**8. References**

24 Will-be-set-by-IN-TECH

−8

−7

−6

log10L2( Error)

TEMPERATURE

−7.8 −7.6 −7.4 −7.2 −7 −6.8 −6.6 −6.4 −6.2 −6 −5.8 −9

Fig. 10. The self-consistent IMEX method versus a classic IMEX method in terms of the time

We have presented a self-consistent implicit/explicit (IMEX) time integration technique for solving the Euler equations that posses strong nonlinear heat conduction and very stiff source terms (Radiation hydrodynamics). The key to successfully implement an implicit/explicit algorithm in a self-consistent sense is to carry out the explicit integrations as part of the non-linear function evaluations within the implicit solver. In this way, the improved time accuracy of the non-linear iterations is immediately felt by the explicit algorithm block and the more accurate explicit solutions are readily available to form the next set of non-linear residuals. We have solved several test problems that use both of the low and high energy density radiation hydrodynamics models (the LERH and HERH models) in order to validate the numerical order of accuracy of our scheme. For each test, we have established second order time convergence. We have also presented a mathematical analysis that reveals the analytical behavior of our method and compares it to a classic IMEX approach. Our analytical findings have been supported/verified by a set of computational results. Currently, we are exploring more about our multi-phase IMEX study to solve multi-phase flow systems that

posses tight non-linear coupling between the interface and fluid dynamics.

log10Δ<sup>t</sup>

−5

−4

−8 −7.5 −7 −6.5 −6 −5.5 −9

log10Δ<sup>t</sup>

Classic IMEX First Order S−Cons. IMEX Sec. Order

Classic IMEX First Order S−Cons. IMEX Sec. Order

VELOCITY

−8 −7.5 −7 −6.5 −6 −5.5 −8.5

log10Δ<sup>t</sup>

Classic IMEX First Order S−Cons. IMEX Sec. Order

DENSITY

−8 −7.5 −7 −6.5 −6 −5.5 −5 −4.5

−8.5 −8 −7.5 −7 −6.5 −6 −5.5 −5

log10L2( Error)

convergence.

**6. Conclusion**

log10L2( Error)

Anderson, J. (1990). *Modern Compressible Flow.*, Mc Graw Hill.


P. Domínguez-García1 and M.A. Rubio<sup>2</sup>

*Spain*

**14**

<sup>1</sup>*Dep. Física de Materiales, UNED, Senda del Rey 9, 28040. Madrid* <sup>2</sup>*Dep. Física Fundamental, UNED, Senda del Rey 9, 28040. Madrid*

**of Low-Confinement Conditions** 

**and Electrostatics Interactions** 

The study of colloidal dispersions of micro-nano sized particles in a liquid is of great interest for industrial processes and technological applications. The understanding of the microstructure and fundamental properties of this kind of systems at microscopic level is also

**Hydrodynamics on Charged Superparamagnetic** 

**Microparticles in Water Suspension: Effects** 

However, a colloidal suspension must be placed somewhere and the dynamics of the micro-particles can be modified as a consequence of the confinement, even if we have a low-confinement system. The hydrodynamics interactions between particles and with the enclosure's wall which contains the suspension are of extraordinary importance to understanding the aggregation, disaggregation, sedimentation or any interaction experienced by the microparticles. Aspects such as corrections of the diffusion coefficients because of a hydrodynamic coupling to the wall must be considered. Moreover, if the particles are electrically charged, new phenomena can appear related to electro-hydrodynamic coupling. Electro-hydrodynamic effects (Behrens & Grier (2001a;b); Squires & Brenner (2000)) may have a role in the dynamics of confined charged submicron-sized particles. For example, an anomalous attractive interaction has been observed in suspensions of confined charged particles (Grier & Han (2004); Han & Grier (2003); Larsen & Grier (1997)). The possible explanation of this observation could be related with the distribution of surface's charges of the colloidal particles and the wall (Lian & Ma (2008); Odriozola et al. (2006)). This effect could be also related to an electrostatic repulsion with the charged quartz bottom wall or to a spontaneous macroscopic electric field observed on charged colloids (Rasa & Philipse (2004)). In this work, we are going to describe experiments performed by using magneto-rheological fluids (MRF), which consist (Rabinow (1948)) on suspensions formed by water or some organic solvent and micro or nano-particles that have a magnetic behaviour when a external magnetic field is applied upon them. Then, these particles interact between themselves forming aggregates with a shape of linear chains (Kerr (1990)) aligned in the direction of the magnetic field. When the concentration of particles inside the fluid is high enough, this microscopic behaviour turns to significant macroscopic

**1. Introduction**

useful for biological and biomedical applications.

