The Del Operator and my own personal vector calculus hell

joshofalltrades

So it has been four years since I had any calculus whatsoever and I am attempting to figure out a variable for use in a simulation.

To calculate this variable, I need to solve:



t is time in seconds, which I know, p is density of a substance, which I have already calculated out and is known in kg/m^3, E is porosity (I assume it's a percentage, as in 1=completely open to flow, 0=no pore space whatsoever), u is Darcy velocity in m/s and since I'm using a substance of known molar weight that was easy to determine.

The whole equation is a variation on Darcy's Law. It has been applied to the continuity equation to give me this thing using derivatives and the Del operator which I have no experience with and now I'm nervous that I'm going to calculate this all wrong and my results won't be real and oh my glooooob.

Qm is a mass source term in units kg/m^3*s. Yeeeeeaaaaah.

So, uh, any help would be appreciated in untangling this monster so I can have an accurate number to plug in.

joshofalltrades

I should also say that I'm aware that the first part is a time derivative, it's really the Del operator that's giving me fits, but if the time derivative is going to be problematic in some way please let me know.

Cowboy-Bebop

http://en.wikipedia.org/wiki/Gradient

Looks like you have a PDE and because of this either the p or the epsilon is probably variable, not something you already know the value of. If you know both then the first bit is zero (derivative of a constant = zero (sorry if you knew this already)).

As far as solving this, I haven't actually seen one with just del, although del^2 is the laplacian and very common which in this case would lead to poisson's equation. You might want to try and find that specific PDE in some text for the solution or alternatively use a numerical solver to get the answer.

Savant

The del operator is not all that bad, more of a notational thing. It looks like in that particular equation you are looking for the divergence of rho times the u vector.

On wikipedia from the del page, these look like the lines you want:

The divergence of a vector field is a scalar function that can be represented as:



The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or repel from a point.

Double check the units to make sure it makes sense, but that looks right to me because the divergence differentiates with respect to distance which should get rid of the extra meter unit in the m/s units for the velocity.

Cowboy-Bebop

Del is contextual, just so you know. If u is a vector field, its divergence. If u is a scalar field it's gradient. You might just want to google solutions to Darcy's equations, I don't think it's as simple as you imagined.

CptHamilton

Cowboy-Bebop wrote: »

Del is contextual, just so you know. If u is a vector field, its divergence. If u is a scalar field it's gradient. You might just want to google solutions to Darcy's equations, I don't think it's as simple as you imagined.

Del isn't contextual. Del(u) is a gradient if u is a scalar field and is undefined for u a vector field. Del . u is a vector field divergence or undefined for u a scalar field. You can't take the direct product of vectors or the dot product of a vector and a scalar. Del's just a vector with components that are operators.

In this case, Del . (rhou) is the dot product of Del and u with a scaling factor rho, so equal to:
rho*(Del . u), which is:
rho*(du/dx + du/dy + du/dz)

So your whole equation is:
Qm/rho = dE/dt + du/dx + du/dy + du/dz

UNLESS

If rho is time or position dependent then matters change. In that case you have:
d/dt(rho*E) = drho/dt*E + rho*dE/dt
and
Del . (rho*u) = rho*(Del . u) + u . (Del*rho)
which is
rho(du/dx + du/dy + du/dz) + u_x*drho/dx + u_y*drho/dy + u_z*drho/dz
for a final equation of:

E*(drho/dt) + rho*(dE/dt + du/dx + du/dy + du/dz) + u . <drho/dx, drho/dy, drho/dz> = Qm

Savant

Yeah, del is a specific thing, a vector type operator. For three dimensional vectors it is as shown on the previously mentioned del page:


It's a little wonky because the partial derivatives are operators rather than numbers, but it is a notational tool so you can encode various vector calculus operations more succinctly using vector notation. So instead of writing the "gradient of f" you can write "del f" or instead of "divergence of the u vector field" you can write "del dot u"

joshofalltrades

Thanks guys, I am at a physics conference right now and am reading through these, I promise.