Chennai Mathematical Institute

Seminars, 3.00 pm
Field theoretic viewpoints on certain fluid mechanical phenomena

Sachin Phatak
Chennai Mathematical Institute.


In this thesis we study field theoretic viewpoints on certain fluid mechanical phenomena.

In the Higgs mechanism, mediators of the weak force acquire masses by interacting with the scalar condensate, leading to a vector boson mass matrix. On the other hand, a rigid body accelerated through an inviscid, incompressible and irrotational fluid feels an opposing force linearly related to its acceleration, via an added-mass tensor. We uncover a striking physical analogy between the two effects and propose a dictionary relating them. The correspondence turns the gauge Lie algebra into the space of directions in which the body can move, encodes the pattern of gauge symmetry breaking in the shape of an associated body and relates symmetries of the body to those of the scalar vacuum manifold. The new viewpoint raises interesting questions, notably on the fluid analogs of the broken symmetry and Higgs particle.

Ideal gas dynamics can develop shock-like singularities which are typically regularized through viscosity. In 1d, discontinuities can also be conservatively smoothed via dispersion. In the second part, we develop a minimal conservative regularization of 3d adiabatic flow of a gas with exponent gamma, by adding to the Hamiltonian a capillarity energy proportional to the square of the density gradient. This leads to a nonlinear body force with 3 derivatives of density, while preserving the conservation laws of mass and entropy. The regularized model admits dispersive sound, solitary and periodic traveling waves, but no steady continuous shock-like solutions. Nevertheless, in 1d, for gamma = 2, numerical solutions in periodic domains show recurrence and avoidance of gradient catastrophes via formation of solitons. This is explained via an equivalence between our model (for homentropic potential flow in any dimension) and a defocussing nonlinear Schrodinger (NLS) equation (cubic for gamma = 2). Thus, our model is a generalization of both the single field KdV and NLS equations to adiabatic gas dynamics in any dimension.