BiconeDrag updated – A data processing application for the oscillating conical bob interfacial shear rheometer,Computer Physics Communications

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BiconeDrag updated – A data processing application for the oscillating conical bob interfacial shear rheometer
Computer Physics Communications ( IF 6.3 ) Pub Date : 2021-06-17 , DOI: 10.1016/j.cpc.2021.108074
Pablo Sánchez-Puga , Javier Tajuelo , Juan Manuel Pastor , Miguel A. Rubio

We present a new improved version of BiconeDrag, a code that allows one to obtain the values of interfacial dynamic moduli from dynamic experiments performed in rotational interfacial shear rheometers with bicone probes. The general structure of the program remains the same: starting from the values of the torque/angular displacement amplitude ratio, and by using the equations of the hydrodynamic field, it is possible to decouple the contribution of the subphase (bulk) drag from the true interfacial drag. In this new version we have improved the implementation of the boundary condition at the interface so that now it is built as second order centered finite differences (SOCFD hereafter) with the help of a line of phantom nodes. The new numerical implementation of the interfacial boundary condition yields smoother velocity profiles at the bicone rim, which result in a more accurate separation of interfacial and subphase drags and, consequently, in a more precise calculation of the interfacial dynamic moduli.

New version program summary

Program Title: BiconeDrag

CPC Library link to program files: https://doi.org/10.17632/c245bmgf5n.2

Code Ocean capsule: https://doi.org/10.24433/CO.5536863.v1

Licensing provisions: GPLv3

Programming language: MATLAB (compatible with GNU Octave)

Journal reference of previous version: Comput. Phys. Commun. 239 (2019) 184–196

Does the new version supersede the previous version?: Yes

Reasons for the new version: Remodeling the implementation of the interfacial boundary condition.

Nature of problem: The different horizontal (second order centered finite differences) and vertical (first order finite differences) discretization schemes at the interface rendered radial and vertical velocity profiles that showed a mild non-smooth behavior at the bicone rim, that resulted in errors in the interfacial torque. Though small, such errors affected the values obtained for the dynamic moduli from the iterative process.

Solution method: We have fully reworked the vertical discretization scheme at the interface, by using a line of fictitious nodes. Now the scheme is SOCFD in both the radial and the vertical coordinates everywhere.

Summary of revisions: BiconeDrag [1] is a computer program for Flow field-based data analysis of the experimental data obtained with interfacial shear rheometers with bicone probes [2–4]. In the new version, we have reformulated the combination of the discretized Boussinesq-Scriven boundary condition and the Navier-Stokes equations for nodes at the interface. The original program was developed by adapting to the bicone geometry the ideas already used in references [5–8] for the magnetic needle ISR, [9] for the double wall-ring interfacial rheometer, and [2] for the oscillating rotational bicone rheometer. A complete account of the so-called Flow field-based data analysis techniques has been given elsewhere [4].

The new discretization is now SOCFD in both spatial directions at the subphase and, more importantly, at the interface. In order to achieve the formulation of the SOCFD at the interface, a line of phantom nodes has been defined above the interface. However, an adequate combination of the discretized Boussinesq-Scriven boundary condition and the Navier-Stokes equations yields expressions for the matrix elements corresponding to the interface nodes in which the values of the velocity field at the phantom nodes are not present and, consequently, the values of the velocity field at the phantom nodes do not appear explicitly in the numerical formulation of the problem. Using the notation in Ref. [3] the new expression reads:(1) $i R e g_{j, 1}^{⁎} = N^{2} (1 + 2 \frac{M}{\bar{h}} B o^{⁎}) (g_{j + 1, 1}^{⁎} + g_{j - 1, 1}^{⁎} - 2 g_{j, 1}^{⁎} + \frac{g_{j + 1, 1}^{⁎} - g_{j - 1, 1}^{⁎}}{2 (j - 1)} - \frac{g_{j, 1}^{⁎}}{{(j - 1)}^{2}}) + 2 {(\frac{M}{\bar{h}})}^{2} (g_{j, 2}^{⁎} - g_{j, 1}^{⁎}), \forall j \in Z / ⌊ N \bar{R_{b}} ⌋ + 2 \leq j \leq N .$ Moreover, the limit of negligible interfacial viscoelasticity, i.e., null complex Boussinesq number ( $B o^{⁎} = 0$ ), is well behaved because in such a case the above expression reduces to the one corresponding to a free interface condition ( $\frac{\partial g^{⁎}}{\partial \bar{z}} = 0$ , at the interface).

The differences in the velocity field with respect to the previous version are located at the interface and nearby. The radial velocity gradient for non-linear interfacial flow-fields is better defined. The new implementation of the vertical discretization and interfacial boundary condition yields more regular velocity fields that allow for a reduction of the computational errors. Such an error reduction may be relevant for interfacial viscosities $η_{s} \leq 10^{- 3}$ Ns/m, where the errors show a strong increase upon decreasing $η_{s}$ . The new version also shows a remarkably increased consistency when calculating the complex interfacial torque, $T_{s u r f}^{⁎}$ , where for $η_{s} > 10^{- 4} Ns/m$ , even low resolution meshes ( $200 \times 100$ nodes) yield errors below 5%.

References

[1]: P. Sánchez-Puga, J. Tajuelo, J.M. Pastor, M.A. Rubio, Comput. Phys. Commun. 239 (2019) 184–196.
[2]: J. Tajuelo, M.A. Rubio, J.M. Pastor, J. Rheol. 62 (1) (2018) 295–311.
[3]: P. Sánchez-Puga, J. Tajuelo, J. Pastor, M. Rubio, Colloids Interfaces 2 (4) (2018) 69.
[4]: P. Sánchez-Puga, J. Tajuelo, J.M. Pastor, M.A. Rubio, Adv. Colloid Interface Sci. 288 (2021) 102332.
[5]: S. Reynaert, C.F. Brooks, P. Moldenaers, J. Vermant, G.G. Fuller, J. Rheol. 52 (1) (2008) 261–285.
[6]: T. Verwijlen, P. Moldenaers, H.A. Stone, J. Vermant, Langmuir 27 (15) (2011) 9345–9358.
[7]: J. Tajuelo, J.M. Pastor, F. Martínez-Pedrero, M. Vázquez, F. Ortega, R.G. Rubio, M.A. Rubio, Langmuir 31 (4) (2015) 1410–1420.
[8]: J. Tajuelo, J.M. Pastor, M.A. Rubio, J. Rheol. 60 (6) (2016) 1095–1113.
[9]: S. Vandebril, A. Franck, G.G. Fuller, P. Moldenaers, J. Vermant, Rheol. Acta 49 (2) (2010) 131–144.

中文翻译：

BiconeDrag 更新 – 振荡圆锥摆式界面剪切流变仪的数据处理应用程序

我们提出了 BiconeDrag 的新改进版本，该代码允许人们从带有双锥探针的旋转界面剪切流变仪中进行的动态实验中获得界面动态模量的值。程序的一般结构保持不变：从扭矩/角位移幅度比的值开始，并通过使用流体动力场方程，可以将子相（体）阻力的贡献与真实的分离界面阻力。在这个新版本中，我们改进了接口处边界条件的实现，现在它是在一系列虚拟节点的帮助下构建为二阶中心有限差分（以下简称 SOCFD）。界面边界条件的新数值实现产生更平滑双锥边缘处的速度分布，从而更准确地分离界面和亚相阻力，从而更精确地计算界面动态模量。

新版本程序汇总

程序名称： BiconeDrag

CPC 库程序文件链接： https : //doi.org/10.17632/c245bmgf5n.2

代码海洋胶囊： https : //doi.org/10.24433/CO.5536863.v1

许可条款： GPLv3

编程语言： MATLAB（兼容GNU Octave）

上一版本的期刊参考： Compute。物理。社区。239 (2019) 184–196

新版本是否取代以前的版本？：是

新版本原因：重构界面边界条件的实现。

问题性质：界面处不同的水平（二阶中心有限差分）和垂直（一阶有限差分）离散化方案呈现径向和垂直速度剖面，在双锥边缘显示出轻微的非平滑行为，导致误差在界面扭矩方面。虽然很小，但这种误差会影响从迭代过程中获得的动态模量的值。

