33 research outputs found
On the bending algorithms for soft objects in flows
International audienceOne of the most challenging aspects in the accurate simulation of three-dimensional soft objects such as vesicles or biological cells is the computation of membrane bending forces. The origin of this difficulty stems from the need to numerically evaluate a fourth order derivative on the discretized surface geometry. Here we investigate six different algorithms to compute membrane bending forces, including regularly used methods as well as novel ones. All are based on the same physical model (due to Canham and Helfrich) and start from a surface discretization with flat triangles. At the same time, they differ substantially in their numerical approach. We start by comparing the numerically obtained mean curvature, the Laplace-Beltrami operator of the mean curvature and finally the surface force density to analytical results for the discocyte resting shape of a red blood cell. We find that none of the considered algorithms converges to zero error at all nodes and that for some algorithms the error even diverges. There is furthermore a pronounced influence of the mesh structure: Discretizations with more irregular triangles and node connectivity present serious difficulties for most investigated methods. To assess the behavior of the algorithms in a realistic physical application, we investigate the deformation of an initially spherical capsule in a linear shear flow at small Reynolds numbers. To exclude any influence of the flow solver, two conceptually very different solvers are employed: the Lattice-Boltzmann and the Boundary Integral Method. Despite the largely different quality of the bending algorithms when applied to the static red blood cell, we find that in the actual flow situation most algorithms give consistent results for both hydrodynamic solvers. Even so, a short review of earlier works reveals a wide scattering of reported results for, e.g., the Taylor deformation parameter. Besides the presented application to biofluidic systems, the investigated algorithms are also of high relevance to the computer graphics and numerical mathematics communities
Creeping motion of a solid particle inside a spherical elastic cavity
On the basis of the linear hydrodynamic equations, we present an analytical
theory for the low-Reynolds-number motion of a solid particle moving inside a
larger spherical elastic cavity which can be seen as a model system for a fluid
vesicle. In the particular situation where the particle is concentric with the
cavity, we use the stream function technique to find exact analytical solutions
of the fluid motion equations on both sides of the elastic cavity. In this
particular situation, we find that the solution of the hydrodynamic equations
is solely determined by membrane shear properties and that bending does not
play a role. For an arbitrary position of the solid particle within the
spherical cavity, we employ the image solution technique to compute the
axisymmetric flow field induced by a point force (Stokeslet). We then obtain
analytical expressions of the leading order mobility function describing the
fluid-mediated hydrodynamic interactions between the particle and confining
elastic cavity. In the quasi-steady limit of vanishing frequency, we find that
the particle self-mobility function is higher than that predicted inside a
rigid no-slip cavity. Considering the cavity motion, we find that the
pair-mobility function is determined only by membrane shear properties. Our
analytical predictions are supplemented and validated by fully-resolved
boundary integral simulations where a very good agreement is obtained over the
whole range of applied forcing frequencies.Comment: 15 pages, 5 figures, 90 references. To appear in Eur. Phys. J.
