A Generalized Two-Phase Free Convection Flow of Dusty Jeffrey Fluid between Infinite Vertical Parallel Plates with Heat Transfer

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Introduction
Two-phase fow occurs when two distinct aggregation states of the same material or two distinct substances exist concurrently. All combinations are feasible, including gaseous and liquid, gaseous and solid, and liquid and solid. Dusty fuid fow can yield a number of forms, including fows that transform from pure liquid to vapor due to outside heat-separated fows and distributed two-phase fows in which one of the phases exists as particles, bubbles, or droplets in a continuous phase (i.e., liquid or gas). Furthermore, bubbles, rain, and sea waves are examples of two-phase fows. Two-phase fows in microgravity are used in a wide variety of critical applications including fuid handling and storage, as well as thermal and power systems on spacecraft (e.g., condensers, evaporators, and piping system). Numerous researchers have conducted a study on the uses of two-phase fows [1][2][3][4][5]. Furthermore, Lee and Mudawar [6] described a high-heat-fux microchannel heat sink with a two-phase fow for chilled applications, as well as its properties of transfer of heat. Mahanthesh et al. [7] studied the efects of second-order heat energy and convective on the two-phase boundary layer fows of dusty fuid via an upright plate. Furthermore, Bhatti et al. [8] reported the infuences of mass and heat transfer on peristaltic propulsion in a Darcy-Forchheimer porous medium via two-phase fow mathematical modeling with MHD. Yu et al. [9] used MHD to analyze the heat transmission of a two phase in a cooling gallery subjected to forced oscillatory motion. Ali et al. [10] scrutinized the infuence of transfer of heat on two-phase viscoelastic dusty fuid fow using nonconducting parallel plates. Te following articles are recommended for a more in-depth look [11][12][13][14].
Fractional calculus refers to the many ways to diferentiate between and integrate powers of real and complex numbers [15]. Ross [16] illustrated the evolution of fractional calculus from 1695 to 1900. Some physical and natural problems cannot be described by the classical derivative, and hence, fractional calculus is used to address these problems. For the last thirty years, scientists have expressed a significant amount of interest in fractional derivatives. Several scientists then presented various defnitions of fractional derivatives in response to these attempts. Riemann-Liouville [17] defnition is frequently used in 18th century. Tere are two important qualities that make the R-L concept of the fractional derivative useable, despite the fact that it has been properly represented in several physical systems. Due to the fact that some of the variables in the Laplace transform have no physical meaning, the diferentiation of the constant term could not result in zero. Numerous scientifc and practical applications of the Caputo fractional derivative may be found in the felds of economics, chemistry, physics, and other physical problems. Using CFD, it is possible to explore signal processing, difusion processes, image processing, material mechanics, damping processes, pharmacokinetics, and bioengineering processes [18][19][20]. A modifed version of fractional calculus called CFD [21]L defnition. However, because the CFD kernel contains a singularity, it cannot appropriately describe some materials with signifcant heterogeneities [22]. It is unable to accurately defne their outcome. According to Caputo-Fabrizio [23], a new defnition with a nonsingular kernel has been proposed to solve the singularity problem in CFD. Tis new concept was utilized in the research of several scholars [24][25][26][27][28]. Te C-F fractional derivative is commonly used by academics to explore the memory efect. Trough the use of time-fractional Caputo and C-F derivatives, Akhtar [29] investigated the fuid fow between two parallel plates. Ali et al. [30] specifcally discussed the investigation of two-phase-generalized MHD fow of dusty fuid between parallel plates.
Te industrial and technological applications of non-Newtonian fuids, such as the transportation of biological fuids and the dyeing of paper, as well as their application in the production of plastics, textiles, and packaged foods, have attracted the attention of scientists [31]. Te Jefrey fuid model is one of them and is the most prominent one. Khan [32] produced an efcient investigation of the Jefrey fuid's free convection fow. Zin et al. [33] investigated the convective fow of the Jefrey fuid with ramping wall temperatures and considered the efects of thermal radiation. Zeeshan and Majeed [34] investigated the impact of the magnetic dipole efect on the Jefrey fuid convective fow through a porous plate with suction and injection. In [35,36], there exist a number of interesting recent studies.
Te Jefrey model is seen as a generalisation of the commonly used Newtonian fuid model because its constitutive equation can be rewritten as a special case of the Newtonian model. Te Jefrey fuid model can describe the way that stress relaxes in non-Newtonian fuids, which is something that the usual viscous fuid model cannot do. Te Jefrey fuid model does a good job of describing the class of non-Newtonian fuids with a characteristic memory time scale, also called a relaxation time. So this article is about the fow of an incompressible Jefrey fuid in a small-width channel with porous walls. Darcy's law is used to fgure out how much water is absorbed in the channel wall [37]. Due to its wide applications, many researchers are attracted to adopt this model in their studies. Awan et al. [38] investigated the heat transfer characteristics of the free convection fow of the Jefrey fuid between two vertical plates, which are either stable or unstable. Te mechanism for heat transmission is created using a generalized and fractional variant of Fourier's equation that provides damping for thermal fux. In this procedure, the Caputo time-fractional derivative (CTFD) with a solitary power law kernel is used. Recently, Khan et al. [39] have investigated the free convection fow of the Prabhakar fractional Jefrey fuid on an oscillated vertical plate with homogeneous heat fux. With the help of the Laplace transform and Boussinesq approximation, precise solutions for dimensionless momentum may be found.
To the best of our knowledge, after a thorough evaluation of the previous research, no attempt has been made to use the defnition of recently obtained CFFD to calculate the optimal solution for Jefrey fuids with heat transfer and dusty fuid between parallel plates. As a result, the current approach takes into consideration heat transfer along vertical parallel plates and the unsteady fow of Jefrey fuids, in which dust particles are uniformly scattered by the efects of free convection. Te Jefrey fuid and particle motion momentum equations are separately modeled and generalized using the CFFD approach when dusty fuid fow is taken into consideration. Te Laplace and FSFTs are used to generate exact solutions for both velocities and temperature profles. Moreover, a number of graphs are used to study the infuence of various factors on fuid fow. Many factors make closed-form solutions important. Tey ofer a benchmark for evaluating the accuracy of several approximations, including asymptotic and numerical techniques. Furthermore, experimentalists and numerical solvers can utilize these solutions as a benchmark to compare their solutions against in order to determine their stability.