解决方法：我们通过使用一排虚拟节点完全重新设计了界面处的垂直离散化方案。现在的方案是在径向和垂直坐标上到处都是SOCFD。

修订摘要： BiconeDrag [1] 是一个计算机程序，用于对使用带有双锥探针的界面剪切流变仪获得的实验数据进行基于流场的数据分析 [2-4]。在新版本中，我们重新制定了离散化的 Boussinesq-Scriven 边界条件和接口处节点的 Navier-Stokes 方程的组合。最初的程序是通过适应双锥几何形状而开发的，参考文献 [5-8] 中已经使用的思想用于磁针 ISR，[9] 用于双壁环界面流变仪，[2] 用于振荡旋转双锥流变仪. 所谓的基于流场的数据分析技术的完整说明已在别处 [4] 中给出。

新的离散化现在是子相的两个空间方向上的 SOCFD，更重要的是，在界面处。为了实现接口处SOCFD的制定，在接口上方定义了一行虚拟节点。然而，离散化 Boussinesq-Scriven 边界条件和 Navier-Stokes 方程的充分组合产生了对应于接口节点的矩阵元素的表达式，其中不存在虚节点处的速度场值，因此，幻像节点处的速度场值没有明确出现在问题的数值公式中。使用参考文献中的符号。[3] 新表达式如下：(1) $一世电阻电子 G_{j, 1}^{⁎} = N^{2} (1 + 2 \frac{米}{\bar{H}} 乙 ○^{⁎}) (G_{j + 1, 1}^{⁎} + G_{j - 1, 1}^{⁎} - 2 G_{j, 1}^{⁎} + \frac{G_{j + 1, 1}^{⁎} - G_{j - 1, 1}^{⁎}}{2 (j - 1)} - \frac{G_{j, 1}^{⁎}}{{(j - 1)}^{2}}) + 2 {(\frac{米}{\bar{H}})}^{2} (G_{j, 2}^{⁎} - G_{j, 1}^{⁎}), \forall j \in Z / ⌊ N \bar{{电阻}_{乙}} ⌋ + 2 \leq j \leq N .$ 此外，可忽略界面粘弹性的极限，即零复数 Boussinesq 数（ $乙 ○^{⁎} = 0$ )，表现良好，因为在这种情况下，上述表达式简化为对应于自由界面条件的表达式 ( $\frac{\partial G^{⁎}}{\partial \bar{z}} = 0$ ，在界面）。

速度场与之前版本的差异位于界面附近。更好地定义了非线性界面流场的径向速度梯度。垂直离散化和界面边界条件的新实现产生了更规则的速度场，从而减少了计算误差。这种误差减少可能与界面粘度有关 $η_{秒} \leq 10^{- 3}$ Ns/m，其中误差在减小时显着增加 $η_{秒}$ . 新版本在计算复杂的界面扭矩时也显示出显着增加的一致性， $吨_{秒你 r F}^{⁎}$ , 哪里 $η_{秒} > 10^{- 4} 纳秒/米$ , 即使是低分辨率的网格 ( $200 \times 100$ 节点）产生低于 5% 的误差。

参考

[1]: P. Sánchez-Puga, J. Tajuelo, JM Pastor, MA Rubio, Comput. 物理。社区。239 (2019) 184–196。
[2]: J. Tajuelo、MA Rubio、JM Pastor、J. Rheol。62 (1) (2018) 295-311。
[3]: P. Sánchez-Puga、J. Tajuelo、J. Pastor、M. Rubio，胶体界面 2 (4) (2018) 69。
[4]: P. Sánchez-Puga, J. Tajuelo, JM Pastor, MA Rubio, Adv. 胶体界面科学。288 (2021) 102332。
[5]: S. Reynaert、CF 布鲁克斯、P. Moldenaers、J. Vermant、GG Fuller、J. Rheol。52 (1) (2008) 261–285。
[6]: T. Verwijlen、P. Moldenaers、HA Stone、J. Vermant、Langmuir 27 (15) (2011) 9345–9358。
[7]: J. Tajuelo, JM Pastor, F. Martínez-Pedrero, M. Vázquez, F. Ortega, RG Rubio, MA Rubio, Langmuir 31 (4) (2015) 1410–1420。
[8]: J. Tajuelo，JM Pastor，MA Rubio，J. Rheol。60 (6) (2016) 1095–1113。
[9]: S. Vandebril、A. Franck、GG Fuller、P. Moldenaers、J. Vermant、Rheol。学报 49 (2) (2010) 131–144。

更新日期：2021-06-22

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