We assume that, at early time stages when the membrane deformations
are small, the membrane height is given by h(x, t) = h0 + h1(x, t) with h1 ≪ h0 where h0 is the height of undeformed membrane. In the
limit ∂h/∂x ≪
1, to a linear order in h1, the pressure
difference p across the lipid membrane is given by where k ∼
10–19 J is the bending modulus of the membrane,
and AH ∼ 10–21 J is the Hamaker constant that characterizes the vdW interactions
between membrane and the aluminum substrate. The local mass conservation
of the incompressible lipid membrane and the incompressible fluid
underneath require that the rate of change of height h be governed by spatial variation of the liquid flux hU in the horizontal direction, where U is horizontal
flow speed. To a linear order in h1, the
mass conservation of a thin liquid film is given by where μ is the dynamic
viscosity of the solvent.Equations 4 along with 3 constitute our
elastohydrodynamic theory. Since the coefficients ineq 4 are independent of x and t, we explored the solutions of the form h1 = |h1|e(iqx+st), where |h1| is the fluctuation amplitude at t =
0 resulting from the thermal energy, q is the wavenumber
of a given mode, and s is the corresponding inverse
deformation time scale. In dimensionless units, substituting this
relation intoeq 4 , we
arrive at the following dispersion relation with critical wavenumber . For numerical implementation we used the
open source finite element analysis (FEA) library FEniCS on Python
3.6.83 (link) At t = 0, the membrane
height profile is given by h(x)
= 1 + 0.001 cos(qx), and the time evolution is subject
to the following boundary conditions: at x = 0 we
set U = 0, , and ; at x = 1 we set h = 1, p = 0, and .
are small, the membrane height is given by h(x, t) = h0 + h1(x, t) with h1 ≪ h0 where h0 is the height of undeformed membrane. In the
limit ∂h/∂x ≪
1, to a linear order in h1, the pressure
difference p across the lipid membrane is given by where k ∼
10–19 J is the bending modulus of the membrane,
and AH ∼ 10–21 J is the Hamaker constant that characterizes the vdW interactions
between membrane and the aluminum substrate. The local mass conservation
of the incompressible lipid membrane and the incompressible fluid
underneath require that the rate of change of height h be governed by spatial variation of the liquid flux hU in the horizontal direction, where U is horizontal
flow speed. To a linear order in h1, the
mass conservation of a thin liquid film is given by where μ is the dynamic
viscosity of the solvent.
elastohydrodynamic theory. Since the coefficients in
0 resulting from the thermal energy, q is the wavenumber
of a given mode, and s is the corresponding inverse
deformation time scale. In dimensionless units, substituting this
relation into
arrive at the following dispersion relation with critical wavenumber . For numerical implementation we used the
open source finite element analysis (FEA) library FEniCS on Python
3.6.83 (link) At t = 0, the membrane
height profile is given by h(x)
= 1 + 0.001 cos(qx), and the time evolution is subject
to the following boundary conditions: at x = 0 we
set U = 0, , and ; at x = 1 we set h = 1, p = 0, and .
