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Gauge freedom and objective rates in fluid deformable surfaces

Published online by Cambridge University Press:  14 March 2025

Joseph Pollard Open the ORCID record for Joseph Pollard [Opens in a new window] ,
Sami C. Al-Izzi Open the ORCID record for Sami C. Al-Izzi [Opens in a new window]  and
Richard G. Morris Open the ORCID record for Richard G. Morris [Opens in a new window]
Joseph Pollard*
Affiliation:
School of Physics, UNSW, Sydney, NSW 2052, Australia EMBL Australia Node in Single Molecule Science, School of Medical Sciences, UNSW, Sydney, NSW 2052, Australia
Sami C. Al-Izzi*
Affiliation:
School of Physics, UNSW, Sydney, NSW 2052, Australia ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems, UNSW, Sydney, NSW 2052, Australia
Richard G. Morris*
Affiliation:
School of Physics, UNSW, Sydney, NSW 2052, Australia EMBL Australia Node in Single Molecule Science, School of Medical Sciences, UNSW, Sydney, NSW 2052, Australia ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems, UNSW, Sydney, NSW 2052, Australia
*
Corresponding authors: Joseph Pollard, [email protected]; Sami C. Al-Izzi, [email protected]; Richard G. Morris, [email protected]
Corresponding authors: Joseph Pollard, [email protected]; Sami C. Al-Izzi, [email protected]; Richard G. Morris, [email protected]
Corresponding authors: Joseph Pollard, [email protected]; Sami C. Al-Izzi, [email protected]; Richard G. Morris, [email protected]
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Abstract

Morphodynamic descriptions of fluid deformable surfaces are relevant for a range of biological and soft matter phenomena, spanning materials that can be passive or active, as well as ordered or topological. However, a principled, geometric formulation of the correct hydrodynamic equations has remained opaque, with objective rates proving a central, contentious issue. We argue that this is due to a conflation of several important notions that must be disambiguated when describing fluid deformable surfaces. These are the Eulerian and Lagrangian perspectives on fluid motion, and three different types of gauge freedom: in the ambient space; in the parameterisation of the surface; and in the choice of frame field on the surface. We clarify these ideas, and show that objective rates in fluid deformable surfaces are time derivatives that are invariant under the first of these gauge freedoms, and which also preserve the structure of the ambient metric. The latter condition reduces a potentially infinite number of possible objective rates to only two: the material derivative and the Jaumann rate. The material derivative is invariant under the Galilean group, and therefore applies to velocities, whose rate captures the conservation of momentum. The Jaumann derivative is invariant under all time-dependent isometries, and therefore applies to local order parameters, or symmetry-broken variables, such as the nematic $Q$-tensor. We provide examples of material and Jaumann rates in two different frame fields that are pertinent to the current applications of the fluid mechanics of deformable surfaces.

JFM classification

Biological Fluid Dynamics: Membranes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1007 , 25 March 2025 , A32
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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