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

Authors: Dolat Khan; Kanayo Kenneth Asogwa; Talal Alqahtani; Salem Algarni; Sultan Alqahtani; M. Ali Akbar
Journal: Journal of Mathematics
 (2022)

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## Abstract

In blood flow and fluid mechanics, Jeffrey fluid has a vital role because of its viscoelastic characteristics. The application of Caputo–Fabrizio time-fractional derivatives for dusty-type Jeffrey fluid is discussed in this article. The concept of free convection flow of dusty Jeffrey fluid between infinite vertical parallel static plates is generalized. Free convection and buoyant force produce the flow. Furthermore, the fluid contains homogeneous dispersion of all spherical dust particles. Heat transmission is therefore taken into account for free convection. Nondimensional variables are used to write the dusty Jeffrey fluid classical model in dimensionless form. Also, the dimensionless model is transformed into a generalized dusty Jeffrey fluid model via a fractional derivative. Using the finite sine and Laplace method, the governing equations of the generalized dusty Jeffrey fluid model have been solved exactly. Numerical computation is used to study the physics of velocity and temperature profiles for a variety of embedded parameters. The collected results are discussed in detail and are shown graphically in this report. Mathcad-15 is used to plot the graphical outcomes for Jeffrey fluid, dust particle, and temperature profiles. Furthermore, skin friction and the Nusselt number are calculated. Table demonstrates how the rate of heat transmission reduces as the Peclet number’s value rises. Similarly, Table demonstrates that skin friction increases as the fractional parameter rises. By increasing the dusty Jeffrey fluid parameterλ, both velocity profiles are retarded.


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## Body

## 1. Introduction

Two-phase flow 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 fluid flow can yield a number of forms, including flows that transform from pure liquid to vapor due to outside heat-separated flows and distributed two-phase flows 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 flows. Two-phase flows in microgravity are used in a wide variety of critical applications including fluid 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 flows [1–5]. Furthermore, Lee and Mudawar [6] described a high-heat-flux microchannel heat sink with a two-phase flow for chilled applications, as well as its properties of transfer of heat. Mahanthesh et al. [7] studied the effects of second-order heat energy and convective on the two-phase boundary layer flows of dusty fluid via an upright plate. Furthermore, Bhatti et al. [8] reported the influences of mass and heat transfer on peristaltic propulsion in a Darcy–Forchheimer porous medium via two-phase flow 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 influence of transfer of heat on two-phase viscoelastic dusty fluid flow using nonconducting parallel plates. The following articles are recommended for a more in-depth look [11–14].Fractional calculus refers to the many ways to differentiate 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 definitions of fractional derivatives in response to these attempts. Riemann-Liouville[17] definition is frequently used in 18th century. There 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 differentiation of the constant term could not result in zero. Numerous scientific and practical applications of the Caputo fractional derivative may be found in the fields of economics, chemistry, physics, and other physical problems. Using CFD, it is possible to explore signal processing, diffusion processes, image processing, material mechanics, damping processes, pharmacokinetics, and bioengineering processes [18–20]. A modified version of fractional calculus called CFD [21]L definition. However, because the CFD kernel contains a singularity, it cannot appropriately describe some materials with significant heterogeneities [22]. It is unable to accurately define their outcome. According to Caputo–Fabrizio [23], a new definition with a nonsingular kernel has been proposed to solve the singularity problem in CFD. This new concept was utilized in the research of several scholars [24–28]. The C-F fractional derivative is commonly used by academics to explore the memory effect. Through the use of time-fractional Caputo and C-F derivatives, Akhtar [29] investigated the fluid flow between two parallel plates. Ali et al. [30] specifically discussed the investigation of two-phase-generalized MHD flow of dusty fluid between parallel plates.The industrial and technological applications of non-Newtonian fluids, such as the transportation of biological fluids 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]. The Jeffrey fluid model is one of them and is the most prominent one. Khan [32] produced an efficient investigation of the Jeffrey fluid’s free convection flow. Zin et al. [33] investigated the convective flow of the Jeffrey fluid with ramping wall temperatures and considered the effects of thermal radiation. Zeeshan and Majeed [34] investigated the impact of the magnetic dipole effect on the Jeffrey fluid convective flow through a porous plate with suction and injection. In [35, 36], there exist a number of interesting recent studies.The Jeffrey model is seen as a generalisation of the commonly used Newtonian fluid model because its constitutive equation can be rewritten as a special case of the Newtonian model. The Jeffrey fluid model can describe the way that stress relaxes in non-Newtonian fluids, which is something that the usual viscous fluid model cannot do. The Jeffrey fluid model does a good job of describing the class of non-Newtonian fluids with a characteristic memory time scale, also called a relaxation time. So this article is about the flow of an incompressible Jeffrey fluid in a small-width channel with porous walls. Darcy’s law is used to figure 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 flow of the Jeffrey fluid between two vertical plates, which are either stable or unstable. The mechanism for heat transmission is created using a generalized and fractional variant of Fourier’s equation that provides damping for thermal flux. 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 flow of the Prabhakar fractional Jeffrey fluid on an oscillated vertical plate with homogeneous heat flux. 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 definition of recently obtained CFFD to calculate the optimal solution for Jeffrey fluids with heat transfer and dusty fluid between parallel plates. As a result, the current approach takes into consideration heat transfer along vertical parallel plates and the unsteady flow of Jeffrey fluids, in which dust particles are uniformly scattered by the effects of free convection. The Jeffrey fluid and particle motion momentum equations are separately modeled and generalized using the CFFD approach when dusty fluid flow is taken into consideration. The Laplace and FSFTs are used to generate exact solutions for both velocities and temperature profiles. Moreover, a number of graphs are used to study the influence of various factors on fluid flow. Many factors make closed-form solutions important. They offer 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.


