Gauge Freedom and Objective Rates in the Morphodynamics of Fluid Deformable Surfaces: the Jaumann Rate vs. the Material Derivative

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.

Figures (4)

• Figure 1: An embedding $r_t$ of an abstract space $B$ into $\mathbb{R}^3$ induces a vector field ${\bf V}_t$ on $B$ called the left Eulerian velocity, as well as a map ${\bf v}_t : M_t \to \text{T}\mathbb{R}^3$ on the image $M_t$ of the embedding. We interpret that latter as a vector field along $M_t$ which has a part tangent to $M_t$ but may also have a part normal to $M_t$, and refer to it as the right Eulerian velocity. Pulling back this vector field along the embedding $r_t$ 'forgets' the normal part, yielding the left Eulerian velocity ${\bf V}_t$. The mathematical relationships between these maps are shown in the diagram on the left, while a more visual representation is shown on the right. Here, $\text{T}r_t$ denotes the tangent map (matrix of partial derivatives) induced by $r_t$, $\pi$ is the projection from the tangent bundle to the underlying manifold, and ${\bf U}_t = \partial_t r_t$ as described in the text.
• Figure 2: Comparison of the Eulerian and Lagrangian pictures for motion in a fixed surface (left) and for a deformable surface (right). In a fixed plane $\mathbb{R}^2$, the Lagrangian picture involves a diffeomorphism $\psi_t$ of $\mathbb{R}^2$ that moves a fluid particle initially the point $p$ to the point $\psi_t(p)$. We may also consider an Eulerian picture, where we stand still at the point $p$ and watch the fluid flow past us, its direction at time $t$ being given by the vector ${\bf v}_t(p)$. A deformable surface $M_t$ embedded in $\mathbb{R}^3$ undergoes a Lagrangian motion described by a diffeomorphism $\lambda_t : M_0 \to M_t$ that maps the initial surface onto the time $t$ surface. The associated right Eulerian velocity field ${\bf v}_t$ lies in the extended tangent space $E_t$ to $M_t$, as described in the text. By pulling back the entire tangent space via $\lambda_t$ (inset) as described in the text we can define a time-varying vector field ${\bf s}_t$ along $M_0$ which plays the role of the Eulerian velocity at a fixed point $p \in M_0$.
• Figure 3: (Left) Construction of a Monge gauge. The evolution of an initially-flat surface (green) can be decomposed as a pair $(\psi_t(p), h_t(p))$ where $\psi_t$ (orange) is a diffeomorphism is the initial surface $M_0$ and the $h_t$ is a height function. By making a time-dependent change of gauge using the inverse $\psi^{-1}_t$ as described in the text, we can ensure the evolution is determined purely by a gauge-transformed height function $h_t \circ \psi^{-1}_t$, which ensures a fluid particle initially at the point $p$ evolves purely in the vertical direction (pink). (Right) Transition from the Monge gauge (pink) to the normal gauge (purple). This involves a diffeomorphism $\psi_t^N$ (blue) whose drive velocity is $-\nabla h^M_t$, where $h^M_t = h_t \circ \psi_t^{-1}$ is the height function in the Monge gauge. In the Monge gauge the velocity vector of the surface is ${\bf v}^M = v^M {\bf e}_z$, while in the normal gauge it is ${\bf v}^N = v^N ({\bf e}_z -\nabla h^M)$.
• Figure 4: Lie dragging a tangent vector field ${\bf P}$ along a flow ${\bf v}$ on the surface results in a new vector field which has a component out of the surface--we illustrate ${\bf P}$ at a single point being dragged along an integral curve of ${\bf v}$. In order to keep the field ${\bf P}$ in the surface, a counteracting force must push back against this effect.