Sarthok Sircar

High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.

Kinetics of swelling gels

We develop a general theory of the swelling kinetics of polymer gels, with the view that a polymer gel is a two-phase fluid. The model we propose is a free boundary problem and can be used to understand both contraction and swelling, including complete dissolving or dehydration of polymeric gels. We show that the equations of motion satisfy a minimum energy dissipation rate principle similar to the Helmholtz minimum dissipation rate principle which holds for a Stokes flow. We also show, using asymptotic analysis and numerical simulation, how the equilibrium swelled state and the swelling rate constant are related to the free energy and rheological properties of the polymer network.

A Note on the Kinematics of Rigid Molecules in Linear Flow Fields and Kinetic Theory for Biaxial Liquid Crystal Polymers

We present a systematic derivation of the extended Jefferys orbit for rigid ellipsoidal and V-shaped polymer molecules in linear incompressible viscous flows using a Lagrange multipliers method based on a constraining force argument. It reproduces the well-known Jefferys orbit for rotating ellipsoids. The method is simple and applicable to any rigid body immersed in a linear flow field so long as a discrete set of representative points on the rigid body can be identified that possess the same rotational degrees of freedom as the rigid body itself. The kinematics of a single V-shaped rigid polymer driven by a linear flow field are discussed, where steady states exist along with time-periodic states in limited varieties. Finally, we show how the kinematics of the rigid V-shaped polymer can be used in the derivation a kinetic theory for the solution of rigid biaxial liquid crystal polymers, where Brownian motion, excluded volume interaction and flow driven kinematics are included.

A Hydrodynamical Kinetic Theory for Self-Propelled Ellipsoidal Suspensions

Using the tools of kinetic theory, we derive a hydrodynamic model for self-propelled particles of an arbitrary shape from first principles, in a sufficiently dilute suspension limit, moving in a 3-dimensional space inside a viscous solvent. The model is then restricted to particles with ellipsoidal geometry to quantify the interplay of the long-range excluded volume and the short range self-propulsion effects. The expression for the constitutive stresses, relating the kinetic theory with the momentum transport equations, are derived using a combination of the virtual work principle (for extra elastic stresses) and symmetry arguments (for active stresses).