Exact solutions of a class of generalized nanofluidic models

Huajun Zeng; Yuduo Ming; Tao Jiang; Cheng Jin

doi:10.1515/phys-2024-0068

Article Open Access

Exact solutions of a class of generalized nanofluidic models

Huajun Zeng , Yuduo Ming , Tao Jiang and Cheng Jin

Published/Copyright: November 12, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

Nanofluid, a significant branch of fluid mechanics, plays a pivotal role in thermal management, optics, biomedical engineering, energy harvesting, and other fields. The nanoparticles present in the fluid render the continuum mechanics ineffective, necessitating the adoption of fractional calculus to elucidate the effects of nanoparticles on the motion properties of the nanofluid. This article applies the modified extended tanh-function technique to solve two classical Schrödinger equations, the fractional Phi-4 model and the conformable fractional Boussinesq model, for nanofluids. Multiple exact solutions are obtained, and the corresponding graphical representations are provided to elucidate the basic properties of the nanofluid. This article provides new research perspectives for the development of nanofluids.

Keywords: nanofluid; the fractional Phi-4 model; the conformable fractional Boussinesq model; modified extended tanh-function technique; exact solution

1 Introduction

Nanofluid is a composite fluid system in which nanoparticles are dispersed in a base fluid [1,2]. The addition of nanoparticles can alter the properties and behavior of the base fluid, thereby imparting unique characteristics to the nanofluid. The effects of nanofluids can be demonstrated in a variety of ways. First, nanofluids can enhance the thermal conductivity of the base fluid, resulting in a higher thermal conductivity [3,4]. This enables the widespread applications of nanofluids in the field of thermal management, such as radiator [5], and energy harvesting [6,7], which is distinct from the traditional spring-pendulum system [8,9]. Second, nanofluids play a significant role in the field of optics, as they can be used to fabricate materials with special optical properties, such as transparent conductive films [10] and optical sensors [11]. Moreover, nanofluids are utilized in the biomedical field for the advancement of nanodrug delivery systems [12], biosensors [13], and blood flow [14], among other applications.

In fact, nanofluid can be considered a branch of fluid mechanics, which is a discipline within the broader field of physics that studies the motion and behavior of liquids and gases. It encompasses aspects such as fluid dynamics, statics, and heat and mass transfer. In contrast, nanofluid research is focused on the dispersion effects of nanoparticles in the base fluid at the nanoscale and their influence on the macroscopic properties of the fluid. The study of nanofluids necessitates the consideration of factors pertaining to the nanoscale, including surface forces, charge effects, and the wetting behavior of nanoparticles. Kou et al. demonstrated that the remarkable phenomena observed in nanofluids can be explained by the fractal boundary-layer theory [15]. A fractal boundary layer can result in minimal friction between the fractal boundary and the flowing fluid, as observed in the case of waving dunes [16,17]. It is, however, important to note that there is a certain connection between nanofluids and traditional fluid mechanics. The foundation of nanofluid research remains firmly rooted in the theoretical and experimental frameworks of traditional fluid mechanics. The behavior of nanofluids can typically be explained and predicted using principles and models of fluid mechanics [18]. Concurrently, the investigation of nanofluids offers a novel avenue for research in fluid mechanics, facilitating enhanced comprehension and exploitation of fluid dynamics. Consequently, there exists a close connection and interdependency between nanofluids and traditional fluid mechanics.

In recent years, fractal thermodynamics [19,20] has emerged as a promising approach to address challenges in fluid mechanics that conventional fluid mechanics has been unable to resolve. Fractional partial differential equations (FPDEs) have also gained significant traction in various applications. FPDEs represent an extension of traditional integer-order partial differential equations, wherein the derivatives assume a fractional-order form. The most commonly employed fractional-order derivatives include the Caputo fractional derivative [21,22], the Caputo–Fabrizio fractional derivative [23,24], and the Jumarie-modified Riemann–Liouville fractional derivative [25]. The application of FPDEs in fluid mechanics encompasses a multitude of facets. First, they are employed to describe the transport behavior of fluids in non-local media. For instance, in the context of non-local diffusion problems, fractional diffusion equations are more effective at describing diffusion phenomena [26]. Second, FPDEs can be employed to describe non-local diffusion phenomena with long-tail distributions [27] and other similar phenomena. In conclusion, FPDEs have made a substantial impact on the advancement of fluid mechanics. They provide a mathematical framework for more accurately describing non-local and non-linear phenomena. The introduction of fractional derivatives enables the characterization of the transport behavior and dynamic characteristics of fluids in complex media. Moreover, the utilization of FPDEs facilitates a more comprehensive and accurate modeling of fluid motion. These applications not only facilitate a more profound comprehension of fluid behavior but also furnish novel mathematical instruments and methodologies for the resolution of practical engineering issues and the formulation of corresponding control strategies.

