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Here, $(n)$ and $(n+1)$ denote the temperature field at the current and next time step, respectively.
Multiplying by test function $w$:
$$\frac{1}{\Delta t} \left(T^{(n+1)}-T^{(n)}\right) \cdot w
= \frac{k}{\rho C_p} \nabla^2 T \cdot w
- (\mathbf{u}\cdot\nabla)\, T \cdot w$$
Then integrate over domain $\Omega$:
$$\frac{1}{\Delta t} \int\limits_\Omega \left(T^{(n+1)}-T^{(n)}\right) \cdot w
= \frac{k}{\rho C_p} \int\limits_\Omega \nabla^2 T \cdot w
- \int\limits_\Omega (\mathbf{u}\cdot\nabla)\, T \cdot w$$
Integration by parts and
Let's take a closure look at right hands side's first term. We can notice that it's a part of integration by parts equation. Let's write its obviously:
where $\partial\Omega$ denotes the boundary of the domain $\Omega$ and $\mathbf{n}$ is the outward unit normal vector on the boundary. Assuming that the temperature is known on the boundary, which means that the test function $w$ equals zero on the boundary, the term becomes zero, and we are left with: