## 3 Results and Discussion  
We have used numerical tools to solve Equation (11) or (24) with the physical parameters of magnetite. For the dependent variable \(\theta(\mathbf{r},t)\), i.e., the relative temperature elevation, we use the initial/boundary conditions,  
\[\theta(\mathbf{r},0)=0,\theta(0,t)=\theta(L,t)=0. \tag{26}\]  
A more realistic situation of three dimensions is supposed to mimic the fact that the ferrofluid is embedded in a medium that is supposed to be kept at a constant temperature \(T_{0}\) (\(\sim\)42-45\({}^{\circ}\) C); recall that \(\theta=(T-T_{0})/T_{0}\). This is, for instance, the situation of a suspension of magnetic nanoparticles injected in a tumor, where the boundary conditions above prevent overheating of the surrounding area. In this preliminary study, we only consider the case of a two-dimensional assembly of NMs, as shown in Figure 4, which may be either organized or randomly distributed in the matrix. In the present work, this does not matter since we do not include dipolar interactions between the NMs. The effect of the latter on the SAR was studied in Refs. [16; 17; 18] and references therein. More precisely, we consider assemblies of magnetite particles as studied experimentally in Ref. [21] and, later, theoretically in Ref. [39]. In Table 1, we collect all the relevant data used in the following figures. The intensity of the AC magnetic field is \(38.2\,\mathrm{kA/m}\) and its frequency is \(123\,\mathrm{kHz}\). Newton's coefficient was obtained in Ref. [39] by fitting the experimental data from Ref. [21]: \(l_{\mathrm{ff}}=330\,(\mathrm{W/K/kg})\). The NMs are lognormal distributed in size with mean diameter \(D_{m}=9\,\mathrm{nm}\) and standard deviation \(\delta=0.25\). [PERSON] et al. [21] used \(1.2\,\mathrm{mL}\) of magnetite ferrofluid (magnetite in water) and a concentration \(\rho_{\mathrm{Fe}}\simeq 8.6\,\mathrm{mg/mL}\simeq 8.6\,\mathrm{kg/m^{3}}\) of magnetite in the ferrofluid. The rest of the data related to the magnetization dynamics are given in Section 2.2. of Ref. [39]. In particular, the relaxation time related to the Brownian motion of NMs in the fluid is given and used to define the effective relaxation time. As discussed earlier, heat Equation (11) can be solved with free boundary conditions or with boundaries at fixed temperatures. The first case would apply in a situation where, for instance, we were interested in the spatial extent of heating. The second case with fixed boundaries would apply to a situation where the assembly is confined in some space brought into contact with a surrounding that is kept \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Substance** & \(\mathbf{c_{v}}\) **(J/K/kg)** & \(\mathbf{\kappa}\) **(W/K/m)** & \(\mathbf{\rho}\) **(kg/m\({}^{3}\))** & \(\mathbf{m}\) **(kg)** & \(\mathbf{V}\) **(m\({}^{3}\))** & \(\mathbf{K}\) **(J/m\({}^{3}\))** & \(\mathbf{M}_{s}\) **(A/m)** \\ \hline \hline Magnetic & 619 & 5.1 & 5200 & \(1.032\times 10^{-5}\) & \(1.985\times 10^{-9}\) & \(3.0\times 10^{4}\) & \(4.8\times 10^{5}\) \\ \hline Water & 4181 & 0.598 & 1000 & \(1.198\times 10^{-3}\) & \(1.198\times 10^{-6}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters from Refs. [21; 39]. Figure 4: Assembly of organized or randomly distributed nanoparticles. Next, the heat equation can also be applied to an assembly of NMs organized or not on a solid matrix, as shown in Figure 4. In the following, all results will be shown for an assembly of NMs that are randomly distributed in space. As already stressed in the Introduction, this work is not an exhaustive study of the effect of the numerous physical parameters of the problem at hand, namely those related with NM assembly (size, shape, distributions, and concentration) and external excitation (amplitude and frequency). The main objective instead is to propose a formalism and a toolbox for investigating the heat generation and transport in assemblies of NMs with the possibility to vary all the relevant parameters.