Formulation and Solution
Unidirectional fow, laminar fow, and one-dimensional fow of the Jefrey fuid through a bounded channel by two infnite parallel vertical plates along with dust particles are taken into consideration in the current fow system. Te two-phase, incompressible fow of the Jefrey fuid is appropriated between parallel plates connected by a distance d. Fluid motion is examined along the x-axis. Moreover, both plates are considered to be at rest. ψ(y, t) indicates the Jefrey fuid, and ψ 1 (y, t) indicates the dusty fuid velocity. Tetemperature of the left plate isT d + (T w − T d )At, on the right plate as shown as in Figure 1. Te usual Boussinesq approximation discovered the following equations to govern fows: Te physical conditions are Using dimensionless variables, we obtain Equations (1)-(3) and (4) become dimensionless when equation (5) is used, with the following form: (for simplicity the " * " sign is dropped) zθ(y, t) zt � 1 Pe Dimensionless physical states consist of

Fractional Model
Here, α is the fractional parameter of the CFFD operator CF D α t , and its defnition is as follows [40]: Here, N(α) is the normalization function, such as Te Laplace transformation for the CFFD of order0 < α ≤ 1andm ∈ Nis [40] L CF D m+α In particular case:

Exact Solutions with the Caputo-Fabrizio Time Fractional Derivative
Te exact solutions for fractional PDEs are obtained by using combined LT and FSFTs.

Solution of the Energy Equation.
Equation (12), after using the Laplace transform, gives Tere is a comparable modifed form of equation (17): Take d 0 andSin(nπ) multiply both sides of equation (15). Ten, using the beginning boundary conditions, we apply the FSFT. We get Equation (17) is more appropriately represented as In equation (18), the inverting LT is obtained as Here, We take inverse FSFT of equation (19). We obtain the following temperature profle solution: Using equation (16) and the LT of equation (11), we obtain qa 0 q + a 1 ψ 1 (y, q) � P m ψ(y, q) − ψ 1 (y, q) .
Here, a 4 � (1 − a 2 /a 2 ), a 5 � (a 1 + a 1 a 3 /1 − a 2 ). Using the FSFT of equation (24) along with the ICs and BCs, we arrived at the following solution: Equation (25) may be represented more clearly and precisely as follows:

Journal of Mathematics
, Inverting the LT of equation (26), we obtain When we inverse the FSFT of equation (28), we get Equation (29) fulflls set BCs, illustrating the correctness of our reported overall solutions.

Nusselt Number and Skin Friction.
Nusselt number terms and skin friction expressions are obtained from equations (21) and Equation 29), respectively, as Saqib et al (29).

