Reduced-order model of the deformation of elastic microcapsules in flow

Reduced-order model of the deformation of elastic microcapsules in flow Carlos Quesada 1a , Pierre Villon 2b and Anne-Virginie Salsac 1 1 Biomechanics & Bioengineering Laboratory (UMR CNRS 7338), Université de Technologie de Compiègne – CNRS, Sorbonne Universités, Compiègne, France 2 Roberval Laboratory (UMR CNRS 7339), Université de Technologie de Compiègne – CNRS, Sorbonne Universités, Compiègne, France Abstract. The objective of this work is the development of reduced-order models for liquid-filled elastic microcapsules in flow. These models must allow not only to predict the deformation (i.e. the shapes) of the capsules when they flow in straight or bifurcated square-section channels, but also to characterize their mechanical properties. The microcapsules considered in this work are spherical particles ranging from 1 µm to 1000 µm, composed of a thin deformable elastic membrane of solid biocompatible material, which are filled with liquid. Although this kind of capsules can be commonly found in nature (eggs, cells), they have aroused considerable interest since they can be artificially produced for a large number of industrial applications. Some examples include the local delivery of therapeutic drugs in the pharmacological industry, or the masking and protection of certain substances in the chemical and cosmetic industries. The characterization of the mechanical properties of the membrane of the microcapsules in flow (which mainly consists in finding the surface elastic shear modulus Gs, and the area-dilation modulus Ks) is a challenging task because of the reduced size of the particles. Through inverse analysis it is possible to infer the membrane mechanical properties by comparing experimental results of capsules flowing in microfluidic channels with the deformations predicted by numerical models (Hu et al. 2013). Existing models accurately solve the strongly coupled fluid-structure interactions (Barthès-Biesel 2016), but require long computational times to determine the capsule deformed shapes (Figure 1a) for different values of the input parameters: the capillary number Ca, ratio of the viscous to elastic forces, and the capsule-to-tube size ratio a/l. We propose to use model order reduction techniques to predict the capsule deformation and drastically reduce the computation time. A Proper Orthogonal Decomposition (POD) is applied on a set of solutions provided by the Hu et al. (2013) numerical model. They correspond to the three-dimensional shapes taken at steady-state by microcapsules flowing in square-section microchannels, when varying the parameters of the problem: the capillary number Ca and the size ratio a/l. POD allows to reduce the dimensionality of the problem and, therefore, its computational complexity. The resulting set of reduced observations lies on a manifold of low intrinsic dimension, and different algorithms have been used to predict and reconstruct the shapes of the microcapsules for any value of the input parameters. A good agreement was obtained on the capsule shape reconstructions (Figure 1b) with impressive gains in computation time (at least 3000 times faster). Keywords: microcapsules in flow, reduced-order models, Proper Orthogonal Decomposition References Hu, X. Q., Sévénié, B., Salsac, A. V., Leclerc, E., Barthès-Biesel, D. (2013), “Characterizing the membrane properties of capsules flowing in a square-section microfluidic channel: Effects of the membrane Corresponding author: Ph.D., E-mail: anne-virg[email protected] a Ph.D., E-mail: carlos.quesada-[email protected] b Professor, E-mail: [email protected] constitutive law.” Physical Review E, 87(6), 063008. Barthès-Biesel, D. (2016). “Motion and deformation of elastic capsules and vesicles in flow”. Annual Review of Fluid Mechanics, 48, 25-52. (a) (b) Fig. 1: (a) 3D shapes of elastic microcapsules flowing in a square-section microchannel (top: Ca = 0.06, a/l = 0.98; bottom: Ca = 0.05, a/l = 0.9). (b) Comparison of the capsule cross-sections predicted by the fully coupled numerical simulation (blue line) and by the reduced-order model (red line) within the midplane of the channel, for the microcapsules in (a).