**3. Solution Analysis**

In this section, we examine the dependence of the determined exact solution (23) on parameters *<sup>µ</sup>*, *<sup>ω</sup>*, and *<sup>δ</sup>*. Parameter *<sup>δ</sup>* is estimated to be <sup>≈</sup> <sup>10</sup><sup>5</sup> by assuming that *<sup>η</sup>*<sup>0</sup> = 10−<sup>3</sup> kg/(m·s), *<sup>ρ</sup>*0= 10<sup>3</sup> kg/m<sup>3</sup> , and *D* = 10−<sup>11</sup> m2/s in Equation (13a). Furthermore, from the findings of previous studies [24–26], it follows that *p* ≈ 1 ÷ 1.5. Then, we must consider that 4*p δ* 2*γ* 2 in Equation (18). Therefore, for sufficiently accurate roots *µ*1,2, we obtain *µ*<sup>1</sup> = −2/*δ*, *ω*<sup>1</sup> = −2, *µ*<sup>2</sup> = −*δ*/*p*, *ω*<sup>2</sup> = −1/*p*.

Furthermore, by substituting the corresponding expressions for parameters and *δ* (their estimates) into Equation (23), we obtain two solutions:

$$\mathcal{C}\_{1}(y,\tau,\mu\_{1}) = \frac{2}{\delta\gamma} \frac{1 + 2c\_{3}\exp(-2(\delta\tau - y))}{(\delta\tau - y) + c\_{2} - c\_{3}\exp} \, \prime \tag{24}$$

$$\mathbb{C}\_2(y, \tau, \mu\_2) = \delta \gamma\_\prime \tag{25}$$

Here, *c*<sup>2</sup> and *c*<sup>3</sup> are the newly redefined constants.

Let us consider Equation (24), the graph of which is shown in Figure 2; we see that it is a function of the variable traveling wave, *z* = (*δτ* − *y*). On the graph, the solution is presented in the form of soliton-like pulses moving to the right (with increasing time). is a function of the variable traveling wave, = ሺ − ሻ. On the graph, the solution is presented in the form of soliton-like pulses moving to the right (with increasing time).

*Nanomaterials* **2021**, *11*, x FOR PEER REVIEW 5 of 8

Using Equation (16), we obtain:

the solution for function ሺ, ሻ as follows:

ሺ, ሻ = ଵ

ሺ, ሻ =

−2 ⁄ , ଵ = −2, ଶ = − ⁄ , ଶ = −1 ⁄ .

where =µ⁄ሺ2 + µሻ.

where ప ~ are constants.

**3. Solution Analysis** 

ఛ

ሺ2 + ሻ௬௬

The solution for this equation can be represented as:

ሺ, ሻ <sup>=</sup> ଵሺሻ

<sup>~</sup> ൭൬1 −

´

(their estimates) into Equation (23), we obtain two solutions:

Here, ଶ and ଷ are the newly redefined constants.

ଵሺ, , ଵሻ <sup>=</sup> <sup>2</sup>

൰ + <sup>1</sup>

ଵ

<sup>ᇱ</sup> = ௬௬

The estimates of parameters and , which are provided below, show that roots µଵ,ଶ

are real. Furthermore, by integrating Equation (15) with respect to variable , we obtain:

ᇱᇱ − ௬

ᇱᇱ + µ௬

ሺሻ can be determined using Equations (20) and (21), and it can be used to express

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

In this section, we examine the dependence of the determined exact solution (23) on pa-

Furthermore, by substituting the corresponding expressions for parameters and

Let us consider Equation (24), the graph of which is shown in Figure 2; we see that it

=

√

The speed depends on the thermodynamic, hydrodynamic, and optical characteris-

tics of the nanofluid + radiation system. It follows from Equations (26) and (13a) that, in

the case of anomalous thermal diffusion (ST < 0—nanoparticles move in a higher-temper-

ature region), the velocity acquires an imaginary term, which has no physical meaning.

We believe that this case requires a separate consideration, which we plan to carry out in

and consider water/silver as a nanofluid. Because the absorption coefficient is present

in Equation (26) through parameter (see Equation (13a)), its evaluation requires the fol-

=

12π

We numerically estimate the wavefront propagation velocity according to Equation (26)

ቆ<sup>ଶ</sup> − 1

Here, by assuming = 0.15 + 3.5, = 1.33 + 0.2*,* = 6.5 ∙ 10ିm, and =1∙

10ିସ , we obtain ≈ 2 ∙ 10ଷ . Furthermore, by substituting = 0.003 kg/m⋅s, =1∙

10ଷ, = 1 ∙ 10ହ W/m2, and = 0.5 W/m⋅K into Equations (26) and (13a), we obtain a

velocity estimate: ≈ 2 ∙ 10ିଷ m/s. The estimated velocity value depends on the initial

concentration distribution, which can be obtained from Equation (24) for τ = 0. It should

be noted that our approach cannot solve the problem analytically under arbitrary initial

