**2. Theoretical Model** tation.

We consider that the particle sizes satisfy the following condition: *a*<sup>0</sup> << λ, where *a*<sup>0</sup> 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 sedimentation. Let us consider a liquid-phase medium with nanoparticles irradiated by a light beam

Let us consider a liquid-phase medium with nanoparticles irradiated by a light beam of intensity *I*<sup>0</sup> that is uniformly distributed over a region (Figure 1). of intensity *I*0 that is uniformly distributed over a region (Figure 1).

**Figure 1.** Geometry of the problem.

**Figure 1.** Geometry of the problem. Temperature and concentration gradients arise as a result of the action of the light Temperature and concentration gradients arise as a result of the action of the light field in the medium, and are then used to determine the heat- and mass-transfer processes (Soret effect). These phenomena are described by a system of balanced equations for the temperature and particles [26,27].

noparticles transferred as follows:

*λ*(*C*) is the thermal conductivity of the medium;

We define the system of balanced equations for heat conduction and the mass of nanoparticles transferred as follows:

$$\mathcal{C}\_{p}\rho\frac{\partial T}{\partial t} = \operatorname{div}(\lambda(\mathbb{C})\,\,(\operatorname{grad}T)) + \mathfrak{a}(\mathbb{C})I\_{0} \tag{1}$$

$$\frac{\partial \mathbb{C}}{\partial t} = \operatorname{div}(\mathsf{D} \mathsf{grad} \mathsf{C}) + D\_T \operatorname{div}(\mathsf{C}(\mathsf{1} - \mathsf{C}) \mathsf{grad} \mathsf{T}) - \vec{V} \cdot \mathsf{grad} \mathsf{C},\tag{2}$$

where *T* is the temperature of the medium; *C* is the volume concentration of the medium; *λ*(*C*) is the thermal conductivity of the medium; *α*(*C*) is the absorption coefficient of the light wave; *D* is the diffusion coefficient of nanoparticles; *D<sup>T</sup>* Tis the thermal diffusion coefficient; *V* is the concentration convection velocity; and *C<sup>p</sup>* and *ρ* are known thermophysical constants. It should be noted that, in Equation (2), we take into account the incompressibility of the nanofluid: *div* → *V* = 0 [27].

We now consider the one-dimensional case, neglecting the Dufour effect owing to its small contribution. We do not consider flows caused by the forces of pressure on the particles from the side of the light field. In further calculations, we assume:

$$
\Delta \operatorname{div} \left( \lambda(\mathbb{C}) \frac{\partial T}{\partial \mathbf{x}} \right) \approx \lambda(\mathbb{C}) \frac{\partial^2 T}{\partial \mathbf{x}^2}, \\
\operatorname{div} \left( D \frac{\partial \mathbb{C}}{\partial \mathbf{x}} \right) \approx D \frac{\partial \mathbb{C}}{\partial \mathbf{x}^\prime} \, \tag{3}
$$

$$\operatorname{div}\left(\mathbb{C}(1-\mathbb{C})\overset{\rightarrow}{grad}T\right) \approx \mathbb{C}(1-\mathbb{C})\frac{\partial^2 T}{\partial \mathfrak{x}^2},\tag{4}$$

The validity of these approximations can be verified by direct calculations. We study the dynamics of nanoparticles against the background of the stationary temperature of the medium, i.e., *∂T*/*∂t* = 0 (thermal processes are assumed to be 2–3 orders of magnitude faster than diffusion). We focus on processes with *C* 1; this inequality ensures that the coagulation (coalescence) of nanoparticles can be disregarded.

According to theoretical and experimental studies [28,29], the concentration dependence of the thermal conductivity of a medium at low concentrations can be considered to be linear, as follows:

$$
\lambda(\mathbb{C}) = \lambda\_0 (1 + p\mathbb{C}), \tag{5}
$$

where *λ*<sup>0</sup> is the value of the thermal conductivity coefficient of the fluid (without nanoparticles), and p is a linear coefficient. We consider the concentration dependence of the light absorption coefficient to be of the form: α = *βC* (where *β* exceeds zero). Given the stationary temperature regime, the approximations (Equations (4) and (5)), and low concentration, we obtain the following from the heat equation:

$$\frac{\partial^2 T}{\partial \mathbf{x}^2} = \frac{-\beta \mathbf{C}}{\lambda\_0 (1 + p\mathbf{C})} I\_0 \approx -\frac{\beta \mathbf{C}}{\lambda\_0} I\_0 (1 - p\mathbf{C}) (p\mathbf{C} < 1),\tag{6}$$