## 2. Formulation and Solution

Unidirectional flow, laminar flow, and one-dimensional flow of the Jeffrey fluid through a bounded channel by two infinite parallel vertical plates along with dust particles are taken into consideration in the current flow system. The two-phase, incompressible flow of the Jeffrey fluid is appropriated between parallel plates connected by a distanced. Fluid motion is examined along the x-axis. Moreover, both plates are considered to be at rest. ψy,t indicates the Jeffrey fluid, and ψ1y,t indicates the dusty fluid velocity. Thetemperature of the left plate isTd+Tw−TdAt, on the right plate as shown as in Figure 1. The usual Boussinesq approximation discovered the following equations to govern flows:(1)∂ψy,t∂t=υ1+λ11+∂∂tλ2∂2ψy,t∂y2−K0N0ρψy,t−ψ1y,t+gβTT−Td,(2)m∂ψ1y,t∂t=K0ψy,t−ψ1y,t,(3)ρcpk∂Ty,t∂t=∂2Ty,t∂y2.Figure 1
The problem’s descriptive schematic.The physical conditions are(4)ψy,t=ψ1y,t=0,att=0,Ty,0=Td,ψy,t|y=0=0,Ty,t|y=0=Tw−TdAt+Td,ψd,t=0,Ty,t|y=d=Td.Using dimensionless variables, we obtain(5)ψ∗=ψψ0,t∗=ψ0dt,y∗=yd,θ∗=T−TdTw−Td,ψ1∗=ψ1ψ0.Equations (1)–(3) and (4) become dimensionless when equation (5) is used, with the following form: (for simplicity the “∗” sign is dropped)(6)Re∂ψy,t∂t=11+λ11+∂∂tλ∂2ψy,t∂y2−Kψy,t−ψ1y,t+Grθy,t,(7)∂ψ1y,t∂t=Pmψy,t−ψ1y,t,(8)∂θy,t∂t=1Pe∂2θy,t∂y2.Dimensionless physical states consist of(9)ψy,0=ψ1y,0=Ty,0=0,ψy,t=0,Ty,t=taty=0,ψy,t=Ty,t=0,aty=1,λ=λ2ψ02υ,Pe=ρcpψ0dk,Gr=gβTd2Tw−Tdψ0υ,Pm=K0dmψ0,K=K0N0d2ρυ,Re=ψ0dυ.


