PHYSICAL REVIEW E 86, 051915 (2012)
Dynamic modes of quasispherical vesicles: Exact analytical solutions
M. Guedda,
1,*
M. Abaidi,
1
M. Benlahsen,
2
and C. Misbah
3
1
Universit´ e de Picardie Jules Verne, LAMFA CNRS UMR 7352, Amiens F-80039, France
2
Universit´ e de Picardie Jules Verne, LPMC, Amiens F-80039, France
3
Universit´ e Grenoble I, CNRS, Laboratoire Interdisciplinaire de Physique, UMR 5588, Grenoble F-38041, France
(Received 2 August 2012; revised manuscript received 18 October 2012; published 26 November 2012)
In this paper we introduce a simple mathematical analysis to reexamine vesicle dynamics in the quasispherical
limit (small deformation) under a shear flow. In this context, a recent paper [Misbah, Phys. Rev. Lett. 96, 028104
(2006)] revealed a dynamic referred to as the vacillating-breathing (VB) mode where the vesicle main axis
oscillates about the flow direction and the shape undergoes a breathinglike motion, as well as the tank-treading
and tumbling (TB) regimes. Our goal here is to identify these three modes by obtaining explicit analytical
expressions of the vesicle inclination angle and the shape deformation. In particular, the VB regime is put
in evidence and the transition dynamics is discussed. Not surprisingly, our finding confirms the Keller-Skalak
solutions (for rigid particles) and shows that the VB and TB modes coexist, and whether one prevails over the
other depends on the initial conditions. An interesting additional element in the discussion is the prediction of
the TB and VB modes as functions of a control parameter , which can be identified as a TB-VB parameter.
DOI: 10.1103/PhysRevE.86.051915 PACS number(s): 87.16.D−, 87.19.U−, 47.15.G−, 47.60.Dx
I. INTRODUCTION
The aim of this investigation is to solve exactly a highly
nonlinear system of coupled ordinary differential equations
that is derived to model the dynamics of a vesicle subject
to unbounded steady shear flow. Vesicles (also known as
fluid membranes) and red blood cells (RBCs) have been, and
remain, the subject of extensive studies (see [1–39] and the
references therein). Nowadays there is an increasing interest
in this research activity in different disciplines ranging from
biology to applied mathematics. It is found that vesicles and
RBCs display at least two main types of dynamics: (i) the
tank-treading (TT) mode, where the vesicle deforms into a
prolate ellipsoid inclined at a stationary angle 0 < ψ < π/4
with the flow direction, while its membrane undergoes a
tank-treading motion, and (ii) the tumbling (TB) mode, in
which the membrane flips like a rigid body, provided its initial
shape is not spherical. These two types of motion are predicted
by the Keller-Skalak (KS) theory [1] which assumed a fixed
ellipsoidal vesicle shape.
To focus the discussion, the dynamics of vesicles and RBCs
under a shear flow depends on three dimensional parameters
(see, for example, [24,26]): (i) The first is the excess area
relative to a sphere = (A − 4πr
2
0
)/r
2
0
, where A is the vesicle
area and r
0
is the effective vesicle radius or the radius of
a sphere having the same volume V as the vesicle ([r
0
=
(3V/4π )
1/3
]. The excess area is non-negative and vanishes
for a sphere. (ii) The second is the ratio λ = η
int
/η
ext
,η
int
and η
ext
being the viscosities of the internal and the external
fluids, respectively. (iii) The third is the capillary number
C
a
= η
ext
˙ γr
3
0
/κ, where ˙ γ and κ are the shear rate and the
membrane bending rigidity, respectively.
In addition to the TT and TB regimes, an intermediate
regime, which has attracted the attention of many researchers,
has been presented by one the authors [12]. In this regime the
main axis of the vesicle oscillates about the flow direction,
*
guedda@u-picardie.fr
whereas its shape makes a breathing motion. This regime,
which is called the vacillating-breathing (VB) mode (later
also described as trembling or swinging), has undergone
considerable numerical and experimental investigation, which
constitutes the initial motivation of the present work.
The theory for the VB dynamic mode is based on the
small excess area approximation (i.e., almost spherical vesicles
or the quasispherical regime) and on spherical harmonic
expansions of the shape deviation. Neglecting membrane
thermal undulations, at leading order (ε =
√
is the small
expansion parameter) Misbah [12] derived the following
coupled nonlinear ordinary differential equations:
d R
dt
= h
1 − 4
R
2
sin (2ψ ),
dψ
dt
=−
1
2
+
h
2R
cos (2ψ ),
(1)
where the unknowns ψ and R represent, respectively, the
vesicle inclination angle and its shape deformation. Here,
lengths are reduced by the vesicle radius r
0
and time by
˙ γ
−1
. System (1) gives the temporal evolutions of ψ and R as
functions of and the parameter h = 60
2π/15/(32 + 23λ)
(or the viscosity ratio λ). This is a generalization of the KS
theory which assumed a fixed ellipsoid shape, R ≡
√
/2.
It is noteworthy that system (1) is free of C
a
. As mentioned
in [24], the approach of [12] has truncated the expansion of
the evolution equations about a spherical shape to leading
order (see also [13]). As a consequence, C
a
is scaled out
from the evolution equations, and only λ and remain. The
insensitivity to C
a
of the vesicle tilt angle in a shear flow was
also reported numerically even for a large enough deformation
[2,5,6]. In [15] it is indicated that the theory of Misbah
corresponds formally to C
a
→∞. Including higher-order
terms leads to the appearance of C
a
in the equations [15,23,24].
In [12] the author showed that system (1) has a critical
viscosity ratio
λ
c
=−32/23 + (120/23)
2π/15, (2)
051915-1 1539-3755/2012/86(5)/051915(11) ©2012 American Physical Society