at rest shows more drastic changes than the others, suggesting that the rheolog- ical response for 1:1 mix [Fig. 5(e)] has a stronger slippage on the wall than the others. This would be caused by the adhesion differ- ences of the tested mixture detailed in Fig. 2. It is important to bear in mind that in sweeping the shear stress, these trends represent the

wall-slip and are generally considered as showing a “yielding behav- ior”;29 this strongly affects the accuracy of the rheological evaluation, as mentioned in Sec. I. Therefore, usually, these critical responses are regarded as the “apparent yield stress” to estimate the spe- cific rheological response when depending on standard rheometer readings.

All together, the rheological characteristics obtained from Figs. 5(a)–5(f), representing the rheological properties of the Fruiche, show large difference depending on the length of time it was left undisturbed, indicating a structure that recovers with time, i.e., thixotropy. Generally, rheological properties of thixotropic flu- ids would be evaluated as the equilibrium state, or as an equilib- rium flow curve.30 Yoshida et al.23 reported, however, that especially in such thixotropic fluids, the flow curves obtained from rotating rheometers do not always provide correct evaluations without the influence of shear localization, wall-slip, and other factors, even when the shear stress apparently assumes equilibrium states. This clearly suggests that alternative tests have to be used to evaluate these test materials to capture the true-rheological properties. The results evaluated by steady rotational tests provided some hints to

‌FIG. 5. Rheological evaluations by the steady rotational tests with the stan- dard rheometer for test materials pre- pared by different recipes: (a)–(c) shear strain controlled measurements and (d)–

(f) shear stress controlled measure- ments; gray and black curves represent flow curved obtained after being left at rest for 1000 s and immediately after the preshearing.

understand the rheological features. For instance, it is speculated that the strength of the gelled structure would be in the order of the mixtures, 1:1, 1:2, and 2:1, here.

