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R. K. Michael Thambynayagam

Publications and source records attributed to R. K. Michael Thambynayagam.

7 recordsLinked to original sources
arXiv2026

Exact Periodic Solutions of the Forced Incompressible Navier-Stokes Equations in Arbitrary Dimensions

Exact periodic solutions of the unforced incompressible Navier-Stokes equations require the convective field to be a pure gradient, absorbed into the pressure; for the cyclic trigonometric families studied previously this occurs only at isolated phases and, over the range so far classified, only at n = 3 and 4. The forced problem, we show, admits an exact construction for every phase vector and every n >= 3. For an n-parameter family of two-term cyclic fields, solenoidal and a Laplacian eigenfunction, the transverse part U^0 of its convective field serves as the body force, giving a closed-form solution with velocity, pressure and force explicit. We prove closed forms for the pressure and for both magnitudes, in every dimension and with no upper bound on n: the forcing is never trivial, and the L^2 magnitude of U^0 stands to that of the whole convective field in the constant ratio 2*sqrt(2)/3, whatever the dimension, phase or units. The forcing does no net work, so the kinetic energy decays purely viscously at arbitrary Reynolds number, giving an exact benchmark in any dimension. A numerical study of vorticity concentration at zero applied work uses it as a reference, and shows that apparent saturation can be a resolution artefact.

physics.flu-dyn↗
arXiv2026

Iterative Construction of n-Dimensional Navier-Stokes Solutions with Non-Gradient Convection

Exact periodic solutions of the incompressible Navier-Stokes equations arise when the convective field is a pure gradient and can be absorbed into the pressure. We construct solutions outside this class, in arbitrary spatial dimension, by an iteration in which each step solves a linear diffusion problem forced by the transverse part of the preceding convective field. The construction is dimension-independent, and for a cyclic family of initial fields the first two iterates are available in closed form. Carried out numerically in three dimensions, the iteration converges: at Re=10 successive iterates contract with an observed factor 0.57 and the limit satisfies the mild equation with relative defect 1.0 x 10^-3. The observed factor exceeds unity by Re=30, which identifies the practical range of the iteration rather than a proved convergence boundary. We also show what goes wrong when the projection is omitted, as it was in an earlier iteration of the author's: the iterates cease to be divergence-free and an apparent growth appears that belongs to the expansion and not to the flow. For an n-dimensional generalisation of the initial field the fraction of the convective field not absorbed by pressure is 2sqrt(2)/3, independently of n over the range examined. Divergence and cell kinetic energy provide simple diagnostics that detect failure at the step where it occurs.

physics.flu-dyn↗
arXiv2026

n-tuple-periodic Solutions of the Navier-Stokes Equations: Existence and Obstructions

We introduce a phase-angle framework for constructing and classifying n-tuple-periodic solutions of the incompressible Navier-Stokes equations, and use it to determine, dimension by dimension, when such solutions exist and why they fail. The velocity field is a difference of two n-fold cyclic trigonometric products, made divergence-free for every phase and every n >= 3 by a telescoping identity. The problem then reduces to two explicit conditions on the n phase angles: a reduction condition, necessary and algebraic, and a pressure-integrability condition, sufficient and differential. Separating them allows a failed construction to be located rather than merely recorded. Unlike the stream-function route on which earlier tri-periodic solutions rest, the framework is not confined to three dimensions, and does not presuppose the Beltrami property. For 3 <= n <= 8 we prove that unforced solutions exist at n=3 and n=4 and nowhere else. In three dimensions the admissible phases form exactly two families, mirror images distinguished by the sign of the helicity; these families are known, and we establish that the construction admits no others, settling a completeness question left open when they were found. In four dimensions integrability reduces a one-parameter curve to a single family, so the quarter-period offset is forced, not assumed. The obstructions divide by parity: arithmetic at n=5,7, where every admissible phase vector annihilates the field; differential at n=6,8, where the field survives but its convective term is not a gradient. Every obstructed dimension nevertheless carries an exact forced solution in closed form. These are not mere self-balancing: the phases are those at which strain and rotation are in exact pointwise balance, so that no pressure is generated at all. What obstructs the unforced problem becomes, in the forced problem, an exactly prescribable input.

math.GM↗
arXiv2023

A class of exact solutions of the Navier-Stokes equations in three and four dimensions

A few basic, intuitive, properties of the Navier-Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier-Stokes equation in a space having an arbitrary number of dimensions. We then derive spatially periodic solutions for the velocity and pressure fields that span an unbounded domain in three and four dimensions, given a smooth solenoidal initial velocity vector field. In these solutions all velocity components depend non-trivially on all coordinate directions.

math.GM↗
arXiv2016

The Navier-Stokes Existence and Smoothness Poser in $\mathbb{R}^n$

In this paper we describe a method to derive solutions of the incompressible Navier- Stokes system of equations for non-stationary initial value problems in $\mathbb{R}^n$. We show that for a given smooth solenoidal initial velocity vector field there exist smooth spatially periodic solutions of pressure and velocity in $\mathbb{R}^n$. An illustrative example in $\mathbb{R}^3$ provides important insights into the ostensible phenomenon of the blowup time.

math.AP↗