**1. Introduction**

In recent years, the optical properties of colloidal suspensions (nanofluids) have been actively studied [1–6]. Researchers are particularly interested in nonlinear optical effects that are realized in such media. In particular, studies have focused on four-wave interactions and the self-action of light waves [7–10]. Without detailed knowledge of the optical properties of nanofluids, it is impossible to create next-generation solar collectors [11–15]. For example, [16] summarized the results of studies on the nanocolloids of ionic liquids (i.e., ionic liquids with nanoparticles in suspension), which can be directly applied to convective heat transfer. In [17], machine learning was used to develop Gaussian process regression models to describe the statistical correlations between the thermal conductivity and physical parameters of two-phase nanofluid components. For this purpose, approximately 300 samples of nanofluids, dispersions of metal, and ceramic nanoparticles in water, ethylenecol, and transformer oil have been investigated. The modeling approach demonstrates a high degree of accuracy and stability, facilitating efficient and inexpensive thermal conductivity estimates. Work [18] considered a liquid consisting of a stable colloidal suspension of magnetic maghemite nanoparticles in water. It has been found that these nanoparticles constitute an excellent absorber of solar radiation and simultaneously an amplifier of thermoelectric power output with a very small volume fraction when the liquid

**Citation:** Livashvili, A.I.; Krishtop, V.V.; Vinogradova, P.V.; Karpets, Y.M.; Efremenko, V.G.; Syuy, A.V.; Kuzmichev, E.N.; Igumnov, P.V. Appearance of a Solitary Wave Particle Concentration in Nanofluids under a Light Field. *Nanomaterials* **2021**, *11*, 1291. https://doi.org/ 10.3390/nano11051291

Academic Editors: Patrice Estellé and S M Sohel Murshed

Received: 7 April 2021 Accepted: 11 May 2021 Published: 14 May 2021

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is heated from above. These results demonstrate that the investigated nanofluid has great potential as a coolant for the co-production of heat and energy in completely new hybrid flat solar thermal collectors, for which top heating geometry is required. The main mechanisms for optical nonlinearity in these cases are the phenomena of thermal diffusion and electrostriction of nanoparticles [19,20]. Despite the many studies on this problem [21–25], several questions still remain. In particular, the dynamics of the concentration of nanofluid particles are unknown in the presence of concentration dependences on the coefficients of the thermal conductivity, viscosity, and absorption of radiation of the medium. Providing a theoretical description of the processes of heat and mass transfer for the nanofluid and radiation system is fraught with serious mathematical difficulties that are associated with the search for analytical solutions of the corresponding nonlinear equations. In this study, we developed a theoretical model for the dynamics of the concentration of nanoparticles in a liquid-phase medium when subjected to constant-intensity laser irradiation. Further, the study considers the dependence of the coefficients of absorption of radiation, thermal conductivity, and viscosity of the medium on the concentration of nanoparticles. It should be noted that, in the works cited above, the dynamics of the concentration of nanoparticles were studied assuming constant values of these coefficients. tration of nanoparticles. It should be noted that, in the works cited above, the dynamics of the concentration of nanoparticles were studied assuming constant values of these coefficients. **2. Theoretical Model**  We consider that the particle sizes satisfy the following condition: *a*0 << λ, where *a*0 is the linear size; and λ is the wavelength of light. Thus, we do not consider diffraction and light scattering processes. We also exclude the processes associated with particle sedimen-

*Nanomaterials* **2021**, *11*, x FOR PEER REVIEW 2 of 8

these nanoparticles constitute an excellent absorber of solar radiation and simultaneously

an amplifier of thermoelectric power output with a very small volume fraction when the

liquid is heated from above. These results demonstrate that the investigated nanofluid has

great potential as a coolant for the co-production of heat and energy in completely new

hybrid flat solar thermal collectors, for which top heating geometry is required. The main

mechanisms for optical nonlinearity in these cases are the phenomena of thermal diffusion

and electrostriction of nanoparticles [19,20]. Despite the many studies on this problem

[21–25], several questions still remain. In particular, the dynamics of the concentration of

nanofluid particles are unknown in the presence of concentration dependences on the co-

efficients of the thermal conductivity, viscosity, and absorption of radiation of the me-

dium. Providing a theoretical description of the processes of heat and mass transfer for

the nanofluid and radiation system is fraught with serious mathematical difficulties that

are associated with the search for analytical solutions of the corresponding nonlinear

equations. In this study, we developed a theoretical model for the dynamics of the con-

centration of nanoparticles in a liquid-phase medium when subjected to constant-inten-

sity laser irradiation. Further, the study considers the dependence of the coefficients of

absorption of radiation, thermal conductivity, and viscosity of the medium on the concen-

field in the medium, and are then used to determine the heat- and mass-transfer processes

where *T* is the temperature of the medium; *C* is the volume concentration of the medium;

We define the system of balanced equations for heat conduction and the mass of na-

= ൫ሺሻ ሬሬሬሬሬሬሬሬሬሬሬሬሬ⃗൯+ሺሻ, (1)

(*C*) is the absorption coefficient of the

= ൫ ሬሬሬሬሬሬሬሬሬሬሬሬሬ⃗൯+் ÷ ൫ሺ1−ሻ ሬሬሬሬሬሬሬሬሬሬሬሬሬ⃗൯−· ሬሬሬሬሬ⃗ ሬሬሬሬሬሬሬሬሬሬሬሬሬ⃗, (2)

α