Consequently, this study aims to identify precise solutions for selected classical fluid particle dynamics equations in fluid mechanics. The objective is to contribute to the development of nanofluids. Specifically, this study selects two Schrödinger equations for investigation: the fractional Phi-4 model and the conformable fractional Boussinesq model. The application of Schrödinger equations in fluid mechanics provides a quantum mechanical framework for the study of wave–particle duality, quantum transport, and the interactions of particles in fluid media [28]. This approach to quantum fluid mechanics enhances our comprehension of fluid phenomena and illuminates the pivotal role of quantum effects in fluid dynamics. The solutions of the nonlinear Schrödinger equation can be obtained through a variety of methods, including the ( G ′ G ) -expansion method [29], the extended tanh-function method [30], the extended rational sin–cos and sinh–cosh methods [31], the new extended direct algebraic method [32], and the exp-function method [33]. In this article, we will employ the modified extended tanh-function technique of fractional complex transformation to identify exact solutions to the subsequent two models:

1) the fractional Phi-4 model [34,35]:

(1) D t 2 α u − D x 2 α u + a 2 u + b u 3 = 0 ,

where a and b are the real constants and 0 < α , β < 1 . The Klein–Gordon equation, a fundamental Schrödinger equation, has found application in a number of fields, including optics, thermal science, nanofluids, and nonlinear vibration. The Phi-4 model represents a specific shape within this equation and has been widely used for modeling purposes. In recent times, researchers have employed a variety of methodologies to investigate the fractional Phi-4 model. For instance, in their work [36], new exact analytical solutions for the time-fractional Phi-4 model were presented using an extended direct algebraic method. Körpinar employed a variety of mapping techniques utilizing the conformal fractional derivative operator to address the aforementioned issue [37]. The construction of new exact solutions for the time-fractional Phi-4 model was achieved by applying the ( G ′ G , 1 G ) expansion technique, as detailed in the study by Hwang et al. [3]. In the study by Akram et al. [35], the traveling wave solutions for the space–time fractional Phi-4 model were constructed using the extended ( G ′ G 2 ) evolution method and the modified auxiliary equation method. In their investigation of computational wave solutions of Eq. (1), Khater employed the novel Kudryashov schemes, as described in the study by Khater [38].

2) The conformable fractional Boussinesq model [39]:

(2) D t 2 α u + D x 2 β u + D x 2 β ( u 2 ) + D x 4 β u = 0 ,

where 0 < α , β < 1 . The Schrödinger–Boussinesq equation, a classic equation in fluid mechanics, occupies a prominent position in the field. Subsequently, the equation was extended by introducing fractional derivatives, leading to extensive applications in various fields. In particular, the fractional Boussinesq equation is employed extensively to model the propagation grid and the nonlinear chain of small-amplitude nonlinear long waves on the water surface. These models have a wide range of engineering applications, including diffraction, shallow water predictions, refraction, and harmonic interactions [40]. A number of methods have been developed to solve this equation in an effective manner. For instance, in their study by Hosseini and Ansari [41], the validated modified Kudryashov method was employed to identify exact solutions to this equation. In order to identify exact solutions for the fractional Boussinesq model [42], the simply improved tan( ϕ ( ξ ) 2 ) method was employed. The study of exact solutions for the new generalized perturbed form of this equation was carried out by Nisar et al. [43], who employed the modified Kudryashov method and the improved generalized Riccati equation mapping method. Moreover, Chen et al. employed the ( G ′ G 2 ) -expansion method and the unified F-expansion method to identify exact wave solutions to this equation [39].

The following section outlines the structure of this article. Section 2 provides a comprehensive introduction to the solution method. Subsequently, Sections 3 and 4 address the fractional Phi-4 model and the conformal fractional Boussinesq model, respectively. Finally, the conclusions are presented in Section 5.

2 Methodology

Considering the FPDE

(3) P ( u , D t α u , D x β u , D t α D t α u , D t α D x β u , D x β D x β u , … ) = 0 , ( 0 < α , β < 1 ) ,

where D t α u , D x β u , D t α D t α u , … are the notations of the fractional derivative. The polynomial P is formulated in terms of the variable u and its corresponding partial derivatives. In this article, we adopt the fractional derivative in the modified Riemann–Liouville sense [25]. Some properties of the proposed derivative are described in the study by Jumarie [44] listed as follows:

(4) D t α t r = Γ ( 1 + r ) Γ ( 1 + r − α ) t r − α ,

(5) D t α ( c f ( x ) ) = c D t α f ( x ) , c is a constant ,

(6) D t α ( f ( w ) + g ( w ) ) = D t α f ( w ) + D t α g ( w ) .

