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0.1. The Basic Partial Differential Equation Models 3

has to be closed by a (phenomenological) constitutive law, postulating a
relation between J and u.
Assume Ω is a container filled with a fluid in which a substance is dis-

solved. If u is the concentration of this substance, then S(u)= u and a
=kg. The description of J depends on the processes involved. If the fluid
is at rest, then flux is only possible due to molecular diffusion, i.e., a flux
from high to low concentrations due to random motion of the dissolved
particles. Experimental evidence leads to
(1)
J = −K∇u (0.4)
2
with a parameter K> 0 [m /s],the molecular diffusivity. Equation (0.4)
is called Fick’s law.
In other situations, like heat conduction in a solid, a similar model occurs.
Here, u represents the temperature, and the underlying principle is energy
conservation. The constitutive law is Fourier’s law, which also has the form
(0.4), but as K is a material parameter, it may vary with space or, for
anisotropic materials, be a matrix instead of a scalar.
Thus we obtain the diffusion equation
∂ t u −∇· (K∇u)= Q. (0.5)
If K is scalar and constant — let K = 1 by scaling —, and f := Q is
independent of u, the equation simplifies further to
∂ t u − ∆u = f,
where ∆u := ∇·(∇u) . We mentioned already that this equation also occurs
in the modelling of heat conduction, therefore this equation or (0.5) is also
called the heat equation.
If the fluid is in motion with a (given) velocity c then (forced) convection
of the particles takes place, being described by
(2)
J = uc , (0.6)
i.e., taking both processes into account, the model takes the form of the
convection-diffusion equation

∂ t u −∇· (K∇u − cu)= Q. (0.7)
The relative strength of the two processes is measured by the P´eclet
number (defined in Section 0.4). If convection is dominating one may ignore
diffusion and only consider the transport equation
∂ t u + ∇· (cu)= Q. (0.8)
The different nature of the two processes has to be reflected in the models,
therefore, adapted discretization techniques will be necessary. In this book
we will consider models like (0.7), usually with a significant contribution
of diffusion, and the case of dominating convection is studied in Chapter
9. The pure convective case like (0.8) will not be treated.