Proving diffusion spreads perpendicular to level curves So this has been bothering me for a little while. When you consider scalar diffusion like $\frac{dc}{dt} = D\nabla^2 c$, where $c=c(x,y)$, most people would say that the scalar will move downhill. Now it is certainly true that the scalar is going to move from regions of high concentration to regions of low concentration. However, most of the time it is said that the diffusion is in the direction of the gradient. (In other words, the level curves move perpendicular to themselves and not at a slight angle) 
This assumption is mentioned off-hand in a lot of places like: http://www.math.umn.edu/~olver/ln_/vc2.pdf (see paragraphs near Proposition 5.3). I however cannot find a proof that it is always true for any scalar concentration field. 
This seems like such a simple thing, but the more I look into it the more I don't know if it's true. When thinking about this, don't just assume a circularly symmetric distribution, consider something like a wedge shape for instance. Shouldn't the point of the wedge grow faster down the opening than along it, thus bending its angle away from the gradient?
I'd appreciate an explanation why diffusion follows the gradient, or a counter-example.
 A: The diffusion equation
$${dc\over dt} = D\Delta c$$
is the result of two equations:
1- continuity equation (conservation of the particles)
 $${dc\over dt}=-\nabla\cdot\textbf{j}$$ 
2 - where $\textbf{j}$ is the flux and is defined as
$\textbf{j} =-D\nabla c$
so that combining the two you get the diffusion equation.
As you can see from (2), the flux of particles is anti-parallel to the gradient of $c$ so that particles really move uphill.
That has a mainly a statistical reason: less particles comes from where the concentration is low with respect to where it is high, so you have a net flux towards smaller concentration.
A: From a global perspective we can think of the diffusion equation as the minimization problem
$$ \frac{dc}{dt} = - \frac{\delta \mathcal{F}}{\delta c} $$ 
Where we have $ \mathcal{F} = \int \frac{1}{2}|\nabla c |^{2} dV $. If nothing else this at least shows that the diffusion equation is driven by minimizing the gradient of the field. 
From a local perspective we can define a normal coordinate($n$) and a coordinate along the level curve$(s)$ of $c$ and expand the laplacian as 
$$ \nabla^{2} c = \partial_{nn} c  + \frac{\kappa \partial_{n} c }{1 + n\kappa} + \frac{\partial_{ss}c}{(1 + n \kappa)^{2}} - \frac{n\partial_{s}\kappa\partial_{s}c }{(1+n\kappa)^{3}} $$
Where $\kappa$ is the curvature of the curve. Since we expanded along a level curve of $c$ the partial derivative along the curve vanish and we find that the evolution of the curve is only dependent on the normal coordinate and the curvature of the curve.
Unfortunately I am not entirely sure this answers your question. 
