User:GraceChen117/sandbox

$$ \vec{v}\ $$ is the velocity filed that the quantity is moving with. It is function of time and space. For example, in advection, c might be the concentration of salt in a river, and then $$\vec{v}$$ would be the velocity of the water flow, $$\vec{v}$$ =f(time, location). Another example, c might be the concentration of small bubbles in a calm lake, and then $$ \vec{v} $$ would be the velocity of bubbles rising towards the surface by buoyancy (see below) depending on time and location of the bubble. For multiphase flows and flows in porous media, $$ \vec{v} $$ is the (hypothetical) superficial velocity.

$$\nabla$$ represents gradient and $$\nabla\cdot$$ represents divergence. In this equation, $$\nabla$$c represents concentration gradient.

The first, $$\nabla \cdot (D \nabla c)$$, describes diffusion. Imagine that c is the concentration of a chemical. When concentration is low somewhere compared to the surrounding areas (e.g. a local minimum of concentration), the substance will diffuse in from the surroundings, so the concentration will increase. Conversely, if concentration is high compared to the surroundings (e.g. a local maximum of concentration), then the substance will diffuse out and the concentration will decrease. The net diffusion is proportional to the Laplacian (or second derivative) of concentration if the diffusivity D is a constant.

In general, D, $$ \vec{v} $$, and  R  may vary with space and time. In cases in which they depend on concentration as well, the equation becomes nonlinear, giving rise to many distinctive mixing phenomena such as Rayleigh–Bénard convection when $$ \vec{v} $$ depends on temperature in the heat transfer formulation and reaction-diffusion pattern formation when  R  depends on concentration in the mass transfer formulation.

When analytical solution is available, it will be more computational efficient comparing with numerical methods.