Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization

TL;DR

This paper introduces a novel slender-body theory for viscous flow that employs dimensional reduction and hyperviscous regularization, enabling simplified modeling of complex geometries with potential for improved computational efficiency.

Contribution

The paper presents a new approach combining dimensional reduction and hyperviscous regularization to model viscous flow around lower-dimensional bodies, improving approximation accuracy and computational feasibility.

Findings

Accurately approximates velocity fields and drag forces for slender bodies.

Reduces geometric complexity in viscous flow simulations.

Applicable to flows with significant inertia.

Abstract

A new slender-body theory for viscous flow, based on the concepts of dimensional reduction and hyperviscous regularization, is presented. The geometry of flat, elongated, or point-like rigid bodies immersed in a viscous fluid is approximated by lower-dimensional objects, and a hyperviscous term is added to the flow equation. The hyperviscosity is given by the product of the ordinary viscosity with the square of a length that is shown to play the role of effective thickness of any lower-dimensional object. Explicit solutions of simple problems illustrate how the proposed method is able to represent with good approximation both the velocity field and the drag forces generated by rigid motions of the immersed bodies, in analogy with classical slender-body theories. This approach has the potential to open up the way to more effective computational techniques, since the complexity of the geometry can be significantly reduced. This, however, is achieved at the expense of involving higher-order derivatives of the velocity field. Importantly, both the dimensional reduction and the hyperviscous regularization, combined with suitable numerical schemes, can be used also in situations where inertia is not negligible.