Particle mobility between two planar elastic membranes: Brownian motion and membrane deformation
We study the motion of a solid particle immersed in a Newtonian fluid and
confined between two parallel elastic membranes possessing shear and bending
rigidity. The hydrodynamic mobility depends on the frequency of the particle
motion due to the elastic energy stored in the membrane. Unlike the
single-membrane case, a coupling between shearing and bending exists. The
commonly used approximation of superposing two single-membrane contributions is
found to give reasonable results only for motions in the parallel, but not in
the perpendicular direction. We also compute analytically the membrane
deformation resulting from the motion of the particle, showing that the
presence of the second membrane reduces deformation. Using the
fluctuation-dissipation theorem we compute the Brownian motion of the particle,
finding a long-lasting subdiffusive regime at intermediate time scales. We
finally assess the accuracy of the employed point-particle approximation via
boundary-integral simulations for a truly extended particle. They are found to
be in excellent agreement with the analytical predictions.Comment: 14 pages, 8 figures and 96 references. Revised version resubmitted to
Phys. Fluid
Slow rotation of a spherical particle inside an elastic tube
In this paper, we present an analytical calculation of the rotational
mobility functions of a particle rotating on the centerline of an elastic
cylindrical tube whose membrane exhibits resistance towards shearing and
bending. We find that the correction to the particle rotational mobility about
the cylinder axis depends solely on membrane shearing properties while both
shearing and bending manifest themselves for the rotational mobility about an
axis perpendicular to the cylinder axis. In the quasi-steady limit of vanishing
frequency, the particle rotational mobility nearby a no-slip rigid cylinder is
recovered only if the membrane possesses a non-vanishing resistance towards
shearing. We further show that for the asymmetric rotation along the cylinder
radial axis, a coupling between shearing and bending exists. Our analytical
predictions are compared and validated with corresponding boundary integral
simulations where a very good agreement is obtained.Comment: 23 pages, 7 figures and 107 references. Revised manuscript
resubmitted to Acta Mec
Brownian motion near an elastic cell membrane: A theoretical study
Elastic confinements are an important component of many biological systems
and dictate the transport properties of suspended particles under flow. In this
chapter, we review the Brownian motion of a particle moving in the vicinity of
a living cell whose membrane is endowed with a resistance towards shear and
bending. The analytical calculations proceed through the computation of the
frequency-dependent mobility functions and the application of the
fluctuation-dissipation theorem. Elastic interfaces endow the system with
memory effects that lead to a long-lived anomalous subdiffusive regime of
nearby particles. In the steady limit, the diffusional behavior approaches that
near a no-slip hard wall. The analytical predictions are validated and
supplemented with boundary-integral simulations.Comment: 16 pages, 7 figures and 161 references. Contributed chapter to the
flowing matter boo
Hydrodynamic interaction between particles near elastic interfaces
We present an analytical calculation of the hydrodynamic interaction between
two spherical particles near an elastic interface such as a cell membrane. The
theory predicts the frequency dependent self- and pair-mobilities accounting
for the finite particle size up to the 5th order in the ratio between particle
diameter and wall distance as well as between diameter and interparticle
distance. We find that particle motion towards a membrane with pure bending
resistance always leads to mutual repulsion similar as in the well-known case
of a hard-wall. In the vicinity of a membrane with shearing resistance,
however, we observe an attractive interaction in a certain parameter range
which is in contrast to the behavior near a hard wall. This attraction might
facilitate surface chemical reactions. Furthermore, we show that there exists a
frequency range in which the pair-mobility for perpendicular motion exceeds its
bulk value, leading to short-lived superdiffusive behavior. Using the
analytical particle mobilities we compute collective and relative diffusion
coefficients. The appropriateness of the approximations in our analytical
results is demonstrated by corresponding boundary integral simulations which
are in excellent agreement with the theoretical predictions.Comment: 16 pages, 7 figures and 109 references. Manuscript accepted for
publication in J. Chem. Phy
Hydrodynamic mobility of a solid particle nearby a spherical elastic membrane. II. Asymmetric motion
In this paper, we derive analytical expressions for the leading-order
hydrodynamic mobility of a small solid particle undergoing motion tangential to
a nearby large spherical capsule whose membrane possesses resistance towards
shearing and bending. Together with the results obtained in the first part
(Daddi-Moussa-Ider and Gekle, Phys. Rev. E {\bfseries 95}, 013108 (2017)) where
the axisymmetric motion perpendicular to the capsule membrane is considered,
the solution of the general mobility problem is thus determined. We find that
shearing resistance induces a low-frequency peak in the particle self-mobility,
resulting from the membrane normal displacement in the same way, although less
pronounced, to what has been observed for the axisymmetric motion. In the zero
frequency limit, the self-mobility correction near a hard sphere is recovered
only if the membrane has a non-vanishing resistance towards shearing. We
further compute the particle in-plane mean-square displacement of a nearby
diffusing particle, finding that the membrane induces a long-lasting
subdiffusive regime. Considering capsule motion, we find that the correction to
the pair-mobility function is solely determined by membrane shearing
properties. Our analytical calculations are compared and validated with fully
resolved boundary integral simulations where a very good agreement is obtained.Comment: 17 pages, 9 figures and 64 references. Manuscript accepted for
publication in Phys. Rev.
