# Boundary conditions

hi pf!

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!

Chestermiller
Mentor
It doesn't make much sense to me either. But, of course, it's in a math book, so who knows what the author knew about mass transfer. The implication is that somehow, there is a diffusive flow of mass from a point source into the stream, with no bulk movement of solvent involved. For this to happen, the concentration L at the point source s = 0 would have to be infinite. Pretty silly, huh. What they really are trying to say is that the mass flow rate of solute into the stream is W.

Chet

1 person

yea, sometimes these conditions are pretty weird. i finally think i do have this one (surface area times surface flux). still uncomfortable, though.

thanks for helping me out a lot lately!