The scaling-law flows: An attempt at scaling-law vector calculus

TL;DR

This paper introduces a novel scaling-law vector calculus framework connecting fractal geometry with classical vector calculus, deriving new theorems and Navier-Stokes-like equations for scaling-law flows.

Contribution

It develops the first comprehensive scaling-law vector calculus based on Leibniz derivatives and Stieltjes integrals, extending classical theorems to fractal-related flows.

Findings

Formulated Gauss-Ostrogradsky-like theorem for scaling-law calculus

Derived Stokes-like and Green-like theorems in the new framework

Obtained Navier-Stokes-like equations for scaling-law flows

Abstract

In this paper, the scaling-law vector calculus, which is related to the connection between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The Gauss-Ostrogradsky-like theorem, Stokes-like theorem, Green-like theorem, and Green-like identities are considered in the sense of the scaling-law vector calculus. The Navier-Stokes-like equations are obtained in detail. The obtained result is as a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.