How can I modify the Eikonal equation to have smooth iso-contours? I am solving the Eikonal Equation in 2D:
$ | \nabla T(x,y)|=1/V(x,y) $
for the traveltime, T(x,y), from a starting point:
$ T(x_0,y_0) = 0$.
The curves $ T(x,y)=C$ forms closed contours around the starting point.
However the Eikonal equation do not impose any smoothness requirements on the contours.
As an example consider the following case:

Figure: Black circle: starting point, color: velocity, V(x,y), red=fast, blue=slow.
The contours would have no problem getting trough the narrow red "gap". 
The way I look at this: the Eikonal equation models sound, and sound have no problems getting trough narrow spaces.
I would like instead to model something more "viscous" so that the contours were smooth and could NOT get trough such a narrow gap.
What would be the PDE for that?
And how would I solve it numerically?
Today I am using the Fast Marching Method. Would it possible to adapt this method to the new equation?
 A: You could model shear viscosity without changing the PDE, just by pre-processing the data $V$. The idea is that since a slow-moving layer of fluid will slow down its neighbors, the velocity $V$ must be continuous, and more precisely Lipschitz continuous: $$|V(x,y)-V(x',y')|\le L\sqrt{(x-x')^2+(y-y')^2}\tag1$$
for all pairs of points $(x,y)$ and $(x',y')$. So, the picture of velocity field should look  like this: 

where purple color indicates intermediate values between red and blue. The velocity being Lipschitz should make iso-contours $C^{1,1}$-smooth.
A natural way to preprocess $V$ is 
$$\widetilde V(x,y) = \min_{x',y'}\left\{V(x',y')+L\sqrt{(x-x')^2+(y-y')^2} \right\} \tag2$$
This guarantees $\widetilde V(x,y)\le V(x,y)$, which means that viscous fluid will not move faster than inviscous one would (makes sense). The value of $L$ is for you to choose: the smaller values of $L$ correspond to greater viscosity. In your example, $\widetilde V$ will be the same as $V$ in the blue region and also in most of the red region, except for some boundary layer in which $\widetilde{V}<V$ due to viscosity.
The cost of pre-processing can be reduced by observing that in (2) you only need to consider $(x',y')$ such that $\sqrt{(x-x')^2+(y-y')^2}\le (\max V-\min V)/L$. 
Also, check out Computational Science SE with its supply of fluid modeling questions. 