In this article, as for fractional calculus, we used the following chain rule [45]:

(7) D t α u = σ t ∂ u ( ξ ) d ξ D t α ξ , D x α u = σ x ∂ u ( ξ ) d ξ D x α ξ , D t 2 α u = ( σ t ) 2 ∂ 2 u ( ξ ) d ξ 2 D t 2 α ξ , D x 2 α u = ( σ x ) 2 ∂ 2 u ( ξ ) d ξ 2 D x 2 α ξ ,

where σ t and σ x are the sigma indices. We take σ t = σ x = L ( L is a constant), without loss of generality.

Based on the aforementioned preliminaries, the general steps to solve Eqs (1) and (2) using the modified extended tanh-function technique are as follows:

Step 1: Using the nonlinear fractional complex wave transformation [45,46]:

(8) u ( x , t ) = u ( ξ ) , ξ = l x β Γ ( β + 1 ) + k t α Γ ( α + 1 ) ,

where l and k are the constants and l , k ≠ 0 . Eq. (8) is the famous fractional complex transform [47,48].

Substituting Eqs (7) and (8) into Eq. (3), we have

(9) Q ( u , u ′ , u ″ , … ) = 0 ,

where u ′ = d u d ξ , u ″ = d 2 u d ξ 2 ⋯ , Q present the polynomial function that contains u and the derivatives of u with respect to ξ .

Step 2: Expressing the solution of ordinary differential Eq. (9) as a polynomial form of ψ :

(10) u ( ξ ) = ∑ i = 0 n a i ψ i ( ξ ) ,

where ψ = ψ ( ξ ) conforming to the following Riccati equation:

(11) ψ ′ = ε + ψ 2 ,

where ε is an arbitrary constant and a i ( i = 0 , 1 , 2 , … , n ) are the indefinite constants. And by equating the highest order of the nonlinear term with that of the derivative term, the value of the positive integer n can be found. For ψ , the following three types of solutions are determined according to different values of the constant ε :

(12) ψ = − − ε tanh − ε ξ , ε < 0 , ψ = − − ε coth − ε ξ , ε < 0 , ψ = ε tan ε ξ , ε > 0 , ψ = ε cot ε ξ , ε > 0 , ψ = − 1 ξ , ε = 0 .

Step 3: Substituting Eq. (10) into Eq. (9), using Eq. (11) to iterate and combining the terms of ε , which has the same power, setting the coefficients and constant terms of each power to zero. Then, we obtain the over-determined algebraic equations about l , k , L , ε and a i ( i = 0 , 1 , 2 , … , n ) . Finally, we acquire multiply different types of exact solutions about Eq. (3) by calculating the parameters and substituting into Eq. (10).

3 Exact solution of the fractional Phi-4 model

Employing the subsequent traveling wave transformation:

(13) u ( x , t ) = u ( ξ ) , ξ = l x α Γ ( α + 1 ) + k t α Γ ( α + 1 ) ,

the original Eq. (1) is transformed into a nonlinear ordinary differential equation:

(14) L 2 k 2 u ″ − L 2 l 2 u ″ + a 2 u + b u 3 = 0 ,

balancing term “ u 3 ,” which is the highest order derivative and the nonlinear term “ u ″ ,” we obtain n = 1 . Thus, Eq. (10) becomes

(15) u ( ξ ) = a 0 + a 1 ψ ( ξ ) .

If we substitute Eqs (11) and (15) into Eq. (14) and then combine the same power terms of ψ and set its coefficient to zero, we obtain the nonlinear algebraic overdetermined systems around a 0 , a 1 , a , b , l , k , L , and ε :

(16) a 0 ( a 2 + a 0 b ) = 0 , a 2 + 3 ( a 0 ) 2 b + 2 ( k 2 − l 2 ) L 2 ε = 0 , 3 a 0 ( a 1 ) 2 b = 0 , ( a 1 ) 3 b + 2 a 1 ( k 2 − l 2 ) L 2 = 0 .

Solving the aforementioned set of equations using Mathematica, the results can be obtained:

Case 1.

a 0 = 0 , a 1 = − 2 − k 2 L 2 + l 2 L 2 b , ε = − a 2 2 ( k 2 − l 2 ) L 2 .

According to the solution, in order for the original equation to have an accurate solution of real numbers, there must be ∣ l ∣ > ∣ k ∣ and l , k ≠ 0 . Therefore, in Case 1, according to the different values of ε , we have

(I) When a = 0 , ε = 0 , which produce

(17) u 1 ( ξ ) = L 2 ( l 2 − k 2 ) b 1 ξ , ε = 0 ,

where ξ = l x α Γ ( α + 1 ) + k t α Γ ( α + 1 ) , l , k are the arbitrary constants and ∣ l ∣ > ∣ k ∣ ≠ 0 .

The numerical simulation image of u 1 ( ξ ) is shown in Figure 1.

