Using Fractal Calculus to Solve Fractal Navier–Stokes Equations, and Simulation of Laminar Static Mixing in COMSOL Multiphysics

Amir Pishkoo,Maslina Darus

doi:10.3390/fractalfract5010016

Abstract

Navier–Stokes equations describe the laminar flow of incompressible fluids. In most cases, one prefers to solve either these equations numerically, or the physical conditions of solving the problem are considered more straightforward than the real situation. In this paper, the Navier–Stokes equations are solved analytically and numerically for specific physical conditions. Using Fα-calculus, the fractal form of Navier–Stokes equations, which describes the laminar flow of incompressible fluids, has been solved analytically for two groups of general solutions. In the analytical section, for just “the single-phase fluid” analytical answers are obtained in a two-dimensional situation. However, in the numerical part, we simulate two fluids’ flow (liquid–liquid) in a three-dimensional case through several fractal structures and the sides of several fractal structures. Static mixers can be used to mix two fluids. These static mixers can be fractal in shape. The Sierpinski triangle, the Sierpinski carpet, and the circular fractal pattern have the static mixer’s role in our simulations. We apply these structures just in zero, first and second iterations. Using the COMSOL software, these equations for “fractal mixing” were solved numerically. For this purpose, fractal structures act as a barrier, and one can handle different types of their corresponding simulations. In COMSOL software, after the execution, we verify the defining model. We may present speed, pressure, and concentration distributions before and after passing fluids through or out of the fractal structure. The parameter for analyzing the quality of fractal mixing is the Coefficient of Variation (CoV).

Highlights

One of the main problems in classical calculus is that the function is discontinuous, or its derivative does not exist at some points
The method used by Yang et al [37] to solve Navier–Stokes equations is different from our work in this paper, which is based on fractal calculus introduced by Parvate and Gangal [7,8,9]
The quality of mixing two fluids in a pipe or channel can be statistically expressed by the Coefficient of Variation (CoV) parameter, which is defined as follows: s

Summary

Introduction

One of the main problems in classical calculus is that the function is discontinuous, or its derivative does not exist at some points. Its reason is that classical mechanics is a local theory in which partial derivatives exist. They have no solution for discontinuous points. Many believe that substituting a non-local theory instead of a local theory is the only solution. In their papers, Parvate and Gangal [7,8,9] introduce another approach based on the local theory entitled F α -calculus, which is convenient for discontinuous point and even non-differentiable continuous functions

Fractal Calculus

Static Mixers

Main Results

To Model Diluting the Fluid with Fractal Mixing Patterns

Results and Discussion

Conclusions