Hydrodynamic coupling and rotational mobilities near planar elastic membranes
We study theoretically and numerically the coupling and rotational
hydrodynamic interactions between spherical particles near a planar elastic
membrane that exhibits resistance towards shear and bending. Using a
combination of the multipole expansion and Faxen's theorems, we express the
frequency-dependent hydrodynamic mobility functions as a power series of the
ratio of the particle radius to the distance from the membrane for the self
mobilities, and as a power series of the ratio of the radius to the
interparticle distance for the pair mobilities. In the quasi-steady limit of
zero frequency, we find that the shear- and bending-related contributions to
the particle mobilities may have additive or suppressive effects depending on
the membrane properties in addition to the geometric configuration of the
interacting particles relative to the confining membrane. To elucidate the
effect and role of the change of sign observed in the particle self and pair
mobilities, we consider an example involving a torque-free doublet of
counterrotating particles near an elastic membrane. We find that the induced
rotation rate of the doublet around its center of mass may differ in magnitude
and direction depending on the membrane shear and bending properties. Near a
membrane of only energetic resistance toward shear deformation, such as that of
a certain type of elastic capsules, the doublet undergoes rotation of the same
sense as observed near a no-slip wall. Near a membrane of only energetic
resistance toward bending, such as that of a fluid vesicle, we find a reversed
sense of rotation. Our analytical predictions are supplemented and compared
with fully resolved boundary integral simulations where a very good agreement
is obtained over the whole range of applied frequencies.Comment: 14 pages, 7 figures. Revised manuscript resubmitted to J. Chem. Phy
Numerical-experimental observation of shape bistability of red blood cells flowing in a microchannel
Red blood cells flowing through capillaries assume a wide variety of
different shapes owing to their high deformability. Predicting the realized
shapes is a complex field as they are determined by the intricate interplay
between the flow conditions and the membrane mechanics. In this work we
construct the shape phase diagram of a single red blood cell with a
physiological viscosity ratio flowing in a microchannel. We use both
experimental in-vitro measurements as well as 3D numerical simulations to
complement the respective other one. Numerically, we have easy control over the
initial starting configuration and natural access to the full 3D shape. With
this information we obtain the phase diagram as a function of initial position,
starting shape and cell velocity. Experimentally, we measure the occurrence
frequency of the different shapes as a function of the cell velocity to
construct the experimental diagram which is in good agreement with the
numerical observations. Two different major shapes are found, namely croissants
and slippers. Notably, both shapes show coexistence at low (<1 mm/s) and high
velocities (>3 mm/s) while in-between only croissants are stable. This
pronounced bistability indicates that RBC shapes are not only determined by
system parameters such as flow velocity or channel size, but also strongly
depend on the initial conditions.Comment: 13 pages, 9 figures (main text). 13 pages, 31 figures (SI
Bending models of lipid bilayer membranes: spontaneous curvature and area-difference elasticity
We preset a computational study of bending models for the curvature
elasticity of lipid bilayer membranes that are relevant for simulations of
vesicles and red blood cells. We compute bending energy and forces on
triangulated meshes and evaluate and extend four well established schemes for
their approximation: Kantor and Nelson 1987, Phys. Rev. A 36, 4020, J\"ulicher
1996, J. Phys. II France 6, 1797, Gompper and Kroll 1996, J. Phys. I France 6,
1305, and Meyer et. al. 2003 in Visualization and Mathematics III, Springer,
p35, termed A, B, C, D. We present a comparative study of these four schemes on
the minimal bending model and propose extensions for schemes B, C and D. These
extensions incorporate the reference state and non-local energy to account for
the spontaneous curvature, bilayer coupling, and area-difference elasticity
models. Our results indicate that the proposed extensions enhance the models to
account for shape transformation including budding/vesiculation as well as for
non-axisymmetric shapes. We find that the extended scheme B is superior to the
rest in terms of accuracy, and robustness as well as simplicity of
implementation. We demonstrate the capabilities of this scheme on several
benchmark problems including the budding-vesiculating process and the
reproduction of the phase diagram of vesicles