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Reynolds transport theorem says that

$ \frac{d\int\phi}{dt}=\int\left(\frac{\partial\phi}{\partial t} + \nabla\cdot(\phi\otimes v) \right) $

Why is the material derivative not defined as what's inside the integral on the right hand side?

Emil
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    The reason is that material derivative, as far as I know, is taken with the system's velocity itself. The term won't make sense in the light that If I choose some arbitrary region instead of a system then the velocity could be anything. – Someone Mar 20 '19 at 20:32

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The Reynolds transport theorem reads $$\frac{\mathrm{d}}{\mathrm{d} t} \int_{\Omega} \phi\, \mathrm{d}V= \int_{\Omega} \frac{\partial \phi}{\partial t} \, \mathrm{d}V + \int_{\partial \Omega} \phi \left[ \mathbf{v} \cdot \mathbf{n} \right] \mathrm{d}A$$ The divergence theorem then says $$ \int_{\partial \Omega} \phi \left[ \mathbf{v} \cdot \mathbf{n} \right] \mathrm{d}A = \int_{\Omega} \mathrm{div} \left( \phi \mathbf{v} \right) \mathrm{d}V $$ Moreover, $$ \mathrm{div}\left( \phi \mathbf{v} \right) = \mathrm{grad}\left( \phi \right) \cdot \mathbf{v} + \phi \, \mathrm{div}\left(\mathbf{v} \right) $$ The term $\phi \, \mathrm{div}\left(\mathbf{v} \right)$ is a source or sink of $\phi$.

Compare the above with the material derivative $$ \frac{\mathrm{D} \phi}{\mathrm{D} t} = \frac{\partial \phi}{\partial t} + \mathrm{grad}\left( \phi \right) \cdot \mathbf{v} $$ It describes the time rate of change of $\phi$. For a conservation law, this change is balanced by the sum of all sources and sinks of $\phi$. The conservation of mass in fluid mechanics is a nice example: $$\frac{\partial \rho}{\partial t} + \mathrm{div}\left( \rho \mathbf{v} \right) = \frac{\partial \rho}{\partial t} + \mathrm{grad}\left( \rho \right) \cdot \mathbf{v} + \rho \, \mathrm{div}\left(\mathbf{v} \right) = 0 $$ The time rate of change of mass density $\rho$ is given by $\partial \rho / \partial t + \mathrm{grad}\left( \rho \right) \cdot \mathbf{v}$, which is balanced by the source/sink $\rho \, \mathrm{div}\left(\mathbf{v} \right)$.

The Reynolds transport theorem and the divergence theorem describe the conservation of a physical quantity that can be transported by a flux. The source/sink of that quantity, that is created by the flux velocity, is already included in this description. The material derivative instead gives the time rate of change of any quantity, even those that are not conserved, for example temperature.

Procyon
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  • Could you explain the source term here https://en.wikipedia.org/wiki/Derivation_of_the_Navier%E2%80%93Stokes_equations , is there a mistake in that article? – Emil Jun 06 '22 at 07:28
  • Is the time derivative of the integral of the quantity $\frac{d\int\phi}{dt}$ equal to net changes (entering, leaving, appearing, disappearing)? – Emil Jun 06 '22 at 07:40
  • (basically the "s" term used for sources and sinks is confusing me, its usage seems inconsistent, and it seems they have a strange version of the transport theorem since they don't get the extra terms) – Emil Jun 06 '22 at 07:54
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    @Emil The $s$ in the Wikipedia article is the sum of all other sources/sinks. For incompressible flow, $\mathrm{div}\left(\mathbf{v}\right)=0$, thus the source/sink due to the flow velocity vanishes. But there are other sources/sinks of momentum: Friction on moving walls, pressure gradients, friction due to molecular or eddy viscosity. The sum of these sources/sinks is called $s$ in the Wikipedia article. – Procyon Jun 06 '22 at 08:38