Nomenclature (in order of appearance)

1. Introduction

2. Materials and Methods

2.1. Soft elastic reactor mixing curves

The SER developed by Chen and the co-workers (Chen and Liu, 2015, Chen, 2016, Liu et al., 2018) consists in promoting mixing through elastic wall movements (Figure 1A), with no limitation regarding corrosive fluids compared to conventional stirred reactors as the agitator is not immerged in the agitated media. It consists of two key elements (Figure 1B): i) a tank made of transparent silicone with soft and elastic properties and ii) a slider crank device constituting the excitatory system responsible for the local deformation of the wall.

In order to assess the mixing performance and improve the design of such reactors, Xiao et al. (2018) developed an effective and robust method based on colorimetric diagnosis. Using this approach, Delaplace et al. (2018) determined the mixing time $t_m$ for highly viscous Newtonian fluids (viscosity $\mu$ ranging from 0.0285 Pa.s to 1.187 Pa.s) and established a pertinent reduced configuration of mixing time number ${\mathsf{\Theta }}_m$ as a function of operating conditions (Since $C_b$/$T$ experimentally had a poor influence on mixing time, it was neglected, Equation (1)):

$${\mathsf{\Theta }}_m=f·t_m=F\left(Re=\frac{\rho ·f·T^2}{\mu },\frac{p}{T}, Ga=\frac{g·T^3·{\rho }^2}{{\mu }^2}\right)$$

(1)

with $f$ as beating frequency (Hz), $t_m$ as mixing time (s), $f·t_m$ as number of beating required to achieve the desired degree of homogeneity, the Reynolds number $Re$ as an internal measurement of viscosity, $\rho$ as density of the liquid (kg.m^-3), $T$ as inner diameter of the tank (m), $\mu$ as viscosity of the liquid (Pa.s), $\frac{p}{T}$ as penetration depth ratio, the Galilei number $Ga$ as a dimensionless number considering the effect of gravity on free surface of liquid and $g$ as acceleration of gravity (m.s^-2).

Mixing curve of dimensionless mixing time number ${\mathsf{\Theta }}_m$ varying $Ga$ and $\frac{p}{T}$ ratios that was established is presented in Figure 2. For the maximum $\frac{p}{T}$ (= 0.42), it was showed that a constant $f·t_m$ was needed to achieve the desired degree of homogeneity under the laminar regime, while a sharp decrease in $f·t_m$ could be identified when the laminar regime is over.

$f·t_m$ required to achieve desired degree of homogeneity increased when $\frac{p}{T}$ decreases but the sigmoidal shape of the mixing curve remained similar, whatever the fixed $\frac{p}{T}$ values. $f·t_m$ required for meeting homogeneity with decreasing $\frac{p}{T}$ values seemed coherent with the theory. Indeed, it is expected that $p$ should have a more pronounced influence under a laminar regime when convection of fluid element is limited rather than in the transition and turbulent regimes. In this sense, the less pronounced influence of $\frac{p}{T}$ values in a transition and turbulent regime is logical. Moreover, the authors highlighted that $Ga$ had no influence on the evolution of ${\mathsf{\Theta }}_m$, in coherence with the absence of free surface deformation for the range of perturbations and liquid viscosities considered.

Three zones in the sigmoidal mixing curve of the SER could be identified:

• In laminar regime (Re<2.7-4.5), a constant value of ${\mathsf{\Theta }}_m$ ranging from 1500 ($\frac{p}{T}$=0.42) to 4600 ($\frac{p}{T}$=0.25) is observed, meaning that the number of strikes required to achieve the desired degree of homogeneity is independent on both viscosity and beating frequency, but still influenced by the penetration depth: the fact that decreasing the value of $\frac{p}{T}$ led to an increase of the number of strikes required to achieve homogeneity under laminar regime is in coherence with the expected physical meaning. Moreover, a large number of strikes is required to achieve homogeneity since the high level of viscosity is an obstacle for dissipating convection;
• In the transition regime (3<Re<60), a decreasing value of ${\mathsf{\Theta }}_m$ meaning that a lower beating frequency is required to reach a desired degree of homogeneity for a given penetration depth;
• For Re>60, ${\mathsf{\Theta }}_m$ reached an asymptotic value close to 40 ($\frac{p}{T}$=0.42) and appeared to be here again independent of Re when getting closer to the turbulent regime. More data would be required to be more precise on this plateau value.