$Figure 1 Three-dimensional plot of u 1 ( ξ ) {u}_{1}\left(\xi ) in Case 1 for α = 1 2 \alpha =\frac{1}{2} , a = 0 a=0 , b = k = L = 1 b=k=L=1 , and l = 2 l=2 .$

Figure 1

Three-dimensional plot of u 1 ( ξ ) in Case 1 for α = 1 2 , a = 0 , b = k = L = 1 , and l = 2 .

(II) When a ≠ 0 , ε > 0 , which produce

(18) u 2 ( ξ ) = − a b tan a L 2 ( l 2 − k 2 ) ξ , ε > 0 ,

(19) u 3 ( ξ ) = − a b cot a L 2 ( l 2 − k 2 ) ξ , ε > 0 ,

where ξ = l x α Γ ( α + 1 ) + k t α Γ ( α + 1 ) , l , k are the arbitrary constants and ∣ l ∣ > ∣ k ∣ ≠ 0 .

The numerical simulation images of u 2 ( ξ ) and u 3 ( ξ ) are shown in Figure 2.

$Figure 2 Three-dimensional plots of u 2 ( ξ ) {u}_{2}\left(\xi ) and u 3 ( ξ ) {u}_{3}\left(\xi ) in Case 1 for α = 1 2 \alpha =\frac{1}{2} , a = b = k = L = 1 a=b=k=L=1 , and l = 2 l=2 .$

Figure 2

Three-dimensional plots of u 2 ( ξ ) and u 3 ( ξ ) in Case 1 for α = 1 2 , a = b = k = L = 1 , and l = 2 .

Case 2.

a 0 = 0 , a 1 = 2 − k 2 L 2 + l 2 L 2 b , ε = − a 2 2 ( k 2 − l 2 ) L 2 .

Similarly, there must be ∣ l ∣ > ∣ k ∣ and l , k ≠ 0 . Therefore, in Case 2, according to the different values of ε , we have

(I) When a = 0 , ε = 0 , which produce

(20) u 4 ( ξ ) = − L 2 ( l 2 − k 2 ) b 1 ξ , ε = 0 ,

where ξ = l x α Γ ( α + 1 ) + k t α Γ ( α + 1 ) , l , k are the arbitrary constants and ∣ l ∣ > ∣ k ∣ ≠ 0 .

The numerical simulation image of u 4 ( ξ ) is shown in Figure 3.

$Figure 3 Three-dimensional plot of u 4 ( ξ ) {u}_{4}\left(\xi ) in Case 1 for α = 1 2 \alpha =\frac{1}{2} , a = 0 a=0 , b = k = L = 1 b=k=L=1 , and l = 2 l=2 .$

Figure 3

Three-dimensional plot of u 4 ( ξ ) in Case 1 for α = 1 2 , a = 0 , b = k = L = 1 , and l = 2 .

(II) When a ≠ 0 , ε > 0 , which produce

(21) u 5 ( ξ ) = a b tan a L 2 ( l 2 − k 2 ) ξ , ε > 0 ,

(22) u 6 ( ξ ) = a b cot a L 2 ( l 2 − k 2 ) ξ , ε > 0 ,

where ξ = l x α Γ ( α + 1 ) + k t α Γ ( α + 1 ) , l , k are the arbitrary constants and ∣ l ∣ > ∣ k ∣ ≠ 0 .

The numerical simulation images of u 5 ( ξ ) and u 6 ( ξ ) are shown in Figure 4.

$Figure 4 Three-dimensional plots of u 5 ( ξ ) {u}_{5}\left(\xi ) and u 6 ( ξ ) {u}_{6}\left(\xi ) in Case 1 for α = 1 2 \alpha =\frac{1}{2} , a = b = k = L = 1 a=b=k=L=1 , and l = 2 l=2 .$

Figure 4

Three-dimensional plots of u 5 ( ξ ) and u 6 ( ξ ) in Case 1 for α = 1 2 , a = b = k = L = 1 , and l = 2 .

4 Exact solution of the conformable fractional Boussinesq model

Supposing that u ( x , t ) = u ( ξ ) , where ξ is given by Eq. (8). Then, Eq. (2) can be turned into the following equation of ordinary differential equations (ODEs):

(23) L 2 k 2 u ″ + L 2 l 2 u ″ + L 2 l 2 ( u 2 ) ″ + L 4 l 4 u ( 4 ) = 0 .

Integrating twice and the integral constant is equal to zero, Eq. (23) turns into

(24) k 2 u + l 2 u + l 2 u 2 + L 2 l 4 u ″ = 0 .

