Viscous Fluids - Fluid Simulation for Computer Graphics - page 107

Then the change in the dot-product, ignoring O (Δ t 2 )terms,is

u Δ x )=Δ t Δ x T ∇ u T Δ y.

Δ t (Δ x ·∇ u Δ y +Δ y ·∇ u + ∇

That is, the rate of change of dot-products of vectors in the flow is de-
termined by the symmetric part of the velocity gradient, the matrix D =
1

u T ). This is called the strain rate tensor or rate of strain, since
it's measuring how fast strain —the total deformation of the continuum—is
changing.
Incidentally, the rest of the velocity gradient, the skew-symmetric part

2 (

∇ u + ∇

u T ) naturally has to represent the other source of velocity dif-
ferences in the flow:

1

2 (

∇ u −∇

rotation. We'll explore this further later in the

topic.
We'll also immediately point out that for incompressible fluids, which
is all we focus on in this topic, the trace of D (the sum of the diagonal
entries, denoted tr( D )) is simply u =0.

We are looking for a symmetric tensor τ to model stress due to viscosity;
the rate of strain tensor D is symmetric and measures how fast the fluid is
deforming. The obvious thing to do is assume τ depends linearly on D .Flu-
ids for which there is a simple linear relationship are called Newtonian .Air
and water are examples of fluids which are, to a very good approximation,
Newtonian. However, there are many liquids (generally with a more com-
plex composition) where a non-linear relationship is essential; they go un-
der the catch-all category of non-Newtonian fluids. 4 Wewon'tgointoany
more detail, but observe that two classes of non-Newtonian fluids, shear-
thickening and shear-thinning fluids, can be easily modeled with a viscosity
coecient η that is a function of

∇·

D F , the Frobenius norm of the strain

rate: 5

3

D i,j .

D F =

i,j =1

4 Also sometimes included in the non-Newtonian class are viscoelastic fluids, which
blur the line between fluid and solid as they can include elastic forces that seek to
return the material to an “undeformed” state—in fact these are sometimes best thought
of instead as solids with permanent ( plastic ) deformations. You might refer to the article
by Goktekin et al. [Goktekin et al. 04] and Irving's thesis [Irving 07] for a fluid-centric
treatment in graphics.
5 Technically this is assuming again that the fluid is incompressible, so the trace of
D is zero, which means it represents only shearing deformations, not expansions or
contractions.