Hyperthermia, an alternative medical approach aiming at locally increasing the temperature of a tumor, can cause the “death” of cancer cells or the sensitization of them to chemotherapeutic drugs and radiation. In contrast with the conventional treatments, hyperthermia provokes no injury to normal tissues. In particular, magnetic hyperthermia can utilize iron oxide nanoparticles, which can be administered intravenously to heat tumors under an alternating magnetic field. Currently, there is no theoretical model in the relative literature for the effective thermal conductivity of blood and magnetic nanoparticles. The scope of the present study is twofold: (a) development of a theoretical relationship, based on experimental findings and blood structure and (b) study of the laminar natural convection in a simplified rectangular porous enclosure, by using the asymptotic expansions method for deriving ordinary differential equations of the mass, momentum and energy balances, as a first approach of investigating heat transfer and providing theoretical guidelines. In short, the thermal conductivity of the resulting bio-nanofluid tends to increase by both increasing the concentration of the nanoparticles and the temperature. Furthermore, the heat transfer is enhanced for more intense internal heating (large Rayleigh numbers) and more permeable media (large Darcy numbers), while larger nanoparticle concentrations tend to suppress the flow.

A water purification method using a static electric field that may drift the dissolved ions of heavy metals is proposed here. The electric field force drifts the positively charged metal ions of continuously flowing contaminated water to one sidewall, where the negative electrode is placed, leaving most of the area of the duct purified. The steady-state ion distributions, as well as the time evolution in the linear regime, are studied analytically and ion concentration distributions for various electric field magnitudes and widths of the duct are reported. The method performs well with a duct width less than 10⁻³ m and an electrode potential of 0.26 V or more. Moreover, a significant reduction of more than 90% in heavy metals concentration is accomplished in less than a second at a low cost.

In this study, we examine the movement of ions that are in a water solution which flows along a duct, due to the existence of an electric field, taking into account the size of the ions, a phenomenon known as the steric effect. We compare the results from the above model with the classical one (the one that uses the Boltzmann distribution where ions are considered dimensionless) for various parameters such as surface charge density, electric field and differential capacitance. It is shown that for dilute water solutions (10¹⁹–10²⁴ ions/m³ final concentration at the center of the duct), with ions of valence $z=1$ (let us say saline water), steric effects become important for potentials greater than 1 V, and the phenomenon is more pronounced at higher concentrations. Furthermore, the steric effect model is applied to the calculation of the percentage of reduction in ion concentration in the main volume of the solution as a function of duct width for various electrode potentials and initial ion concentrations. Removal times are also calculated using Modified PNP equations which take into account steric effects. It is found that with a potential of 2.6 V, a 96% reduction in ions is achieved in the main volume of the solution for duct width 0.1 mm for 10²¹ ions/m³ final concentration at the center of the duct within approximately 1.6 s, while the percentage drops to 80% for duct width 1 mm. For smaller potentials, no noticeable decrease in concentration is observed, while for higher potentials, there are more impressive results, but we must be very careful because there is the case of other electrochemical phenomena taking place. The results are better when reducing the width of the duct, but relatively large widths are considered for the method to be practically applicable. With the increase in the concentration of the ions, their reduction percentage in the main volume of the solution decreases but remains significant up to 10²³ ions/m³ final concentration at the center of the duct. In addition, the completion time is shown to be proportional to the duct width. Therefore, for example, with the other parameters the same (2.6 V, 10²¹ ions/m³) but with $L~1\mathrm{mm}$, the completion time can be estimated to be approximately 16 s. This observation enables us to estimate the completion time for different duct widths, eliminating the need for repeated numerical computation of the MPNP equations.

An algorithm for the calculation of steady-state flowing under uncertain conditions is introduced in this work in order to obtain a probabilistic distribution of uncertain problem parameters. This is particularly important for problems with increased uncertainty, as typical deterministic methods are not able to fully describe all possible flow states of the problem. Standard methods, such as polynomial expansions and Monte Carlo simulations, are used for the formation of the generalized problem described by the incompressible Navier-Stokes equations. Since every realization of the uncertainty parameter space is coupled with non-linear terms, an incremental iterative procedure was adopted for the calculation. This algorithm adopts a Jacobi-like iteration methodology to decouple the equations and solve them one by one until there is overall convergence. The algorithm was tested in a typical artery geometry, including a bifurcation with an aneurysm, which consists of a well-documented biological flow test case. Additionally, its dependence on the uncertainty parameter space, i.e., the inlet velocity distribution, the Reynolds number variation, and parameters of the procedure, i.e., the number of polynomial expansions, was studied. Symmetry exists in probabilistic theories, similar to the one adopted by the present work. The results of the simulations conducted with the present algorithm are compared against the same but unsteady flow with a time-dependent inlet velocity profile, which represents a typical cardiac cycle. It was found that the present algorithm is able to correctly describe the flow field, as well as capture the upper and lower limits of the velocity field, which was made periodic. The comparison between the present algorithm and the typical unsteady one presented a maximum error of ≈2% in the common carotid area, while the error increased significantly inside the bifurcation area. Moreover, “sensitive” areas of the geometry with increased parameter uncertainty were identified, a result that is not possible to be obtained while using deterministic algorithms. Finally, the ability of the algorithm to tune the parameter limits was successfully tested.

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