Results and Discussion
Te unsteady, incompressible, unidirectional fow of the Jefery fuid along with dust particles and parallel plates and the time-fractional model are investigated in this article. It has also considered the efects of free convection and heat transmission. Te exact solutions are obtained by using LT and FSFTs. Figures 2-14 illustrate the efect of diferent physical parameters on the Jefery fuid velocity distribution, dust particle velocity distribution, and temperature profle. Te examination of several parameters for the base fuid containing dust particles is performed using Mathcad-15 software, where K � 10, Pm � 1.5, Gr � 10, t � 1, α � 0.5, λ � 5, Re � 5, and Pe � 10.

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Journal of Mathematics In order to investigate the efects of α on Jefery fuid and dusty fuid velocities, Figures 2 and 3 are drawn. If we use the classical order derivative, we only get one solution of the velocity outline; however, if we use the fractional-order derivative (α), it is obvious from the fgures that we have several velocity outlines for diferent values. Although the other parameters are constant, the variation of fractional parameters α has a signifcant memory impact on both velocities, which shows that the fractional model is more realistic than the classical one.
Te efect of the dusty fuid parameter K on both velocities profles is shown in Figures 4 and 5. It is clear from the fgure that the velocity declines with the rising values of K. Physically this behavior is true, as, by Stocks drag formula(K � 6πrμ), it is clear that increasingK increase viscous forces which resists the fow and consequently decreases both the velocities profle. Figures 6 and 7 are drawn to study the impact of the Jefrey fuid parameter λ on Jefrey fuid and dusty fuid velocities. From these fgures, it is clear that increasing the Jefrey fuid parameter λ retards both velocity profles.
Physically, the Jefrey fuid parameter λ increases non-Newtonian behavior or causes viscous forces, which raises the thickness of the momentum boundary layer. Tis results in a drop in both velocities.
Te Gr behavior of dust particles and Jefrey fuid velocities is shown in Figures 8 and 9. It is clear from these fgures that increasing the values of Gr enhances both velocity profles. Physically, due to the increasing Gr, buoyant forces rise and viscous forces reduces, which enhance both velocities. Figures 10 and 11 show the efect of the Jefrey fuid and dust particle velocity profles. By raising the values of Re, the profle of both the Jefrey fuid and dusty fuid velocities is produced. As the Reynolds number rises, more inertial forces are generated, slowing fow behavior because Re is the ratio of inertial to viscous forces. Figures 12 and 13 depict the impact of Pm on dust particle velocity and Jefrey fuid velocity. Due to the Pm inverse relationship with dust particle mass, a rise in Pm results in a decrease in particle mass, which raises both dust particle and Jefrey fuid velocity.   Te result of Pe on the temperature outline is plotted in Figure 14. By raising the values of Pe the temperature profle has also decreased. Physics behind this behavior is that Pe has the inverse relationship with thermal conductivity. When the values of Pe increase, the thermal conductivity of the fuid decrease, and therefore, temperature distribution decreases.
Variations in the rate of heat transmission in Pe are seen in Table 1. According to how temperature distribution occurs, it is seen that the rate of heat transmission reduces as Pe increases. When t � 0.6, Table 2 shows variations in skin friction for various values of α. Table 2 clearly demonstrates that skin friction decreases for high values of α, which is in great agreement with velocity profles.

. Conclusions
Te purpose of this study is to analyze the fractional model of dusty fuid fow of the dusty Jefrey fuid. Flow and varying temperatures are transmitted between parallel plates. Te model is fractionalized using the CFFD method without a single kernel. Te family of PDEs that govern the fow is solved using the Laplace and FSFT techniques. Software MATHCAD-15 is used to implement parametric studies that make use of fgures and tables.
(i) Te increase in Gr improves the Jefrey fuid and particle velocity. (ii) In order to examine the impact of α on both velocities (Jefrey fuid and dusty particle), if we use the classical order derivative, we only get one solution of the velocity profle; however, if we use the fractional-order derivative (α), it is obvious from the fgures that we have several velocity profles for diferent values. (iii) Te transformation (Laplace and FSFT) reduces the computational time required to obtain exact solutions to such problems. (iv) Te Jefrey fuid and particle velocity lowering tendency is seen with enhancing the values of K and λ. (v) Skin friction and the rate of heat transfer decrease with an increase of α and Pe, respectively.

Data Availability
Te data associated with the study are present in the paper.

Conflicts of Interest
Te authors declare that there are no conficts of interest.