From a general point of view, the process relationship illustrated in Figure 2 demonstrated the strong influence of the viscosity of the liquid contained in the SER, the beating frequency and maximum penetration depth on the overall homogenization performance of the SER in the range of operating conditions investigated (0.5<$f$<3 s^-1 and 0.03<$\mu$<1.25 Pa.s). Indeed, depending on the value of the Reynolds number and penetration extent, the number of strikes required to achieve homogenization could be largely different, ranging from 40 to 4600).

Figure 2. Mixing time characteristics of the SER using the reduced configuration defined in Equation (1) obtained at varying Galilei number and penetration depth ratio ($\frac{p}{T}$) ratios. Each symbol represents a $\frac{p}{T}$. Dotted line, plotted for mixing time experiments performed at constant $\frac{p}{T}$ value of 0.42, is a guide to the eye (from Delaplace et al., 2018).

Figure 2. Mixing time characteristics of the SER using the reduced configuration defined in Equation (1) obtained at varying Galilei number and penetration depth ratio ($\frac{p}{T}$) ratios. Each symbol represents a $\frac{p}{T}$. Dotted line, plotted for mixing time experiments performed at constant $\frac{p}{T}$ value of 0.42, is a guide to the eye (from Delaplace et al., 2018).

2.2. Hypotheses for considering the stomach as soft elastic reactor

The human stomach and the pattern of the ACWs during digestion are illustrated in Figure 3. It is recognized that after a typical meal, an average-sized human stomach has a width of 0.10 m at its widest point and a capacity of about 0.94 L, with the greater curvature being around 0.30 m long. The pyloric ring diameter is 1.1 cm or less. The average diameter of the distal region under relaxed conditions was considered to be 0.052 m (Ferrua et al., 2014).

The motility pattern of the stomach wall is still not fully characterized, due to the experimental limitations and despite recent advances in medical imaging technologies that allowed a more accurate description of the ACWs (Pal et al., 2004; Kwiatek et al., 2006; Schwizer et al., 2006; Treier et al., 2006; Pal et al., 2007). From these studies, it can be retained that ACWs were initiated every 20 s (3 cycles per minute) at around 15 cm from the pylorus. Consequently, the ACWs frequency is $f=\frac{1}{20}=0.05$ s^-1. The average amplitude of the initial antral contraction, that may be viewed as corresponding to the maximum penetration depth of the SER, was shown to be equal to 7.0 ±0.8 10^-3 m (Kwiatek et al., 2006; Figure 4). The ACWs propagated down with an average horizontal speed velocity of 0.002 m·s^-1.

The fact that deformation location and liquid free surface deformation have few impacts on the mixing time for the soft reactor supports the comparison of the SER to the stomach. Although different exciting modes were used (crank slider for the soft reactor and ACW for the stomach), the resulting effect of deformation may be viewed as being similar, i.e., the propagation of a contraction wave along the stomach/soft reactor wall.

Figure 4. Computed analysis of ACWs as seen by MRI. The average distance ($w_1$ and $w_2$, cm) between the bases (x) of the indentations in the antral wall, representative of ACWs, denoted the width of ACWs, $w_{av}$. The average distance ($a_1$ and $a_2$, cm) between the tips (o) of these indentations and the line linking the bases (x) of the indentations corresponds to the amplitude of ACWs, $p_{av}$. Extracted from Kwiatek et al. (2006).

Figure 4. Computed analysis of ACWs as seen by MRI. The average distance ($w_1$ and $w_2$, cm) between the bases (x) of the indentations in the antral wall, representative of ACWs, denoted the width of ACWs, $w_{av}$. The average distance ($a_1$ and $a_2$, cm) between the tips (o) of these indentations and the line linking the bases (x) of the indentations corresponds to the amplitude of ACWs, $p_{av}$. Extracted from Kwiatek et al. (2006).