1. ‌Results of oscillatory testsFrom the measurements of steady rotational tests (Sec. III A), it is found that the rheological characteristics of the completed Fruiche are time dependent, i.e., thixotropy.30 To evaluate the equilibrium state in oscillatory tests, long duration measurements at different shear strain amplitudes (1–300%) with the same frequency (1.0 Hz) for the 1:1 mix were conducted (Fig. 6). The G′ and G′′ ampli- tudes here were measured twice, immediately after preshearing and at 5000 s elapsed after the start of oscillation. Immediately after the preshearing [solid line in Fig. 6(a)], the G′ amplitudes decrease monotonically as the shear amplitude increases, while G′′ ampli- tudes remain constant. After 5000 s, the two amplitudes are very dif- ferent to those immediately after the preshearing; the G′′ amplitude displays a local maximum at around γ0 = 100%.It is hard to guarantee exact linearity of the test material defor- mation in the oscillatory test shown in Fig. 6(a), because there may be shear banding effects in the gap of the rheometer; the photo- graph in Fig. 6(b) shows the condition of Fruiche when removing the shaft of the standard rheometer immediately after the long dura- tion measurements at γ0 = 50%. As shown in Fig. 6(b), a thin layer with O(0.1 mm) thickness was formed, and the diameter was almost the same as that of the parallel plate. Here, the material sticks to the bottom plate forming a thin layer, whereas the material at the top can be deformed because of shearing by the upper plate. This is clear evidence that the Fruiche was influenced by shear banding, result- ing in thixotropic characteristics here with the time scale longer than O(103 s). In such cases, the obtained G′ and G′′ should be considered as the “apparent” value.To estimate the relaxation characteristic of the test materials with different frequencies (0.1, 0.3, 1, and 3 Hz), oscillatory tests in a shear amplitude sweep from γ0 = 1% to 300% were conducted after preshearing and leaving at rest for 5000 s. The storage and loss mod- uli (G′ and G′′) obtained are shown in Fig. 7(a). For the 2:1 mix, the G′ and G′′ gradually change with respect to the strain amplitude, while more abrupt variations are obtained in the other mixtures (1:1 and 1:2). For the 1:1 mix, an abrupt variation is observed at around γ0 = 50% in all conditions at 0.1–3 Hz. For the 2:1 mix, theabrupt variation appears at very similar γ0 as for the 1:1 mix, there with a frequency lower than 0.3 Hz. Although the amplitude sweep measurements would be influenced by shear history effects such as the oscillations, the abrupt variation would suggest that phenomena invisible here possibly arise from the wall-slip or some other effects, suggesting that there may be significant differences in the relaxation behaviors of the test materials at O(0.1–1 s).To understand details of the rheological response during oscil- lations, LAOS measurements were conducted simultaneously with the G′ and G′′ evaluations in Fig. 7(a). Figure 7(b) shows the changes in Lissajous curves (shear strain vs shear stress) with respect to the oscillation frequency f (0.1, 0.3, 1, and 3 Hz) and the maximumshear strain amplitude γ0 (10%, 30%, 100%, and 300%). As shown in Sec. II B, the Lissajous curves have been used to distinguish the rhe- ological properties from the curve shape as three kinds of features representing viscous, viscoelastic, and elastic.At the high frequency (3 Hz), in Fig. 7(b), the Lissajous curves in each test material suggest an elastic response irrespective of the shear amplitude, and at f = 1 Hz and γ0 = 100%, the shapes of Lis- sajous curves represent more viscoelastic characteristics compared to the others of shear amplitude γ0. At f = 0.3 Hz, all the curves for all test materials are less regular with kinks than at higher frequen- cies; these results can be explained by the qualitative deformation features described in Fig. 2 for the behavior on the inclined glass plates.For the 2:1 mix, the Lissajous curve at f = 0.3 Hz and γ0 = 30% has a nearly circular shape representing a viscous reaction coex- isting with irregular features [Fig. 7(b)]. Here, Fig. 2 shows easily deformable characteristics of the material. For the 1:1 mix, the Lis- sajous curve at f = 0.3 Hz and γ0 = 30% shows a close to elastic response with some kinks [Fig. 7(c)], similar to the more viscoelastic 1:2 mix [Fig. 7(d)]. The factors causing this difference can be under- stood by the qualitative differences shown in Fig. 2. The 1:2 mix [the right-hand column in Fig. 7(a) and Fig. 7(d)] may more easily be exposed to wall-slip than the 1:1 mix, and the Lissajous curves at the 1:2 mix suggest results influenced by the wall-slip. This sugges- tion is supported by the results in the steady rotation tests under the shear stress sweep (Sec. III A); the critical yielding behavior of the 1:1 mix was larger than that of the 1:2 mix, as shown in Figs. 5(e) and 5(f). It may be concluded that the gelled appearance in Fig. 2 is accompanied by the wall-slip.‌FIG. 6. (a) Storage and loss modulus (G′, G′′) vs the maximum shear strain amplitude, where the black and gray symbols representing the elapsed time and circle and triangle symbol represent- ing storage and loss modulus (G′, G′′), and (b) the photography of the condi- tion of Fruiche after removing the shaft of the standard rheometer immediately after the long duration measurements at γ0 = 50%.

‌FIG. 7. (a) Storage and loss modu- lus (G′, G′′) of the shear amplitude sweep for different oscillating frequency and test conditions for different materi- als, and [(b)–(d)] the Lissajous curves (shear strain vs shear stress normal- ized by the maximum value) for differ- ent shear amplitudes and oscillating fre- quencies that were obtained from the measurements plotted in (a).

The wavy features observed on the curves at frequencies lower than f = 1 Hz at low shear strain amplitudes, and the wavy features of the 2:1 and 1:2 mixtures are larger than those of the 1:1 mix. These features are observed at lower shear strain amplitudes than those resulting in abrupt changes in the Lissajous curves, for example, at the 2:1 mix at f = 0.3 Hz from γ0 = 10%–30%. According to Fardin et al.,14 unstable shear banding would arise from elastic instabilities, when shear stress in flow curves exhibits a plateau with respect to the shear rate under unsteady shear conditions. The geometry used in this study is different from the Taylor-Couette system used in the previous study, and it is of importance to be aware that elastic instability would be triggered by a transient state in the rheological characteristics of complex materials.