This equation with a quadratic nonlinearity has some amazing properties as discussed in the study by He and Liu [49]. Balancing the term “ u 2 ” and the term “ u ″ ,” we obtain n = 2 . Thus, Eq. (10) becomes

(25) u ( ξ ) = a 0 + a 1 ψ ( ξ ) + a 2 ψ 2 ( ξ ) .

Substitute Eqs (11) and (25) into Eq. (24), and collect all terms with the same power of ψ . Equating each coefficient to zero yields the following set of algebraic equations for a 0 , a 1 , a 2 , l , k , L , and ε :

(26) a 0 2 l 2 + a 0 ( k 2 + l 2 ) + 2 a 2 l 4 L 2 ε 2 = 0 , a 1 ( k 2 + l 2 ( 1 + 2 a 0 + 2 l 2 L 2 ε ) ) = 0 , a 1 2 l 2 + a 2 ( k 2 + l 2 + 2 a 0 l 2 + 8 l 4 L 2 ε ) = 0 , a 1 l 2 ( a 2 + l 2 L 2 ) = 0 , a 2 + 6 l 2 L 2 = 0 .

Solving the aforementioned algebraic equations, we obtain:

Case 1.

a 0 = k 2 + l 2 2 l 2 , a 1 = 0 , a 2 = − 6 l 2 L 2 , ε = − k 2 + l 2 4 l 4 L 2 < 0 ,

which produces

(27) u 1 ( ξ ) = k 2 + l 2 2 l 2 − 3 ( k 2 + l 2 ) 2 l 2 tanh 2 k 2 + l 2 2 l 2 L ξ , ε < 0 ,

(28) u 2 ( ξ ) = k 2 + l 2 2 l 2 − 3 ( k 2 + l 2 ) 2 l 2 coth 2 k 2 + l 2 2 l 2 L ξ , ε < 0 ,

where ξ = l x β Γ ( β + 1 ) + k t α Γ ( α + 1 ) , l , k are the arbitrary constants and l , k ≠ 0 .

For more convenience, the graphical representations of Eq. (27) and Eq. (28) are shown in Figure 5.

$Figure 5 Three-dimensional plots of u 1 ( ξ ) {u}_{1}\left(\xi ) and u 2 ( ξ ) {u}_{2}\left(\xi ) in Case 1 of Eq. (2), with α = β = 1 2 \alpha =\beta =\frac{1}{2} , l = k = L = 1 l=k=L=1 .$

Figure 5

Three-dimensional plots of u 1 ( ξ ) and u 2 ( ξ ) in Case 1 of Eq. (2), with α = β = 1 2 , l = k = L = 1 .

Case 2.

a 0 = − 3 ( k 2 + l 2 ) 2 l 2 , a 1 = 0 , a 2 = − 6 l 2 L 2 , ε = k 2 + l 2 4 l 4 L 2 > 0 ,

which produces

(29) u 3 ( ξ ) = − 3 ( k 2 + l 2 ) 2 l 2 sec 2 k 2 + l 2 2 l 2 L ξ , ε > 0 ,

(30) u 4 ( ξ ) = − 3 ( k 2 + l 2 ) 2 l 2 csc 2 k 2 + l 2 2 l 2 L ξ , ε > 0 ,

where ξ = l x β Γ ( β + 1 ) + k t α Γ ( α + 1 ) , l , k are the arbitrary constants and l , k ≠ 0 .

The graphical representations of Eqs (29) and (30) are shown in Figure 6, respectively.

$Figure 6 Three-dimensional plots of u 3 ( ξ ) {u}_{3}\left(\xi ) and u 4 ( ξ ) {u}_{4}\left(\xi ) in Case 2 of Eq. (2), with α = β = 1 2 \alpha =\beta =\frac{1}{2} , l = k = L = 1 l=k=L=1 .$

Figure 6

Three-dimensional plots of u 3 ( ξ ) and u 4 ( ξ ) in Case 2 of Eq. (2), with α = β = 1 2 , l = k = L = 1 .

5 Discussion and conclusion

As illustrated earlier, the dynamical properties of nanofluids can be controlled by the fractional order. Nanofluids have been successfully applied to fractal pattern dynamic [50], fractal financial system [51], and MEMS systems [52], where the nanoparticles or nanoparticle-induced boundaries play an important role in the systems’ efficiency and reliability. For instance, when a nanofluid flows through a nano/microdevice as shown in Figure 7, it can significantly enhance the thermal conduction, thereby enabling the temperature to be maintained below an unsafe threshold.

Figure 7

Nanofluid in a micro-channel.

As shown earlier, the dynamical properties of a nanofluid are a function of t α and x β , rather than t and x . The values of α and β can be determined by the He–Liu fractal dimension formulation [53], which allows for the reflection of the nanoparticles’ size and distribution. This is not possible in traditional fluid mechanics, and it can save a significant amount of time during theoretical analysis compared to the multi-scale numerical approach to the two-phase fluid [54].

