to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Advanced Mathematical Methods for Scientists and Engineers

A large quantity of small molecules may migrate into a network of long polymers, causing the network to swell, forming an aggregate known as a polymeric gel. This paper formulates a theory of the coupled mass transport and large deformation. The free energy of the gel results from two molecular processes: stretching the network and mixing the network with the small molecules. Both the small molecules and the long polymers are taken to be incompressible, a constraint that we enforce by using a Lagrange multiplier, which coincides with the osmosis pressure or the swelling stress. The gel can undergo large deformation of two modes. The first mode results from the fast process of local rearrangement of molecules, allowing the gel to change shape but not volume. The second mode results from the slow process of long-range migration of the small molecules, allowing the gel to change both shape and volume. We assume that the local rearrangement is instantaneous, and model the long-range migration by assuming that the small molecules diffuse inside the gel. The theory is illustrated with a layer of a gel constrained in its plane and subject to a weight in the normal direction. We also predict the scaling behavior of a gel under a conical indenter.

https://imechanica.org/files/Kinetics%202007%2010%2022%20correct.pdf

A theory of coupled diffusion and large deformation in polymeric gels

https://imechanica.org/files/gel_in_equilibrium_2008_05_06%20submit.pdf

Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load

The effect of growth in the stability of elastic materials is studied. From a stability perspective, growth and resorption have two main effects. First a change of mass modifies the geometry of the system and possibly the critical lengths involved in stability thresholds. Second, growth may depend on stress but also it may induce residual stresses in the material. These stresses change the effective loads and they may both stabilize or destabilize the material. To discuss the stability of growing elastic materials, the theory of finite elasticity is used as a general framework for the mechanical description of elastic properties and growth is taken into account through the multiplicative decomposition of the deformation gradient. The formalism of incremental deformation is adapted to include growth effects. As an application of the formalism, the stability of a growing neo-Hookean incompressible spherical shell under external pressure is analyzed. Numerical and analytical methods are combined to obtain explicit stability results and to identify the role of mechanical and geometric effects. The importance of residual stress is established by showing that under large anisotropic growth a spherical shell can become spontaneously unstable without any external loading.

http://www.phys.ens.fr/~benamar/paper/2005-JMPS(growth).pdf

Growth and instability in elastic tissues

The deformation of a rectangular block into an annular wedge is studied with respect to the state of swelling interior to the block. Nonuniform swelling fields are shown to generate these flexure deformations in the absence of resultant forces and bending moments. Analytical expressions for the deformation fields demonstrate these effects for both incompressible and compressible generalizations of conventional hyperelastic materials. Existing results in the absence of a swelling agent are recovered as special cases.

Swelling Induced Finite Strain Flexure in a Rectangular Block of an Isotropic Elastic Material

In this paper the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. Lett., 73 (1994), pp. 1311-1315; Phys. Rev. E, 54 (1996), pp. 376-394] is presented in a pedagogical way to increase its visibility in applied mathematics and to argue favorably for its incorporation into the corresponding graduate curriculum. The method is illustrated by some linear and nonlinear singular perturbation problems.

https://epubs.siam.org/doi/pdf/10.1137/080731967

The Renormalization Group: A Perturbation Method for the Graduate Curriculum

Thin viscous liquid films sitting on a solid substrate support nonlinear capillary waves, driven by surface shear stresses at a liquid–gas interface. When surface tension is spatially dependent other mechanisms, such as the thermocapillary effect, influence the dynamics of thin films. In this article we show that in liquids with broken time-reversal symmetry the character of the aforementioned waves and of the thermocapillary effect are significantly modified due to the presence of odd or Hall viscosity in the liquid. This is because odd viscosity gives rise to new terms in the pressure gradient of the flow thus modifying the evolution equation of the liquid–gas interface accordingly. These terms in turn break the reflection symmetry of the evolution equation leading the system to evolve from a pitchfork to a Hopf bifurcation. The odd-viscosity incipient waves can stabilize unstable thin liquid films. For instance, we show that they can suppress the thermocapillary instability. We establish the parameter ranges that odd viscosity has to satisfy in order to initiate those waves that will lead to stability.

Odd-viscosity-induced stabilization of viscous thin liquid films

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach [L.-Y. Chen, N. Goldenfeld, and Y. Oono, Phys. Rev. E 54, 376 (1996)] is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence ${{{Z}_{i}}}_{i=1}^{\ensuremath{\infty}}$. Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation.

Reduction of amplitude equations by the renormalization group approach.

In this paper the method of renormalization group (RG) [Phys. Rev. E54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution.

E. Kirkinis

Papers

Swelling Induced Finite Strain Flexure in a Rectangular Block of an Isotropic Elastic Material

The Renormalization Group: A Perturbation Method for the Graduate Curriculum

Odd-viscosity-induced stabilization of viscous thin liquid films

Reduction of amplitude equations by the renormalization group approach.

Renormalization group interpretation of the Born and Rytov approximations.