Quick check: TG vortices

TG vortices go turbulent

Onset of 2D turbulence

Onset of 2D turbulence

Membranes: 2D fluids

Flow in membranes

Overdamped regime

Stokes flow

Flow driven only by forces. Stokes equation:

$\displaystyle {\eta \nabla^2 \mathbf{u} -\nabla p = - \mathbf{f} }$

$\displaystyle { \nabla \cdot \mathbf{u} = 0 }$

Solution: $\displaystyle {\mathbf{u}_{\mathbf{q},a} = \frac 1{\eta q^2} \sum_b \left[ \delta_{ab} - \frac{ \mathbf{q}_a \mathbf{q}_b}{q^2} \right] \mathbf{f}_{\mathbf{q},a} =: \sum_b T_{ab} \mathbf{f}_{\mathbf{q},b} }$ $\displaystyle {T_{ab} := \frac 1{\eta q^2 } \left[ \delta_{ab} - \frac{ \mathbf{q}_a \mathbf{q}_b}{q^2} \right] }$

Phase segregation in membranes

Modeling phase segregation: CH

$\displaystyle {\frac{d \phi}{d t} = M \nabla^2 \mu }$

$\displaystyle {\mu = \frac{\delta}{\delta \phi} \int \left[ \color{blue}{ -\frac{a}{2} \phi^2 + \frac{b}{4} \phi^4} + \color{red}{ \frac{c}{2} |\nabla \phi|^2 } \right] dV }$

Landau double-well

Equilibrium profile: $\mu=0$

Closure: connection with 2D Stokes flow

Force is due to concentration gradients:

$\displaystyle { \mathbf{f} = -\phi \nabla \mu }$

Also, the surrounding liquid modifies the Oseen tensor: $\displaystyle {T_{ab} := \frac 1{\eta_\mathrm{m} (q^2 + \color{red} {q / L_\mathrm{SD}} ) } \left[ \delta_{ab} - \frac{ \mathbf{q}_a \mathbf{q}_b}{q^2} \right] \qquad L_\mathrm{SD} = \frac{\eta_\mathrm{m} }{ 2 \eta_\mathrm{f} } }$

Membrane segregation

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