1. ‌Summary: The sources of physical inaccuracies in a standard rheometerThe standard rheometer has great advantages in the measure- ment of viscous or elastic materials without abrupt phase tran- sitions such as yielding behavior. As shown by the experimental findings in Secs. III A and III B, the torque sensitivity in mea- suring the rheological response will augment inaccuracies aris- ing from bias errors in the understanding of the rheology of testmaterials in the standard rheometer; the influence from the pres- ence of a depleted layer arising from abrupt phase transitions is very significant here; the data output is a “specific-mechanical response” influenced by the geometry of the rheometer. Many reports have attempted to explain such shear banding problems as mentioned in Sec. I, with most focusing on measuring the shear rate profiles in the gap of the rheometer. However, the problems arising from the standard rheometer cannot be solved by looking at only the influence of elastic instability, even under steady state variations, as there are 3-dimensional unsteady flows, as reported elsewhere.11 Furthermore, it may not be possible to correct the data obtained from oscillatory tests because the unsteady changes depend on the rheological characteristics of complex fluids. In oscillatory tests, inaccuracies will be significantly magnified as the basic principle of oscillatory tests does not consider the “fluid inertia.” Experimental proof of such inaccuracies will be discussed in Sec. IV by compar- ing the rheological evaluations of standard rheometers with those by USR.
1. ‌USR TESTS OF THE RHEOLOGY OF GELLED FOODIn this section, inaccuracies in the rheological properties evalu- ated by standard rheometers (Sec. III) will be explained using USR.‌FIG. 8. Time variations in the azimuthal velocity distributions of a test material (the milk:source = 1:1 mix) at differ- ent elapsed oscillating times, with the oscillation frequency, amplitude, temper- ature, and the maximum angular velocity, f o = 1.0 Hz, Θ = 60○, T0 = 15 ○C, and Uwall = 477 mm/s, respectively.

The USR has the great advantage that it can evaluate materials with dispersed ingredients contained in the original source. Furthermore, based on the equation of motion and rheological model detailed in Sec. II C, rheological evaluations can be realized in unsteady shear conditions by frequency domain analysis (detailed explanations of the algorithm can be found in Refs. 21 and 23).

1. ‌Rheological evaluation on phase lag in velocity distributionsSpatiotemporal velocity distributions measured for the 1:1 mix using UVP are shown in Fig. 8. The vertical and horizontal axes indi- cate the radial positions normalized by the radius of the container, R, and the spin-cycle period tf o. The contours represent the azimuthal velocity normalized by its maximum value at the cylinder wall (r/R = 1), Uwall (= 2πf oRΘ). The distributions here were obtained at different elapsed oscillation times with f o = 1.0 Hz in the oscillation frequency and Θ = 60○ in amplitude. The resolutions of the UVP are 25 ms in time, 0.7–0.8 mm in space, and 1.35–1.4 mm/s in velocity. To eliminate the shear history effects, UVP measurements were con- ducted leaving the test materials at rest for longer than 15 min after stirring the fluid medium.Immediately after the start of oscillation (tf o = 0–1), the test material oscillates in the azimuthal direction almost in phase from the wall to the center of the cylinder as in a rigid body. As the gel structures of the test material are gradually broken and become fluidized due to the shear stress from the oscillating wall, the oscillations of the test material assume small phase lags in the radial direction, for example, at tf o = 60 (center column), and this is significant after tf o = 1800. This phase lag development can be used to distinguish the rheological characteristics of test fluids.20From Eq. (4), and after calculating the Fourier components from the velocity distribution, radial profiles of the phase lag for the test materials can be quantified as shown in Figs. 9(a)–9(c). The vertical axis represents the radial position normalized by the cylindrical wall, R, where the horizontal axis shows the phase lag. Here, the range of phase lags are different for the three materials in Figs. 9(a)–9(c) to show the differences in the phase lags. The pro- file shape does not drastically change at least after tf o = 1800, and the profiles are assumed to reach the terminal state at tf o = 1800. The figures show some similarities in the changes accompany- ing the development of the phase lag with the oscillation period tf o. These suggest shear thinning of the test material with the‌FIG. 9. Radial profiles of the phase lag of the local velocity fluctuations from the cylinder wall for 10 s times averaging at different elapsed times: (a) milk:source = 2:1 mix,(b) milk:source = 1:1 mix, and (c) milk:source = 1:2 mix, with the oscillation frequency, amplitude, temperature, and the maximum angular velocity, f o = 1.0 Hz, Θ = 60○, T0= 15 ○C, and Uwall = 477 mm/s, respectively.

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shear deformations taking place during oscillations. For the 2:1 mix in Fig. 9(a), the development is more rapid than for the other mixtures in Figs. 9(b) and 9(c). The terminal phase lag profile (tf o = 1800–1810) for the 1:2 mix [Fig. 9(c)] shows a knee around r/R = 0.7–0.75, while the profile for the 1:1 mix [Fig. 9(b)] has smoother variation around the radial position. It suggests that the 1:1 mix has a viscoelastic layer functioning as a buffer for the momentum propagation between the liquid and gelled regions. There are also large difference in the terminal phase lag profile between the 1:1 and 1:2 mixtures, that is, the gradient of the termi- nal phase lag for the 1:2 mix [Fig. 9(c)] near the wall (0.8 r/R 1.0) shows as much larger than that at the 1:1 mix [Fig. 9(b)]. This can be interpreted as showing a lower viscosity of the 1:2 mix [Fig. 9(c)] than that of the 1:1 mix [Fig. 9(b)].