1. Two exact analytical solutions of a nonlinear one-dimensional Burgers–Huxley-type

2. One of the solutions found was represented as a solitary wave. Both solutions were

tion of radiation by particles were found to be concentration dependent.

expressed in the form of traveling single-phase waves.

equation were obtained. These solutions describe the dynamics of the concentration

of nanoparticles in a liquid-phase medium by taking into account concentration con-

vection. In this case, the coefficients of thermal conductivity, viscosity, and absorp-

ଶ + 2ቇ,

. (26)

1 + 2ଷ ൫−2ሺ − ሻ൯

rameters , , and . Parameter is estimated to be ≈ 105 by assuming that = 10−<sup>3</sup>

kg/(m∙s), = 103 kg/m3, and *D* = 10−11 m2/s in Equation (13). Furthermore, from the find-

ings of previous studies [24–26], it follows that ≈ 1 ÷ 1.5. Then, we must consider that

4 ≪ ଶଶ in Equation (18). Therefore, for sufficiently accurate roots ଵ,ଶ, we obtain ଵ =

~ − ଷ~ ሺሺ+ሻ − ሻ

ଵ൫ሺµ−ሻ + ൯ + ~ଶ + ଷ~ ሺሺ+ሻ − ሻ

<sup>ᇱ</sup> + ଵሺሻ, (19)

<sup>ᇱ</sup> + ଵሺሻ = 0, (20)

<sup>µ</sup> +ଶሺሻ + ଷሺሻሺ−ሻ, (21)

൱ + ଶ~ + ଷ~ሺሺ+ሻ − ሻ, (22)

ሺ − ሻ + ଶ − ଷ , (24)

ଶሺ, , ଶሻ = , (25)

, (23)

**Figure 2.** Solution of Equation (24). ሺሺ ⁄ − 1ሻ−ሻ term. Clearly, the nature of curve ሺሻ in Equation (25) strongly de-

**Figure 2.** Solution of Equation (24). A distinctive feature of Equation (25) is the presence in the denominator of the (*δ*(*γ*/*p* − 1)*τ* − *y*) term. Clearly, the nature of curve *C*(*y*) in Equation (25) strongly depends on the *γ*/*p* ratio. When plotting the function, we set *γ*/*p* = 3 (see Figure 3). Note that the wave-pulse profiles are not similar in this case. pends on the ⁄ ratio. When plotting the function, we set ⁄ = 3 (see Figure 3). Note that the wave-pulse profiles are not similar in this case.

**Figure 3.** Solution profiles of the concentration wave obtained from Equation (25) with increasing **Figure 3.** Solution profiles of the concentration wave obtained from Equation (25) with increasing time.

(13a). As a result, we obtain:

time.

the future.

conditions.

**4. Conclusions** 

lowing equation [37]:

where =௧௦⁄௨ௗ , = + .

The velocity of the wave front of Equation (24) can be determined using Equation (13b). As a result, we obtain:

$$\nu = \frac{\eta\_0}{\rho\_0 \sqrt{b}}.\tag{26}$$

The speed depends on the thermodynamic, hydrodynamic, and optical characteristics of the nanofluid + radiation system. It follows from Equations (26) and (13b) that, in the case of anomalous thermal diffusion (S<sup>T</sup> < 0—nanoparticles move in a higher-temperature region), the velocity acquires an imaginary term, which has no physical meaning. We believe that this case requires a separate consideration, which we plan to carry out in the future.

Wenumerically estimate the wavefront propagation velocity according to Equation (26) and consider water/silver as a nanofluid. Because the absorption coefficient *β* is present in Equation (26) through parameter *b* (see Equation (13b)), its evaluation requires the following equation [37]:

$$\beta = \frac{12\pi}{\lambda} C I m (\frac{m^2 - 1}{m^2 + 2}) \lambda$$

where *m* = *mparticles*/*mf luid*, *m* = *n* + *ik*.

Here, by assuming *<sup>m</sup><sup>p</sup>* <sup>=</sup> 0.15 <sup>+</sup> 3.5*i*, *<sup>m</sup><sup>f</sup>* <sup>=</sup> 1.33 <sup>+</sup> 0.2*i*, *<sup>λ</sup>* <sup>=</sup> 6.5·10−<sup>7</sup> m, and *<sup>C</sup>* <sup>=</sup> <sup>1</sup>·10−<sup>4</sup> , we obtain *<sup>β</sup>* <sup>≈</sup> <sup>2</sup>·10<sup>3</sup> . Furthermore, by substituting *η*<sup>0</sup> = 0.003 kg/m·s, *<sup>ρ</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup>·10<sup>3</sup> , *<sup>I</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup>·10<sup>5</sup> W/m<sup>2</sup> , and *λ*<sup>0</sup> = 0.5 W/m·K into Equations (26) and (13b), we obtain a velocity estimate: *<sup>v</sup>* <sup>≈</sup> <sup>2</sup>·10−<sup>3</sup> m/s. The estimated velocity value depends on the initial concentration distribution, which can be obtained from Equation (24) for τ = 0. It should be noted that our approach cannot solve the problem analytically under arbitrary initial conditions.
