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I apologize if this question is too elementary for mathoverflow; I asked it (unsuccessfully) on MATH.SE first.

As a bit of background: one way to study the mechanics of deformation of a continuous solid body is by defining a reference body $B_0$, a connected, well-behaved subset of $R^2$ or $R^3$. In 3D one assumes that the boundary $\partial B_0$ is a closed, orientable surface; in 2D it is a closed curve.

A motion is then a map $\psi : B_0 \rightarrow B$, where $\psi = \psi(\bar{X},t)$, $t > 0$ and $\bar{X}$ is some label (usually a position vector) for points in the reference body $B_0$. In particular, motions are treated as diffeomorphisms parametrized by $t$.

A particular subset of motions involve those that conserve volumes / areas. For instance, in a planar motion, the area $dv^a \times dv^b$ is conserved by the motion for all time $t$. A general incompressible motion would, if I am not mistaken, preserve a volume form in 3D.

In 2D, the area preserving motions are symplectomorphisms, not just diffeomorphisms. One expects that this imposes strong restrictions on the map $\psi(\bar{X},t)$. In the odd-dimension (3D), one expects similarly strong constraints from the conservation of volume.

The governing PDE for continuous bodies is the Cauchy equation of motion

$$ \nabla \cdot \sigma + \rho \bar{b} = \rho \left ( \frac{\partial \bar{v}}{\partial t} + \bar{v} \cdot \nabla \bar{v} \right) $$

The velocity $\bar{v}$ is defined as $\left. \dfrac{\partial \psi}{\partial t} \right|_X$ and $\bar{b}$ is a specified vector field.

In general, the stress tensor $\sigma$ at a point is a functional of the history of motion of a material point. Specifying it completes the physical description of the continuous body e.g. a fluid, elastic solid etc.

My questions are as follows:

(1) Is the presence of an incompressibility condition associated with any non-trivial, scalar, conserved quantities in the Cauchy momentum equation?

(2) It is known that there are some obvious restrictions on the velocity field, for instance $\nabla \cdot \bar{v} = 0$. However, does incompressibility also impose other, non-obvious restrictions of any kind e.g. on geometric quantities associated with the motion.

Alternatively, is there a reason that symplectomorphisms / conserved volume vs diffeomorphisms just does not yield too much more useful information as far as the Cauchy momentum equation goes?

Added: I was able to find a lot of material for a very particular case, that of incompressible fluid flow (the Euler equations), not much for the more general situation I am interested in.

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  • $\begingroup$ 'Alternatively, is there a reason that symplectomorphisms / conserved volume vs diffeomorphisms ' .. Can you clarify what you are asking here ? What do you mean 'conserved volume Vs diffeos' ? $\endgroup$
    – Piyush Grover
    May 2, 2016 at 20:06
  • $\begingroup$ @PiyushGrover I meant ordinary diffeomorphisms vs volume conserving diffeomorphisms. $\endgroup$
    – user_of_math
    May 4, 2016 at 2:23

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1) conserved quantity for incompressible flow:

$$\frac{d\rho}{dt}=\frac{\partial\rho}{\partial t}+\bar{v}\cdot\nabla\rho=0$$

so if the flow is stationary, $\partial\rho/\partial t=0$, the density $\rho$ is conserved along streamlines

2) restrictions to the velocity field:

in full generality you only have the continuity equation,

$$\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\bar{v})=-\bar{v}\cdot\nabla\rho-\rho\nabla\cdot\bar{v},$$

which in combination with the incompressibility condition gives you $\rho\nabla\cdot\bar{v}=0$; no other restrictions.

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  • $\begingroup$ Thanks. I knew of the divergence free condition. So, would it be correct to state that conserved quantities that do arise in special cases (e.g. in the Euler equation) have to do purely with the functional relating $\sigma$ with the motion? $\endgroup$
    – user_of_math
    May 4, 2016 at 2:22
  • $\begingroup$ yes, for example, if the viscosity is zero, then in 2D (but not in 3D) you have $d\bar{\omega}/dt=0$ with $\bar{\omega}=\nabla\times\bar{v}$ (irrotational flow) $\endgroup$
    – Carlo Beenakker
    May 4, 2016 at 6:23
  • $\begingroup$ Yes, but suppose one talks of constitutive relations in the abstract without explicitly specifying a continuum that is known to conserve certain properties on physical grounds (a non-viscous fluid in your example) - what is the most general class of relations $\sigma = functional(motion)$ that are associated with a conserved quantity? I suppose I should ask this separately. $\endgroup$
    – user_of_math
    May 4, 2016 at 14:44
  • $\begingroup$ I would think this is $\sigma =$ scalar (giving you conservation of momentum) $\endgroup$
    – Carlo Beenakker
    May 4, 2016 at 14:50
  • $\begingroup$ Well, the Cauchy equation is itself a statement of linear momentum balance. Also, how can $\sigma$ (a tensor) equal a scalar? $\endgroup$
    – user_of_math
    May 4, 2016 at 14:53

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