The rheological characteristics of the different test materials can be summarized from the above experimental results as fol- lows: the rheological characteristic of the 2:1 mix is the one that is most like a liquid and where it is easiest for the gelled struc- ture to break down; in the 1:1 mix, the rheological characteristics show a strong viscoelastic response at the interface between the liq- uid and gelled regions; in the 1:2 mix, the rheological characteris- tic has a more abrupt change at the interface between the regions than is the case for the 1:1 mix. These results may then be inter- preted to be in close agreement with both the qualitative features of each mixture (Fig. 2), as well as that they display the differ- ences suggested by the standard rheometer measurements (Sec. III), and overall, the rheological characteristics may be described by the terminal shape of the phase-lag and the spatial gradient of the phase-lag.

From the gradient of the phase differences calculated by numer- ical differential of the phase lag in the radial direction, the fluid motions in the oscillations suggests a division into the liquid and gelled regions. The borderline of the division is now defined as the appearance of “yielding” as determined in this study. The thresh- old value of the gradient here is set at 10 rad/m corresponding to around 104 mm2/s.20 Figure 10 shows the phase lag with respect to the oscillation period tf o. Here, the phase lag profiles in Fig. 9 were averaged by 10 periods of oscillations, while those in Fig. 10 are not averaged but are as measured profiles calculated from the velocity distributions for 1 s intervals.

For the 2:1 mix shown in Fig. 10(a), the phase lag and the yield- ing border described with a dotted curve vary more rapidly, immedi- ately after the oscillation starts than in the case for the other mixtures shown in Figs. 10(b) and 10(c). Here, the gray scale in Fig. 10(a) showing the phase lag is different from the others [Figs. 10(b) and 10(c)] because of the drastic changes in the rheological characteris- tics. In the 1:1 [Fig. 10(b)] and 1:2 [Fig. 10(c)] mixtures, the varia- tions in the positions where yielding starts show similar changes and vary more gradually, even immediately after the oscillation starts. The phase lags with the 1:2 mix [Fig. 10(c)], however, display larger values than that for the 1:1 mix [Fig. 10(b)].

1. Results of the linear viscoelastic analysisIn the linear viscoelastic analysis by USR, the values of the rheo- logical properties, such as viscosity, elasticity, shear strain rate, shear strain, and shear stress, are obtained from the measured spatiotem- poral velocity distributions (e.g., Fig. 8).23 Furthermore, by utilizing the gradient of phase-lag as described in Sec. IV A, the physical borderline between liquid and gelled states can be simultaneously estimated for every oscillation cycle. To quantify the rheology of the test materials, the rheological evaluations were performed by calcu- lating flow curves from the viscosity and elasticity obtained via the linear viscoelastic analysis.Figure 11 shows the flow curves obtained from averaged Fourier components for each of the test materials, with the ver- tical and horizontal axes representing the shear rate and shear stress, respectively. The gradation of the plots indicates the cor- responding radial position in the cylindrical container normalized by R. Each symbol in the plots shows the period of the oscillation cycle, tf o, circles: 0–4, triangles: 4–8, squares: 16–20, diamonds: 32–>36, pentagons: 64–68, and crosses: 1800–1900. Figures 11(a)–11(c) show the flow curves in each period for each of the mixtures, and Fig. 11(d) plots the terminal flow curves (tf o = 1800–1900) obtained from each material together. Such “transient flow curve” is typi- cal in the linear viscoelastic analysis by USR and is not obtained by standard rheometers. Here, the ranges of r/R used for the flow curves are from r/R = 0.95 to the radial position corresponding to the physically determined borderline between liquid and gel, r/R 0.6.‌FIG. 10. Radial-time distribution of phase lags in the local velocity fluctuations from a cylinder wall for (a) milk:source = 2:1 mix, (b) milk:source = 1:1 mix, and (c) milk:source = 1:2 mix, where the dotted curves represent the border between the liquid and gelled regions estimated from the gradient of the phase lag, with the oscillation frequency, amplitude, temper- ature, and the maximum angular velocity, f o = 1.0 Hz, Θ = 60○, T0 = 15 ○C, and Uwall = 477 mm/s, respectively.

