Analytical investigation of vesicle dynamics via the modified Riemann-Liouville fractional derivative: Mittag-Leffler function solution and comparative analysis with Caputo's derivative.

The solution of fractional differential equations is a significant focus of current research, given their prevalence in various fields of application. This paper introduces an innovative exploration of vesicle dynamics using Jumarie's modified Riemann-Liouville fractional derivative within a five-dimensional fractional rigid sphere model. The study reveals an exact solution through the Mittag-Leffler function, providing a deep understanding of intricate vesicle dynamics, including alternative motions, such as tank-treading with over-damped and under-damped vesicle oscillations, respectively, TT-OD and TT-UD.

Analytical solutions and classification of vesicle motion and deformation in shear flow: Uncovering new tank-treading modes.

Using a small deformation approach, a fractional ordinary differential system is proposed to investigate the motion and deformation of a vesicle in shear flow. Closed analytical expressions of the orientation angle and the ellipticity of the vesicle contour (shape deformation) are provided. Three different motions are identified, the classical tank-treading state, and two new types of motions, namely, the over-damped tank-treading mode, in which the vesicle's orientation angle ψ and its shape deformation R tend more slowly toward equilibrium, and the under-damped tank-treading mode, in which ψ oscillates all the time along the flow direction with decreasing amplitude, while R starts making a breathing motion and then tends to an attractive amplitude.