Using the approximations in Equations (3), (4) and (6), Equation (2) can be rewritten as follows:

$$\frac{\partial \mathcal{C}}{\partial t} = D \frac{\partial^2 \mathcal{C}}{\partial \mathbf{x}^2} - \frac{D\_T \beta I\_0}{\lambda\_0} (1 - p\mathcal{C}) \mathcal{C}^2 - V \frac{\partial \mathcal{C}}{\partial \mathbf{x}'} \tag{7}$$

For a complete description of the transport processes in the system under consideration, Equation (7) must be supplemented by the Navier-Stokes equation (to determine the velocity, *V*). In this case, the formulated problem can be solved numerically [25]. However, here, we use a different approach to derive the analytical solution. In particular, we represent the convective velocity in the following form:

$$V(\mathbb{C}) = \frac{\eta(\mathbb{C})}{\rho(\mathbb{C})l'} \tag{8}$$

where *η*(*C*) is the dynamic viscosity coefficient of the nanofluid; *ρ*(*C*) is its density; and *l* is the characteristic length of the system, the value of which is determined later.

We consider the dependence of the viscosity coefficient on concentration to be linear, such that:

$$
\eta(\mathbb{C}) = \eta\_0 (1 + \gamma \mathbb{C}),
\tag{9}
$$

where *η*<sup>0</sup> is the value of the viscosity coefficient of the base fluid devoid of nanoparticles. A similar dependence was obtained theoretically and experimentally, as confirmed in previous studies [28–30]. As for *ρ*(*C*), a linear dependence on concentration is also permissible here [31,32]:

$$
\rho = \rho\_0 (1 + \chi \mathbb{C}),
\tag{10}
$$

where *ρ*<sup>0</sup> is the average density of the medium, and *χ* is the coefficient of the concentration expansion. As *γ χ* is real, we consider the density dependence on concentration to be insignificant.

Therefore, the expression for the velocity (Equation (8)) can be represented using Equations (9) and (10):

$$V(\mathbb{C}) = \frac{\eta\_0 (1 + \gamma \mathbb{C})}{\rho\_0 l (1 + \chi \mathbb{C})} \approx \frac{\eta\_0}{\rho\_0 l} (1 + \gamma \mathbb{C}),\tag{11}$$

As a result, the diffusion equation (Equation (7)) can be rewritten as follows:

$$\frac{\partial \mathcal{C}}{\partial t} = D \frac{\partial^2 \mathcal{C}}{\partial \mathbf{x}^2} - \frac{\eta\_0}{l \rho\_0} (1 + \gamma \mathcal{C}) \frac{\partial \mathcal{C}}{\partial \mathbf{x}} - \frac{D\_T \beta I\_0}{\lambda\_0} \mathcal{C}^2 (1 - p\mathcal{C}),\tag{12}$$

We now introduce the dimensionless variables and parameterize Equation (12). As a result, we obtain:

$$\frac{\partial \mathbb{C}}{\partial \mathbf{r}} = \frac{\partial^2 \mathbb{C}}{\partial y^2} - \delta \frac{\partial \mathbb{C}}{\partial y} - \delta \gamma \mathbb{C} \frac{\partial \mathbb{C}}{\partial y} - \mathbb{C}^2 (1 - p\mathbb{C}), \tag{13a}$$

The following notation is accepted here:

$$\tau = \frac{\mathbf{S}\_T D \beta I\_0}{\lambda\_0} t, y = \frac{1}{\sqrt{b}} \mathbf{x}, b = \frac{\lambda\_0}{\mathbf{S}\_T \beta I\_0}, \sqrt{b} = l, \delta = \frac{\eta\_0}{\rho\_0 D} \tag{13b}$$

Thus, we demonstrate that light-induced thermal diffusion in nanofluids, in the lowparticle-concentration approximation, against the background of a steady temperature, and taking into account concentration convection, can be described by nonlinear Equation (13a), which differs from the Burgers–Huxley equation [33] owing to the derivative in the last linear term.