‌FIG. 11. Flow curves (γ˙ vs τ) at dif- ferent oscillating cycles; (a) milk:source

= 2:1 mix, (b) milk:source = 1:1 mix, and

(c) milk:source = 1:2 mix, and (d) ter- minal flow curves at tf o = 1800–1900 in each test material, with the oscilla- tion frequency, amplitude, temperature, and the maximum angular velocity, f o

= 1.0 Hz, Θ = 60○, T0 = 15 ○C, and Uwall

= 477 mm/s, respectively.

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For the 2:1 mix [Fig. 11(a)], the flow curves have almost mono- tonically increasing trends with respect to the shear rate, but the shear stress shows larger values locally at the lowest shear rates in the periods from tf o = 0–4 to tf o = 64–68. The tendencies of these changes are quite similar to the typical flow curves observed in shear banding fluids,7 where most of the results were evaluated by steady rotational tests with standard rheometers in combination with a technique to visualize the inside of the gap between the plates. In the terminal flow curve for the 2:1 mix, the gradients of the flow curve are almost constant in the double logarithmic expression in the present range of shear rates. Comparing the initial and termi- nal flow curves, it is noteworthy that the flow curves overlap over a wide range of shear rates. For the 1:1 mix [Fig. 11(b)], the ini- tial flow curve (tf o = 0–4) has a negative gradient at smaller shear stresses, and after some periods of oscillations (tf o 4–16), the flow curve appears similar to that of the 2:1 mix. At the terminal state (tf o = 1800–1900), however, the flow curve bends around γ˙ = 4 s−1, and this may be physically understood to show that the viscous resis- tance of the test material changes greatly at a specific critical shear rate. For the 1:2 mix [Fig. 11(c)], the shear stresses in the flow curves from tf o = 0–4 to tf o = 64–68 are much larger than those of the other materials, and after reaching the terminal state (tf o = 1800–1900), the flow curve becomes linear except around the lowest shear rates.

In comparison of terminal flow curves [Fig. 11(d)], the gradient of the flow curve of the 1:2 mix is the smallest of the three mixtures, and the gradient of the 1:1 mix in 4 s−1 γ˙ 10 s−1 is in good agree- ment with that of the 2:1 mix. Furthermore, at shear rates lower than 4 s−1, the flow curve of the 1:1 mix shows the steepest gradient. Relat- ing these differences in the flow curve to the qualitative observations for each mixture (Fig. 2) suggests rheological explanations of behav- iors for each of the mixtures. For the 2:1 mix, the mixture behaves as a liquid due to the lower yield stresses; however, the mixture sticks to the glass surface after placing the glass plate on the flat surface. For the 1:1 mix, the main feature is a large viscous resistance in the range of low shear rates, which contributes to sticking to the inclined glass plate (Fig. 2). For the 1:2 mix, the agglutinating property is much smaller than in the other mixtures because of the decrease in viscous resistance after yielding.

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Figure 12(a) shows both flow curves for the 1:1 mix in the rheological evaluations of USR and with the standard rheometer, where the flow curves (stars) were calculated from the experiment in Fig. 6. The vertical and horizontal axes represent the shear rate and shear stress, respectively, but the values for the standard rheometer are apparent values, as mentioned in Sec. III. The shear stresses at t = 5000 s by the standard rheometer are much larger than those by USR, but at t = 0 s, the standard rheometer values show good agreement with the 5 s−1 γ˙ values by USR at the terminal state.

‌FIG. 12. (a) Flow curves of the test material (milk:source

= 1:1 mix) in LAOS plotted with USR results and (b) the Lissajous curves (shear stress vs shear strain) plotted with oscillating time vs maximum shear strain γ0.

Figure 12(b) shows the Lissajous curves, obtained by LAOS at the experiments in Fig. 12(a), and help to elucidate the factors caus- ing the differences in the flow curves. The amplitude of the appar- ent shear stresses increases with time because of the restoration of the structures present in the mixtures, i.e., thixotropy, however, an increased influence of the viscous characteristics can be observed from the shape of the Lissajous curves at γ0 = 50, 80%, and 100%. If the test materials display thixotropy, the rheological properties would shift from somewhat viscous to elastic, but the results of the Lissajous curves show an opposite change. It may be suggested that the rheological evaluations of the standard rheometer are influenced by shear banding causing the occurrence of a depleted layer at the oscillating wall. The apparent shear rate would increase if there is a thin layer causing the shear banding, and the presence of this can be assumed from the changes in the rheological properties displayed in the flow curves obtained by USR.