This article focuses on nanofluids and applies the modified extended Tanh function technique to solve two classical fractional Schrödinger equations in fluid mechanics: the Phi-4 fractional model and the conformal fractional Boussinesq model. The exact solutions of these two equations are presented graphically. These solutions have direct practical applications in the field of nanofluids. We posit that this article offers numerous opportunities to advance the development of nanofluids and can serve as a good example for future research.

Funding information: This research was supported by the National Natural Science Foundation of China (11971433), First Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics), the Characteristic & Preponderant Discipline of Key Construction Universities in Zhejiang Province (Zhejiang Gongshang University-Statistics), Collaborative Innovation Center of Statistical Data Engineering Technology and Application.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] He JH, Elgazery NS, Elagamy K, Abd-Elazem NY. Efficacy of a modulated viscosity-dependent temperature/nanoparticles concentration parameter on a nonlinear radiative electromagneto-nanofluid flow along an elongated stretching sheet. J Appl Comput Mech. 2023;9(3):848–60. Search in Google Scholar

[2] Kumar K, Chauhan PR, Kumar R, Bharj RS. Irreversibility analysis in Al2O3-water nanofluid flow with variable property. Facta Univ-Ser Mech. 2022;20(3):503–18. 10.22190/FUME210308050KSearch in Google Scholar

[3] Hwang YJ, Ahn YC, Shin HS, Lee CG, Kim GT, Park HS, et al. Investigation on characteristics of thermal conductivity enhancement of nanofluids. Curr Appl Phys. 2006;6(6):1068–71. 10.1016/j.cap.2005.07.021Search in Google Scholar

[4] Asshaari I, Jedi A, Abdullah S. Brownian motion and thermophoresis effects in co-flowing carbon nanotubes towards a moving plate. Results Phys. 2023;44:106165. 10.1016/j.rinp.2022.106165Search in Google Scholar

[5] Bhogare RA, Kothawale BS. Performance investigation of automobile radiator operated with Al2O3 based nanofluid. J Mech Civ Eng. 2014;11(3):23–30. 10.9790/1684-11352330Search in Google Scholar

[6] He JH, Abd-Elazem NY. The carbon nanotube-embedded boundary layer theory for energy harvesting. Facta Univ-Ser Mech. 2022;20(2):211–35. 10.22190/FUME220221011HSearch in Google Scholar

[7] Wang QL, He JH, Liu Z. Intelligent nanomaterials for solar energy harvesting: from polar bear hairs to unsmooth nanofiber fabrication. Front Bioeng Biotech. 2022;10:926253. 10.3389/fbioe.2022.926253Search in Google Scholar PubMed PubMed Central

[8] He CH, Amer TS, Tian D, Abolila AF, Galal AA. Controlling the kinematics of a spring-pendulum system using an energy harvesting device. J Low Freq Noise V A. 2022;41(3):1234–57. 10.1177/14613484221077474Search in Google Scholar

[9] He CH, El-Dib YO. A heuristic review on the homotopy perturbation method for non-conservative oscillators. J Low Freq Noise V A. 2022;41(2):572–603. 10.1177/14613484211059264Search in Google Scholar

[10] Ilatovskii DA, Gilshtein EP, Glukhova OE, Nasibulin AG. Transparent conducting films based on carbon nanotubes: rational design toward the theoretical limit. Adv Sci. 2022;9(24):2201673. 10.1002/advs.202201673Search in Google Scholar PubMed PubMed Central

[11] Hu XY, Abbasi R, Wachsmann-Hogiu S. Microfluidics on lensless, semiconductor optical image sensors: challenges and opportunities for democratization of biosensing at the micro-and nano-scale. Nanophotonics. 2023;12(21):3977–4008. 10.1515/nanoph-2023-0301Search in Google Scholar

[12] Thakare Y, Dharaskar S, Unnarkat A, Sonawane SS. Nanofluid-based drug delivery systems. Appl Nanofluids Chem Bio-med Process Industry. 2022;13:303–34. 10.1016/B978-0-323-90564-0.00005-2Search in Google Scholar

[13] He JH, He CH, Qian MY, Alsolami AA. Piezoelectric Biosensor based on ultrasensitive MEMS system. Sens Actuators A Phys. 2024;376:115664. 10.1016/j.sna.2024.115664Search in Google Scholar

[14] He JH, Elgazery NS, Abd-Elazem NY. Gold nanoparticles’ morphology affects blood flow near a wavy biological tissue wall: an application for cancer therapy. J Appl Comput Mech. 2024;10(2):342–56. Search in Google Scholar

[15] Kou SJ, He CH, Men XC, He JH. Fractal boundary layer and its basic properties. Fractals. 2022;30(9):2250172. 10.1142/S0218348X22501729Search in Google Scholar

