Correct Expression of the Material Derivative in Continuum Physics
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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 solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"
2020 ◽
2022 ◽
pp. 106223
1998 ◽
Vol 115
(1)
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pp. 18-24
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2021 ◽
Vol 375
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pp. 113600
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2015 ◽
Vol 17
(1)
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pp. 14-19
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1973 ◽
Vol 59
(2)
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pp. 391-396
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