3. Results and discussions

3.2. Application of the soft elastic reactor model to data from litterature

In order to reinforcing the fundamental understanding of gastric fluid mixing by antral contraction in the stomach, we have evaluated how far the $Re$ values and the penetration depth ratios induced by ACWs when mixing digesta of various viscosity are to the range of experimental mixing runs on the SER. For that, we have used the experimental data from Ferrua et al. (2014) who studied the mixing of water (10^-3 Pa·s), tomato juice (0.17 Pa·s) and honey (1 Pa·s) in the stomach. These authors calculated the corresponding $Re$ values using $\left(\rho ,c,L_{av}\right)$ as repeated physical parameters (Equation (2)), where $c$ is the average horizontal linear speed of ACW propagation considered as the characteristic velocity:

$$Re=\frac{\rho ·c·L_{av}}{\mu }$$

(2)

Considering the average diameter of the distal region under relaxed conditions $L_{av}=0.052$ m as characteristic length and an average horizontal linear speed of ACW propagation $c=0.002$ m.s^-1, these authors obtained $Re$ values of 116, 0.7 and 0.1 for water, tomato juice and honey, respectively.

For the SER, the characteristic speed of propagation $c$ was not experimentally determined, as the beating frequency $f$ was the only accessible parameter. Therefore, a $Re$ definition with $\left(\rho ,f,L_{av}\right)$ as physical repeated variables was adopted:

$$Re=\frac{\rho ·f·L_{av}^2}{\mu }$$

(3)

In order to compare stomach peristalsis and SER mixing, $Re$ values encountered in the distal region were re-calculated using Equation (3) for water, tomato juice and honey, considering the corresponding density values given in Ferrua et al. (2014), an average antral contraction frequency $f=0.05$ s^-1 and a characteristic length $L_{av}=0.052$ m. We obtained $Re$ values equal to 150, 0.91 and 0.13 for water, tomato juice and honey respectively, in the range of those covered in SER.

Penetration depth ratio of the distal region was found to be $\frac{p_{av}}{L_{av}}=0.13$ and was thus lower although in the same order of magnitude as the range of contraction covered in the SER experiments (i.e., 0.25-0.42; Figure 2).

From these $Re$ and $\frac{p_{av}}{L_{av}}$, it is expected that the gastric mixing of both tomato juice and honey in the distal region was performed under laminar regime ($Re<3$). Consequently, it is expected from the process relationship (Figure 2) that a constant number of contractions should be performed for these two digesta, independently of their viscosity values. The expected number of contractions ${\mathsf{\Theta }}_m$ is probably higher than $4600$ since $\frac{p_{av}}{L_{av}}=0.13$ is lower than $\frac{p}{T}=0.25$, for which this plateau value was obtained in SER mixing time experiments (Figure 5). Given an average frequency of 3 cycles.min^-1 ($f=0.05$ s^-1), it could be expected that 1530 min is required to achieve complete homogeneity.

On the contrary, the number of contractions ${\mathsf{\Theta }}_m$ required to achieve homogeneity should decrease for water, as the $Re$ value is higher in this case ($Re=150$). Once again, this value could not be exactly determined since experimental study for the SER was not performed at this penetration depth $p$. However, at least 40 antral contractions should be necessary according to the mixing curve, which means more than 13 min at the same average frequency of 3 cycles.min^-1 (Figure 5).

As the number of contractions was limited to 15 (corresponding to 5 min of residence time) in the study of Ferrua et al. (2014), these authors logically concluded to a poor ability of the stomach to advect gastric contents, regardless of the nature of the digesta. Indeed and as underlined above, homogeneity could not be achieved for water, tomato juice and honey after such residence time in human stomach.

Figure 5. ${\mathsf{\Theta }}_m$ deduced for honey (⬤), tomato juice (⭘) and water (◍) using the mixing time characteristics of the SER from Figure 2.

Figure 5. ${\mathsf{\Theta }}_m$ deduced for honey (⬤), tomato juice (⭘) and water (◍) using the mixing time characteristics of the SER from Figure 2.

5. Conclusions

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