[16] Mei Y, Liu YQ, He JH. On the mountain-river-desert relation. Therm Sci. 2021;25(6):4817–22. 10.2298/TSCI211010330MSearch in Google Scholar

[17] Mei Y, Liu YQ, He JH. The yellow river-bed evolution a statistical proof of the mountain-river-desert conjecture. Therm Sci. 2023;27(3A):2075–79. 10.2298/TSCI2303075MSearch in Google Scholar

[18] Wang LQ, Fan J. Nanofluids research: key issues. Nanoscale Res Lett. 2010;5(8):1241–52. 10.1007/s11671-010-9638-6Search in Google Scholar PubMed PubMed Central

[19] Qian MY, He JH. Two-scale thermal science for modern life: making the impossible possible. Therm Sci. 2022;26(3B):2409–12. 10.2298/TSCI2203409QSearch in Google Scholar

[20] He JH, Ain QT. New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle. Therm Sci. 2020;24(2A):659–81. 10.2298/TSCI200127065HSearch in Google Scholar

[21] Tuan NH, Mohammadi H, Rezapour S. A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Soliton Fract. 2020;140:110107. 10.1016/j.chaos.2020.110107Search in Google Scholar PubMed PubMed Central

[22] Baleanu D, Mohammadi H, Shahram S. The existence of solutions for a nonlinear mixed problem of singular fractional differential equations. Adv Differ Equ. 2013;1:359. 10.1186/1687-1847-2013-359Search in Google Scholar

[23] Baleanu D, Aydogn SM, Mohammadi H, Rezapour S. On modelling of epidemic childhood diseases with the Caputo-Fabrizio derivative by using the Laplace Adomian decomposition method. Alex Eng J. 2020;59(5):3029–39. 10.1016/j.aej.2020.05.007Search in Google Scholar

[24] Baleanu D, Jajarmi A, Mohammadi H, Rezapour S. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Soliton Fract. 2020;134:109705. 10.1016/j.chaos.2020.109705Search in Google Scholar

[25] Jumarie G. Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput Math Appl. 2006;51(9-10):1367–76. 10.1016/j.camwa.2006.02.001Search in Google Scholar

[26] Correa-Escudero IL, Gómez-Aguilar JF, López-López MG, Alvarado-Martínez VM, Baleanu D. Correcting dimensional mismatch in fractional models with power, exponential and proportional kernel: Application to electrical systems. Results Phys. 2022;4:105867. 10.1016/j.rinp.2022.105867Search in Google Scholar

[27] Zhang Y, Sun HG, Stowell HH, Zayernouri M, Hansen SE. A review of applications of fractional calculus in Earth system dynamics. Chaos Soliton Fract. 2017;102(1):29–46. 10.1016/j.chaos.2017.03.051Search in Google Scholar

[28] Kaur L, Adel W, Inc M, Rezazadeh H, Akinyemi L. Gaussian solitary wave solutions for nonlinear perturbed Schrödinger equations with applications in nanofibers. Int J Mod Phys B. 2024;38(24):2450318. 10.1142/S0217979224503181Search in Google Scholar

[29] Zaman UHM, Arefin MA, Akbar MA, Uddin MH. Solitary wave solution to the space-time fractional modified Equal Width equation in plasma and optical fiber systems. Results Phys. 2023;52:106903. 10.1016/j.rinp.2023.106903Search in Google Scholar

[30] Zeng HJ, Wang YX, Xiao M, Wang Y. Fractional solitons: New phenomena and exact solutions. Front Phys. 2023;11:1177335. 10.3389/fphy.2023.1177335Search in Google Scholar

[31] Montazeri S, Nazari F, Rezazadeh H. Solitary and periodic wave solutions of the unstable nonlinear Schrödinger as equation. Optik. 2024;297:171573. 10.1016/j.ijleo.2023.171573Search in Google Scholar

[32] Rezazadeh H. New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity. Optik. 2018;167:218–27. 10.1016/j.ijleo.2018.04.026Search in Google Scholar

[33] Lv GJ, Tian D, Xiao M, He CH, He JH. Shock-like waves with finite amplitudes. J Appl Mech. 2024;55(1):1–7. Search in Google Scholar

[34] Zaman UHM, Arefin MA, Akbar MA, Uddin MH. Analyzing numerous travelling wave behavior to the fractional-order nonlinear Phi-4 and Allen-Cahn equations throughout a novel technique. Results Phys. 2022;37:105486. 10.1016/j.rinp.2022.105486Search in Google Scholar

[35] Akram G, Sadaf M, Zainab I. Observations of fractional effects of β-derivative and M-truncated derivative for space time fractional Phi-4 equation via two analytical techniques. Chaos Soliton Fract. 2022;154:111645. 10.1016/j.chaos.2021.111645Search in Google Scholar