First, we consider the two spatially homogeneous stationary states derived from *C* 2 (1 − *pC*) = 0, which correspond to the roots of the equation, namely, *C*<sup>1</sup> = *C*<sup>2</sup> = 0, *C*<sup>3</sup> = 1/*p*(*p* > 1). The kinetics of a dissipative system strongly depend on the stabilities of the stationary states. In our case, the states *C* = *C*1,2 are twofold degenerate and unstable (they contain derivatives from the source *F* 0 (*C*) > 0), whereas state *C* = *C*<sup>13</sup> is stable. Thus, the medium studied herein is not bistable, unlike that studied by Ognev et al. [34].

We note that similar parabolic equations with cubic nonlinearities have been considered in previously published studies, in which they were applied to a model dissipative medium with arbitrary parameters [34], and to a nanofluid + radiation system [35,36]. We look for particular solutions in the form of the Cole-Hopf transform [36]:

$$\mathbf{C}(y,\tau) = \frac{\mathcal{W}'\_y}{\mathcal{W}} \cdot \mu\_\prime \mathcal{W} = \mathcal{W}(y,\tau),\tag{14}$$

where *µ* is a parameter, and 0 denotes the derivative.

By substituting Equation (14) into (13a) and equating the coefficients for the various powers of *W* to zero, we obtain an overdetermined system of equations for function *W*(*y*, *µ*):

$$\mathsf{W}\_{y\tau}^{\prime\prime} = \mathsf{W}\_{yyy}^{\prime\prime} - \delta \mathsf{W}\_{yy\prime}^{\prime\prime} \tag{15}$$

$$\mathcal{W}'\_{\tau} = \Im \mathcal{W}''\_{yy} + \delta \gamma \mu \mathcal{W}''\_{yy} + (\mu - \delta) \mathcal{W}'\_{y\prime} \tag{16}$$

$$p\mu^2 + \delta\mu + \mathcal{2} = 0,\tag{17}$$

From the last equation of this system, we obtain the values of parameter *µ*:

$$
\mu\_{1,2} = \frac{1}{2p'} \tag{18}
$$

The estimates of parameters *γ* and *δ*, which are provided below, show that roots *µ*1,2 are real. Furthermore, by integrating Equation (15) with respect to variable *y*, we obtain:

$$\mathcal{W}'\_{\tau} = \mathcal{W}''\_{yy} - \delta \mathcal{W}'\_y + \mathcal{C}\_1(\tau), \tag{19}$$

Using Equation (16), we obtain:

$$(\mathfrak{L} + \delta \gamma) \mathcal{W}\_{yy}^{\prime\prime} + \mu \mathcal{W}\_y^{\prime} + \mathbb{C}\_1(\tau) = \mathbf{0},\tag{20}$$

The solution for this equation can be represented as:

$$\mathcal{W}(y,\tau) = \frac{\mathbb{C}\_1(\tau)}{\mu}y + \mathbb{C}\_2(\tau) + \mathbb{C}\_3(\tau)\exp(-\omega y),\tag{21}$$

where *ω* = *µ*/(2 + *δγµ*).

*Ci*(*τ*) can be determined using Equations (20) and (21), and it can be used to express the solution for function *W*(*y*, *τ*) as follows:

$$\mathcal{W}(y,\tau) = \widetilde{\mathcal{C}\_1} \left( \left( 1 - \frac{\delta}{\mu} \right) + \frac{1}{\mu} y \right) \tau + \widetilde{\mathcal{C}\_2} + \widetilde{\mathcal{C}\_3} \exp(\omega(\omega + \delta)\tau - \omega y), \tag{22}$$

where *C*e *<sup>i</sup>* are constants.

According to Equations (14) and (22), the desired concentration can be represented as:

$$\mathcal{C}(y,\tau) = \mu \frac{\widetilde{\mathcal{C}\_1} - \omega \widetilde{\mathcal{C}\_3} \exp(\omega(\omega + \delta)\tau - \omega y)}{\widetilde{\mathcal{C}\_1}((\mu - \delta)\tau + y) + \widetilde{\mathcal{C}\_2} + \widetilde{\mathcal{C}\_3} \exp(\omega(\omega + \delta)\tau - \omega y)},\tag{23}$$
