Boundary conditions

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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!
 
19,351
3,838
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
 
1,766
69
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!
 

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