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Let $d\in\left\{2,3\right\}$ and $\Omega_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $c\in\Omega_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto X_t(c)\in\Omega_t\tag 1$$ be the movement of $c$ over time. Let $x:=X_t(c)$. Then, the speed of movement of $c$ along the path $(1)$ is described by $$v_t(x):=\frac{\partial X_t}{\partial t}(c)\;.\tag 2$$ [Note, that $(2)$ is a well-defined mapping $\Omega_t\to\mathbb R^d$]. Now, let $\eta_t:\Omega_t\to[0,\infty)$ be the concentration of imaginary things in the fluid at time $t\ge 0$. Then, $$N_t:=\int_{\Omega_t}\eta_t\;{\rm d}\lambda$$ is the amount of things in $\Omega$ at time $t\ge 0$.

It's easy to show, that \begin{equation} \begin{split} \frac{{\rm d}N_t}{{\rm d}t}&=\int_{\Omega_t}\frac{\partial\eta_t}{\partial t}+\nabla\cdot\eta_tv_t\;{\rm d}\lambda\\ &=\int_{\Omega_t}\left(\frac{\partial}{\partial t}+v_t\cdot\nabla\right)\eta_t+\eta_t(\nabla\cdot v_t)\;{\rm d}\lambda \end{split}\tag 3 \end{equation} [Note, that $(3)$ is Reynolds' transport theorem in its form for a material element].

Let's assume, that we're considering an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. By the instationary Navier-Stokes equations, $$\left\{\begin{matrix}\displaystyle\left(\frac\partial{\partial t}+v_t\cdot\nabla\right)v_t&=&\displaystyle\nu\Delta v_t-\frac 1\rho\nabla p_t+f_t&&\text{in }\Omega_t\\\nabla\cdot v_t&=&0&&\text{in }\Omega_t\end{matrix}\right.\;,\tag 4$$ for all $t>0$, where $p_t:\Omega_t\to\mathbb R$ is the pressure of the fluid and $f_t:\Omega_t\to\mathbb R^d$ is the sum of all external forces.

Substituting the second equation of $(4)$ into $(3)$ yields $$\frac{{\rm d}N_t}{{\rm d}t}=\int_{\Omega_t}\left(\frac{\partial}{\partial t}+v_t\cdot\nabla\right)\eta_t\;{\rm d}\lambda\tag 5$$

Now, I want to come up with an advection-diffusion equation of the form $$\left(\frac\partial{\partial t}+v_t\cdot\nabla\right)\eta_t=\kappa\Delta\eta_t+s_t\;\;\;\text{in }\Omega_t\tag 6$$ where $\kappa\in\mathbb R$ is somehow a diffusion rate and $s_t:\Omega_t\times[0,\infty)\to\mathbb R^d$ is the sum of all external sources.

However, $(5)$ is an "advection equation", which doesn't take diffusion into account. Taking a look at $(6)$, it seems like we only need to show, that $$\frac{{\rm d}N_t}{{\rm d}t}=\int_{\Omega_t}\kappa\Delta\eta_t+s_t\;{\rm d}\lambda\;.\tag 7$$


Let me try to do something stupid: Assuming conservation of mass, $$N_t=N_0\;\;\;\text{for all }t\ge 0$$ and thereby $$0=\frac{{\rm d}N_t}{{\rm d}t}\stackrel{(3)}=\int_{\Omega_t}\frac{\partial\eta_t}{\partial t}+\nabla\cdot\eta_tv_t\;{\rm d}\lambda\;.\tag 8$$ Using Fick's first law, we obtain $$\eta_tv_t=-d_t\nabla\eta_t\;,\tag 9$$ where $d_t:\Omega_t\to\mathbb R$ is the diffusion coefficient of the corresponding fluid. Combination of $(8)$ and $(9)$ yields $$\frac{\partial\eta_t}{\partial t}=\nabla\cdot(d_t\nabla\eta_t)\;.\tag{10}$$ However, we don't want to assume conservation of mass. If we would have $$\int_{\Omega_t}s_t-(v_t\cdot\nabla)\eta_t=\frac{{\rm d}N_t}{{\rm d}t}\tag{11}$$ instead of $(8)$, the same argumentation would yield $$\frac{{\rm d}N_t}{{\rm d}t}=\int_{\Omega_t}\nabla\cdot(d_t\nabla\eta_t)+s_t\;{\rm d}\lambda\tag{12}$$ which is exactly what I'm looking for.

Now, I need somebody to make sense of $(11)$.

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  • $\begingroup$ I don't think you can, Reynold's transport theorem is not a balance equation but rather an identity; it can be used to rewrite the accumulation and convection term but it completely neglects any diffusion and production/consumption terms which you only get when writing a conserved balance for $\eta_t$. $\endgroup$
    – nluigi
    Dec 19, 2015 at 14:08
  • $\begingroup$ @nluigi If you take a look at $(5)$ and $(6)$, it seems like we only need to show, that $$\frac{{\rm d}N_t}{{\rm d}t}=\int_{\Omega_t}\kappa\Delta\eta_t+s_t\;{\rm d}\lambda\;.$$ So, we need to derive a "diffusion equation" for $N_t$. Do you think, that's possible? $\endgroup$
    – 0xbadf00d
    Dec 19, 2015 at 14:15
  • $\begingroup$ well physically $\frac{dN_t}{dt}$ is the rate of change of $N_t$ as the reference frame is moving with flow, so clearly the only way something moving with the flow can increase in time is if it gets diffused in or out and/or produced/consumed. I can't make much more of it than that, i guess i am just not mathematically inclined enough :) $\endgroup$
    – nluigi
    Dec 19, 2015 at 14:26

1 Answer 1

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I don't know if this is the sort of answer you're looking for, but it seems like you've taken the right approach for proving (6), which consists of:

  1. Defining an additional diffusive flux in your system $\kappa\nabla\eta_t$, which can be derived from Fick's law (which itself can be derived from an Ehrenfest-Afanassjewa model). I assume $\kappa$ is isotropic, but you can generalize to anisotropy by turning it into a tensor.

  2. Inserting this diffusive flux into your model in addition to the convective flux and applying the Reynolds transport theorem, noting the linearity of the divergence operation splits things up neatly.

Using the diffusive flux from the get-go should have you avoid any pitfalls regarding what is or isn't conserved; if your intent is to model diffusion into a control volume that is convectively moving with the fluid, you should recover a simple diffusion equation (in very funky coordinates).

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