PHY.6:5.2 - Obtain a total without solving its spatial distribution

Let c(x,t) be the concentration of one chemical species in a fixed region. Its prescribed velocity field is u(x,t). Use diffusion coefficient D>0, a constant first-order consumption rate k>=0, and the flux law

J = c*u - D*grad(c).

The species balance supplies

partial_t(c) = -div(J) - k*c.

These equations state the response assumptions: advection, Fickian diffusion and first-order conversion. Their applicability is a physical premise. Impose no flux of this species through the region’s boundary, J dot n = 0, where n is the outward normal.

For the total amount N(t)=integral_region c(x,t) dx, integration of the balance gives

N' = -integral_boundary J dot n dS - k*N = -k*N,

and therefore N(t)=N(0)*exp(-k*t). The requested total follows from its initial total and the stated boundary and reaction laws. Internal transport need not be solved. Consumption of this species can coexist with conservation of the atoms it transfers into products.

Changed question. A local concentration maximum requires the initial spatial distribution and its evolution. The total alone no longer answers. Alternatively, if the reaction rate varies with position, the total rate becomes -integral_region k(x)*c(x,t) dx; replacing it by a constant times N now needs grounds for that reduction. Return to the spatial distribution or a justified bound when the new use needs it.

This case shows how the intended consequence determines which physical relations must be completed. It also separates a global balance result from a local transport prediction.