Solution

Problem 3: We are asked to explain why there is no diffusion term in the mass continuity equation. The mass continuity equation (eq 1.33b pg 10) is given by:

$$\frac{D \rho}{Dt} + \rho (\nabla \cdot \vec{v}) = 0$$

or equivalently,

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$$

We start by deriving the local form of the mass continueity equation. We define the mass $M_i$ in a volume $V$ as:

$$M_i = \int_V \rho_i dV$$

Now we impose the conservation of mass. Thus we have:

$$\frac{dM_i}{dt} = 0$$

Now we can evaluate this time derivative using Reynolds transport theorem derived in problem 2.3 as follows:

$$\frac{d}{dt} \int_V \rho_i dV = \int_V \frac{\partial \rho_i}{\partial t} + \nabla \cdot (\rho_i \vec{v}_i) dV = 0$$

Since this holds for any arbitrary volume, the integrand must be zero. Thus we have the local form of the mass continuity equation:

$$\frac{\partial \rho_i}{\partial t} + \nabla \cdot (\rho_i \vec{v}_i) = 0$$

We continue by rewriting this equation in terms of the diffusion flux of species $i$. We start by adding and subtracting $\rho_i \vec{v}$ inside the divergence operator.

$$\frac{\partial \rho_i}{\partial t} + \nabla \cdot (\rho_i \vec{v}+(\rho_i \vec{v}_i - \rho_i \vec{v}) = 0$$

$$\frac{\partial \rho_i}{\partial t} + \nabla \cdot (\rho_i \vec{v} + \vec{J}_i) = 0$$

where $\vec{J}_i$ is the diffusion flux of species $i$ relative to bulk velocity $\vec{v}$. Diffusion refers to the extra movement of a particular species beyond just moving with the surroundings driven by a physical process (i.e. Ficks Law is driven by the consentration gradient $\rho_i/\rho$). Summing over all species (each term in the sum is zero), we have:

$$\sum_i \left( \frac{\partial \rho_i}{\partial t} + \nabla \cdot (\rho_i \vec{v} + \vec{J}_i) \right) = 0$$

Using the linearity of the derivative and divergence operators, we can rewrite this as:

$$\frac{\partial}{\partial t} \left( \sum_i \rho_i \right) + \nabla \cdot \left( \sum_i \rho_i \vec{v} + \sum_i \vec{J}_i \right) = 0$$

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho \vec{v} + \sum_i \vec{J}_i\right) = 0$$

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho \vec{v} + \sum_i \left( \rho_i \vec{v}_i - \rho_i \vec{v} \right)\right) = 0$$

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \left(\sum_i \rho_i \vec{v}_i\right) = 0$$

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$$

Now we see that this works because the diffusion from each species cancels out when we sum over all species. The Diffusion term is simply a redistribution of mass, not the creation or destruction of mass. Thus, there is no diffusion term in the mass continuity equation for all species combined.