Linear stability analysis

In order to perform a linear stability analysis we perturb the radius of the initially cylindrical interface of radius R0 in a periodic fashion

R(z, t) =R₀+δR(z, t) = R₀+R₀₀e^ωtcos(kz), (4.5)

4.1 Rayleigh-Plateau instability of anisotropic interfaces

with ₀ being a small amplitude,k = ^2π_λ the wave number, i.e., the magnitude of a wave vector pointing along the axisz of the interface, andωthe growth rate of the perturbation mode. The result of the linear stability analysis is a relation for the growth rate depending on the wave number, which allows to identify perturbations to which the interface is unstable (with positive growth rate) and the dominant mode that evolves fastest in time. The anisotropic interfacial tension, our key ingredient, modifies the Young-Laplace equation (4.1) in the way that both curvature components now are weighted differently by the two components of the interfacial tension in equation (4.4)

p= γ^φ Rφ

+ γ^z Rz

. (4.6)

This relation follows by contraction of the curvature tensor in equation (2.13) with c^φ_φ = _R¹

φ andc^z_z = _R¹_z and the in-plane components of the surface stress in equation (4.4).

This contraction equals the pressure difference according to the normal force balance equation (2.26) with the fluid force in equation (2.24) proportional to the pressure, cf.

equation (2.33). In the modified Young-Laplace equation (4.6) the destabilizing and the stabilizing mechanism of the Rayleigh-Plateau instability as introduced at the beginning of this chapter are now weighted differently.

We first consider the suspending fluid in the limit of a small Reynolds number as often applicable to vesicles and cells [68, 151]. Therefore, we can solve the linear Stokes equation (2.36) which we modify to account for the presence of the interface

− ∇p+η∇²v+δ(r−R₀)γ^φ₀

R0 1− γ^z

γ^φ(R0k)²

!

cos(kz)er =0, (4.7) where the interfacial force due to anisotropic interfacial tension enters via a ring force [53]. The system of equations is closed by the continuity equation (2.38). We solve this system of equations by introducing the Hankel transformationHν and the inverse Hankel transformation H⁻¹ν of order ν of a general function f, which is defined by [258]

F(s) = H^ν[f] =

Z∞ 0

f(r)rJν(sr) dr, f(r) = H⁻¹ν [F] =

Z∞ 0

F(s)sJν(sr) ds, (4.8) with the Bessel function Jν(x) of first kind and order ν. By introducing the Hankel transform of the velocity components and the pressure, we obtain the components of the modified Stokes equation (4.7) and the continuity equation (2.38) in the Hankel space.

The equations in Hankel space can be solved analytically and the obtained velocity and pressure is transformed back into real space.

At the interface the no-slip boundary condition to the velocity field applies, which relates the radial motion of the interface to the radial fluid velocity vr according to equation (2.39), which in linear order leads to

∂R

∂t =R₀₀ωe^ωtcos(kz) =vr(r =R), (4.9) where the derivative of the interface perturbation given by equation (4.5) has been inserted. Evaluating the transform of the radial velocity from Hankel space to real space

4 Rayleigh-Plateau instability for anisotropic tension and blood platelet formation

at the interface, eventually leads to an equation which can be solved for the growth rate.

This finally leads to the dispersion relation ω_Stokes(k) = γ^φ

R₀η 1−γ^z

γ^φ(R0k)²

!"

I₁(kR0)K1(kR0) +kR0

2

I1(kR0)K0(kR0)−I0(kR0)K1(kR0)

#

,

(4.10)

with Iν(x) the modified Bessel function of first kind and Kν(x) of second kind and of order ν. Equation (4.10) is the dispersion relation for the Rayleigh-Plateau instability in the limit of the Stokes equation with the same viscosity of the inner and outer fluid.

The dispersion relation gives the growth rate for each mode of wave numberk. In case of a negative growth rate the interface is stable with respect to the corresponding mode, in case of a positive growth rate the interface is unstable and the corresponding mode grows.

Because the prefactor and the factor containing the Bessel functions are positive for positive wave number k, the root of the dispersion relation is determined by

1− γ^z

γ^φ(R0k)² = 0, (4.11)

and therefore strongly influenced by the ratio ^γ_γ^zφ that accounts for the anisotropy of the interfacial tension. This is in stark contrast to the classical Rayleigh-Plateau instability, which is covered as well by setting ^γ_γ^zφ = 1, where the root of the dispersion relation only depends on the radius of the fluid tube. For the axial tension γ^z exceeding the azimuthal tensionγ^φthe range of unstable modes shrinks and vice versa. All in all, the axial tension dampens the dispersion relation and stabilizes the interface, while the azimuthal tension destabilizes the interface.

Utilizing the concept of the ring force and the Hankel transform, we further consider an interface enclosing an ideal fluid and surrounded by an ideal fluid with same density.

Here, the Euler equation (2.37) in linear order of the velocity has to be solved, which can be done in a similar way as solving the Stokes equation. In this limit we end up with the dispersion relation

ω_Euler² (k) = γ^φ

ρR³₀(kR0)² 1− γ^z

γ^φ(kR0)²

!

I1(kR0)K1(kR0). (4.12) Furthermore, we derived the dispersion relation for a liquid jet of an ideal fluid immersed in air [pub3], which is given by

ω_jet² (k) =γ^φkR₀ ρR³₀

1− γ^z

γ^φ(kR0)²I₁(kR0)

I0(kR0). (4.13)

Both equations in the ideal fluid limit differ from the one in the Stokes regime by a different prefactor and distinctive combinations of the Bessel functions. However, the tension anisotropy enters in the same way. As a consequence, the discussion on unstable modes above holds for the limit of the Euler equation, as well.

4.1 Rayleigh-Plateau instability of anisotropic interfaces

ideal fuid jet w/o ambient fuid ideal fuid

Fig. 4.1:Tension anisotropy strongly alters the dominant wavelength of the Rayleigh-Plateau instability. We compare the analytically obtained wavelength to simulation results both in the limit of a) the Stokes equation together with simulations using boundary integral method and in the limit of b) an ideal fluid governed by the Euler equation together with lattice-Boltzmann simulations. Analytical results for the ideal fluid limit with (red) and without (blue) an outer fluid differ only slightly. Simulation snapshots as insets show the interface shape for various tension anisotropies. c) Lattice-Boltzmann/immersed boundary (LBM/IBM) simulations covering fluid inertia show a transition in the dominant wavelength for typical vesicle parameters.