[36] Rezazadeh H, Tariq H, Eslami M, Mirzazadeh M, Zhou Q. New exact solutions of nonlinear conformable time-fractional Phi-4 equation. Chinese J Phys. 2018;56(6):2805–16. 10.1016/j.cjph.2018.08.001Search in Google Scholar

[37] Körpinar Z. Some analytical solutions by mapping methods for non-linear conformable time-fractional PHI-4 equation. Therm Sci. 2019;23(S6):S1815–22. 10.2298/TSCI190108341KSearch in Google Scholar

[38] Khater MMA. De Broglie waves and nuclear element interaction; Abundant waves structures of the nonlinear fractional Phi-four equation. Chaos Soliton Fract. 2022;163:112549. 10.1016/j.chaos.2022.112549Search in Google Scholar

[39] Chen HY, Zhu QH, Qi JM. Further results about the exact solutions of conformable space-time fractional Boussinesq equation (FBE) and breaking soliton (Calogero) equation. Results Phys. 2022;37:105428. 10.1016/j.rinp.2022.105428Search in Google Scholar

[40] Peregrine DH. Long waves on a beach. J Fluid Mech. 1967;27(4):815–27. 10.1017/S0022112067002605Search in Google Scholar

[41] Hosseini K, Ansari R. New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method. Wave Random Complex. 2017;27(4):628–36. 10.1080/17455030.2017.1296983Search in Google Scholar

[42] Çerdik-Yaslan H, Girgin A. SITEM for the conformable space-time fractional Boussinesq and (2.1)-dimensional breaking soliton equations. J Ocean Eng Sci. 2020;6(3):228–36. 10.1016/j.joes.2020.11.010Search in Google Scholar

[43] Nisar KS, Akinyemi L, Inc M, Şenol M, Mirzazadeh M, Houwe A, et al. New perturbed conformable Boussinesq-like equation: soliton and other solutions. Results Phys. 2022;33:105200. 10.1016/j.rinp.2022.105200Search in Google Scholar

[44] Jumarie G. Fractional partial differential equations and modified Riemann–Liouville derivative new methods for solution. J Appl Math Comput. 2007;24(1–2):31–48. 10.1007/BF02832299Search in Google Scholar

[45] He JH, Elagan SK, Li ZB. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys Lett A. 2012;376(4):257–9. 10.1016/j.physleta.2011.11.030Search in Google Scholar

[46] Li ZB, He JH. Fractional complex transform for fractional differential equations. Math Comput Appl. 2010;15(5):970–3. 10.3390/mca15050970Search in Google Scholar

[47] Ain QT, He JH, Anjum N, Ali M. The fractional complex transform: a novel approach to the time-fractional Schrodinger equation. Fractals. 2020;28(7):2050141. 10.1142/S0218348X20501418Search in Google Scholar

[48] Han C, Wang YL, Li ZY. A high-precision numerical approach to solving space fractional Gray-Scott model. Appl Math Lett. 2022;125:107759. 10.1016/j.aml.2021.107759Search in Google Scholar

[49] He CH, Liu C. Fractal dimensions of a porous concrete and its effect on the concreteas strength. Facta Univ-Ser Mech. 2023;21(1):137–50. 10.22190/FUME221215005HSearch in Google Scholar

[50] Gao XL, Zhang HL, Wang YL, Li ZY. Research on pattern dynamics behavior of a fractional vegetation-water model in arid flat environment. Fractal Fract. 2024;8(5):264.10.3390/fractalfract8050264Search in Google Scholar

[51] Gao XL, Li ZY, Wang YL. Chaotic dynamic behavior of a fractional-order financial system with constant inelastic demand. Int J Bifurcat Chaos. 2024;34(9):2450111.10.1142/S0218127424501116Search in Google Scholar

[52] Tian D, Ain QT, Anjum N, He CH, Cheng B. Fractal N/MEMS: from pull-in instability to pull-in stability. Fractals. 2021;29(2):2150030. 10.1142/S0218348X21500304Search in Google Scholar

[53] He JH, Yang Q, He CH, Alsolami AA. Pull-down instability of the quadratic nonlinear oscillators. Facta Univ-Ser Mech. 2023;21(2):191–200. 10.22190/FUME230114007HSearch in Google Scholar

[54] Li XJ, Wang D, Saeed T. Multi-scale numerical approach to the polymer filling process in the weld line region. Facta Univ-Ser Mech. 2022;20(2):363–80. 10.22190/FUME220131021LSearch in Google Scholar

Received: 2023-12-13

Revised: 2024-06-25

Accepted: 2024-07-22

Published Online: 2024-11-12

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0068

Keywords for this article

nanofluid; the fractional Phi-4 model; the conformable fractional Boussinesq model; modified extended tanh-function technique; exact solution

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