Understanding Boundary Conditions in ODEs: A Guide for Beginners

[member 428835]

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]

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

 

[member 428835]

thanks for the reply, chet.

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!