- #1

- 2,017

- 113

i was reading a sample problem in a text on ode's and came across a boundary condition that didnt really make sense to me.

the physical scenario is: a liquid ##L## measured in moles/cubic meter (##mol / m^3##) is injected into a stream of water. ##L## is being injected at a rate ##W## measured in (g-moles)/sec (##(g-mol) / s##). the following boundary condition is presented: $$s \to 0 \implies -4 \pi s^2 C \frac{\partial L}{\partial s} \to W$$ where ##s^2 = x^2 + y^2 + z^2##. this boundary condition physically represents that the injection rate at ##s=0## is ##W## (the coordinate system is centered at the injection site). ##C## is a constant, who's units are square meters per second (##m^2 / s##)

now i know ##4 \pi r^2## is the surface area of a sphere. also, we are given that molar flux, ##\vec{n}## is ##\vec{n}=-C\nabla L## which has units ##mol / (m^2 \times s)##.

thanks for any help on the help!