# HowTo Adding a new transport equation

Suppose you want to solve and additional scalar transport equation for the scalar ψ by adding it to an existing solver. The equation has the form

$\frac{\partial}{\partial t} \left(\rho \psi \right) + \nabla \cdot \phi \psi - \nabla \cdot \Gamma \nabla \psi = S_{\psi}$

where ρ and Γ are defined as dimensionedScalar and are retrieved from a dictionary using the lookup member function. Sφ is the source term.

To implement the equation, just add the line:

dimensionedScalarField psi;

to the createFields.H file of the solver,

Then, solve the equation by adding the following lines in the point of the solver where you want the equation to be solved.


solve
(
fvm::ddt(rho, psi)
+ fvm::div(phi, psi)
- fvm::laplacian(gamma, psi)
==
S_psi
);

In this example, the source term is treated explicitly. To manage it implicitly, OpenFOAM provides the Sp and the SuSp functions. Suppose we can write it as Sk = Κψ. The code lines to solve the same equation with an implicit treatment of the source term are:


solve
(
fvm::ddt(rho, psi)
+ fvm::div(phi, psi)
- fvm::laplacian(gamma, psi)
==
fvm::Sp(kappa, psi)
);

SuSp(kappa, psi) can be used to discretise the source term implicitly or explicitly according to the sign of kappa.