Table 1 Minimum dissipation theorems
From: Minimum entropy production by microswimmers with internal dissipation
Dissipation function | Motivation | Theorem | V-problem | F-problem |
|---|---|---|---|---|
External dissipation38 | Ideal swimmer | \({P}_{{{{{{{{\rm{A}}}}}}}}}\ge {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}} \cdot {\left({{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{PS}}}}}}}}^{-1}-{{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{NS}}}}}}}}^{-1}\right)}^{-1}\cdot {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\) | Perfect slip | No-slip |
Surface-driven droplet (Fig. 1a) | Active droplets (Marangoni) | \({P}_{{{{{{{{\rm{A}}}}}}}}}\ge {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\cdot {\left({{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{Droplet}}}}}}}}^{-1}-{{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{NS}}}}}}}}^{-1}\right)}^{-1}\cdot {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\) | Droplet | No-slip |
External tangential forces (Fig. 1b) | Model of cilia | \({P}_{{{{{{{{\rm{A}}}}}}}}}\ge {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\cdot {\left({{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{NSi}}}}}}}}^{-1}-{{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{CP}}}}}}}}^{-1}\right)}^{-1}\cdot {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\) | No-slip core | No-slip core, no tangential slip shell |
External normal forces (Fig. 1c) | Phoretic swimmers | \({P}_{{{{{{{{\rm{A}}}}}}}}}\ge {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\cdot {\left({{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{NSi}}}}}}}}^{-1}-{{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{DC}}}}}}}}^{-1}\right)}^{-1}\cdot {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\) | No-slip core | No-slip core, zero normal velocity shell |
External unconstrained forces (Fig. 1d) | Idealized external propulsion | \({P}_{{{{{{{{\rm{A}}}}}}}}}\ge {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\cdot {\left({{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{NSi}}}}}}}}^{-1}-{{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{NS}}}}}}}}^{-1}\right)}^{-1}\cdot {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\) | No-slip core | No-slip (outer shell) |
Surface dissipation (Fig. 1e) | Coarse-grained (cilia, phoretic) | \({P}_{{{{{{{{\rm{A}}}}}}}}}\ge {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\cdot {\left({{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{Navier}}}}}}}}^{-1}-{{{{{\boldsymbol{\sf{R}}}}}}}_{{{{{{{\rm{NS}}}}}}}}^{-1}\right)}^{-1}\cdot {{{{{\boldsymbol{\sf{V}}}}}}}_{{{{{{{{\rm{A}}}}}}}}}\) | Navier slip | No-slip |
