Governing equationsΒΆ
I consider the incompressibility constraint:
\[\der{u_i}{x_i}
=
0\]
and the momentum balance:
\[\der{u_i}{t}
=
-
u_j \der{u_i}{x_j}
-
\der{p}{x_i}
+
\frac{1}{Re}
\der{}{x_j} \der{u_i}{x_j}
+
a_i\]
to describe the motion of the fluid, where \(Re\) is the Reynolds number. Also a passive scalar field \(T\) is transported, which is governed by the advection-diffusion equation:
\[\der{T}{t}
=
-
u_j \der{T}{x_j}
+
\frac{1}{Re Sc}
\der{}{x_j} \der{T}{x_j},\]
where \(Sc\) is the Schmidt number (the ratio of the fluid diffusivity to the scalar diffusivity).
For later convenience, I consider the advective terms in the divergence form:
\[\der{u_j q}{x_j},\]
where the incompressibility constraint is used:
\[\der{u_j q}{x_j}
\equiv
q \der{u_j}{x_j}
+
u_j \der{q}{x_j}.\]