The depleted layer as the result of shear banding can be seen on the photograph [Fig. 12(a)] taken after removing the shaft of the standard rheometer following the long duration measurements at γ0 = 50%. As also described in Sec. III B, the material was attached

to the bottom plate in the form of a solid structure of a thin layer of O(0.1 mm) thickness, while the material at the top was softer and could be deformed because of shearing by the upper plate. The spec- ulations of a shear banding effect in the gap are strongly supported by this observation.

1. Discussion: The rheology of a better-prepared dessertFrom the rheological evaluations by USR based on fluid mechanics, the relations of the shear strain rate and shear stress around the critical shear rate for yielding were quantified. These findings are significant as it is difficult to evaluate the true- rheological properties mentioned in Sec. III with the standard rheometer. The schematic illustrations in Fig. 13 summarize the findings: The figure represents the gelling behavior schematically with the ratio of included Ca2+ and pectin.When the ratio is low [Fig. 13(a)], for the whole milk:Fruiche source = 2:1, there is a low concentration of structured pectin, and it does not contribute to gelling, but it increases the viscos- ity. With moderate ratios [Fig. 13(b)] for whole milk:Fruiche source‌FIG. 13. Schematic visualization of the gelling features for different ratios of included pectin and calcium ions:(a) Pectin/Ca2+ ratio is low, e.g., whole milk:Fruiche source = 2:1, (b) pectin/Ca2+ ratio is moderate, e.g., whole milk:Fruiche source = 1:1, and(c) pectin/Ca2+ ratio is high, e.g., whole milk:Fruiche source = 1:2.

= 1:1, the flow curve indicates a secondary critical shear rate that should be distinguished from the critical shear rate for yielding. It is thought that the pectin inclusion caused the formation of struc- tures of chained networks that lead to the viscoelastic behaviors at lower shear rates than the critical shear rate for yielding. The result of this would be that the material shows strong viscoelasticity aris- ing from the electrical attraction between the included calcium ions and dispersed pectin. Finally, for higher pectin ratios [Fig. 13(c)], for whole milk:Fruiche source = 1:2, the yield stress estimated by USR shows larger values, and the viscosity is lower than the mixture with intermediate ratios; this may be interpreted to suggest that the gelled structures were fully developed and further that the material has only a small influence on the viscosity increase arising from the electrical attraction after yielding.

If the recipe suggested by the food company is assumed to pro- vide the best eating quality (or texture), evaluations of these are given as the rheological property of moderate viscoelasticity and yield stress as described above. Furthermore, the stability of gelled structures with moderate yield stress is also required to be evaluated, and the stability might be supported by the thixotropic behavior as the structures need to be maintained after stirring.

1. CONCLUDING REMARKSRheological evaluations on three test mixtures, prepared by adding whole milk to Fruiche (a commercially available dessert) source in different ratios, were conducted by both a standard rheometer and USR for better understanding of qualities and phys- ical features evaluated as better quality for eating. Using the stan- dard rheometer with parallel plate geometry, the evaluations showed inaccuracies in measuring accurate rheological properties caused by commonly existing problems in the rheometers (wall-slip, shear banding, shear localization, elastic instability, etc.); for steady rota- tional tests, there are the problems around the critical shear causing yielding or slipping of the test material on walls, and the rheolog- ical evaluations showed significant differences between the shear rate and stress sweep data. In oscillatory tests, drastic changes in the storage and loss moduli in shear amplitude sweeps were found, especially at low oscillation frequencies, and the Lissajous curves suggested unexplained problems by LAOS measurements.To better understand the inaccuracies in measuring the true- rheological properties by the standard rheometer, the rheology rep- resenting the textures of complex food materials was suggested using USR based on the equation of motion and spatiotemporal velocity information. In conclusion, the flow curves for all the test materi- als evaluated by USR showed more reliable rheological character- istics than the data obtained by the standard rheometer. The word “reliable” here means guaranteed precisions with physical mean- ingfulness. Some unclarities in the rheological evaluations using the standard rheometer were found from the experimental results (Sec. III). Not surprisingly, such complex test fluid showing rheo- logical changes drastically against shear deformations must affect the rheological tests. 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