Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

# Correct Expression of the Material Derivative in Continuum Physics

Version 1 : Received: 1 March 2020 / Approved: 2 March 2020 / Online: 2 March 2020 (15:39:55 CET)
Version 2 : Received: 19 April 2020 / Approved: 22 April 2020 / Online: 22 April 2020 (09:51:56 CEST)
Version 3 : Received: 27 April 2020 / Approved: 28 April 2020 / Online: 28 April 2020 (09:40:46 CEST)
Version 4 : Received: 14 June 2020 / Approved: 15 June 2020 / Online: 15 June 2020 (09:55:06 CEST)

How to cite: Sun, B. Correct Expression of the Material Derivative in Continuum Physics. Preprints 2020, 2020030030 (doi: 10.20944/preprints202003.0030.v4). Sun, B. Correct Expression of the Material Derivative in Continuum Physics. Preprints 2020, 2020030030 (doi: 10.20944/preprints202003.0030.v4).

## Abstract

The material derivative is important in continuum physics. This Letter shows that the expression $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$, used in most literature and textbooks, is incorrect. The correct expression $\frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is formulated. The solution existence condition of Navier-Stokes equation has been proposed from its form-solution, the conclusion is that "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

## Keywords

material derivative; continuum physics; solution existence condition; the Navier-Stokes equation

## Subject

PHYSICAL SCIENCES, Fluids & Plasmas

Comment 1
Commenter: Bohua Sun
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