## 3. Fractional Model

Applying equations (6), (7), and (8) to the CFFD parameter α, we obtain(10)DtαCFReψy,t=11+λ11+λDtαCF∂2ψy,t∂y2+Kψ1y,t−ψy,t+Grθy,t,(11)DtαCFψy,t=Pmψy,t−ψ1y,t,(12)DtαCFPeθy,t=∂2θy,t∂y2.Here,α is the fractional parameter of the CFFD operator DtαCF, and its definition is as follows [40]:(13)DtαCFzτ=−Nαα−1∫0τexp−ατ−t1−αz′τdτ.Here,Nα is the normalization function, such as N1=N0=1, 0<α<1.The Laplace transformation for the CFFD of order0<α≤1andm∈Nis [40](14)LDτm+αCFzτq=11−αLzm+1τLe−α/1−αt=1q+α1−qqm+1Lzτ−qmz0−qm−1z′0−………zm0In particular case:LDταCFψy,t,q=qq+α1−qLψy,t,m=0,.


## 4. Exact Solutions with the Caputo–Fabrizio Time Fractional Derivative

The exact solutions for fractional PDEs are obtained by using combined LT and FSFTs.


### 4.1. Solution of the Energy Equation

Equation (12), after using the Laplace transform, gives(15)qa0Peq+a1θ¯y,q=∂2θ¯y,q∂y2.There is a comparable modified form of equation (17):(16)ψ¯y,q|q=0=0,ψ¯1y,q|q=0=0,ψ¯y,q=0,ψ¯1,q=0,θ¯y,q|q=0=0,θ¯0,q=1q2,θ¯1,q=0.Take∫0d andSinnπ multiply both sides of equation (15). Then, using the beginning boundary conditions, we apply the FSFT. We get(17)θ¯FTη,q=m1q2q+a1q+m2.Equation (17) is more appropriately represented as(18)θ¯FTη,q=m1a1q2m2−a1−m2m1m22q+a1−m2m1m2+qm22.In equation (18), the inverting LT is obtained as(19)θFTη,t=a1m1m2t+m3exp−t⋅m2−1.Here,(20)a0=11−α,a1=α1−α,m1=nπn⋅π2+a0Pe,m2=a1nπ2nπ2+Pea0,m3=a1−m2m22m1.We take inverse FSFT of equation (19). We obtain the following temperature profile solution:(21)θy,t=t1−y+2∑n=1∞sinnπym31+exp−t⋅m2.Using equation (16) and the LT of equation (11), we obtain(22)qa0q+a1ψ¯1y,q=Pmψ¯y,q−ψ¯1y,q.After some simplification, we have(23)a2q+a3ψ¯1y,q=q+a1ψy,q.Here,a2=Pm+a0/Pm, a3=Pma1/a0+Pm.


### 4.2. Solution of the Momentum Equation

After applying the LT to equations (10) and using (16) and (23), we get(24)Rea0qq+a1ψ¯y,q=11+λ1d2ψ¯y,qdy2+λa0qq+a1d2ψ¯y,qdy2+Ka4q+a5q+a3ψ¯y,q+Grθ¯y,q.Here,a4=1−a2/a2, a5=a1+a1a3/1−a2.Using the FSFT of equation (24) along with the ICs and BCs, we arrived at the following solution:(25)ψ¯FTη,q=b9m1q+a1q2q+m2⋅q+a1q+a3q2+b7q+b8.Equation (25) may be represented more clearly and precisely as follows:(26)ψ¯FTη,q=m4q2+m5q+m6q+b10+m7q−b11−m8q+m2,where(27)b0=1+λ1Rea0,b1=λa0,b2=K1+λ1a4,b3=Gr1+λ1,b4=b0+nπ2−b1nπ2+b2,b5=b0a3+nπ2a1+a3−nπ2b1a3+b2a1+a5,b6=nπ2a1a3+b2a1a5,b7=b5b4,b8=b6b4,b9=b3m1b4,b10=−b7+b72−4b82,b11=−b7−b72−4b82,m4=a12a3b9b10b11m2,m5=b9a12a3b10b11−a12a3b10m2+a12a3b11m2−a12m2b10b11−2a1a3b10b11m2b102b112m22,m6=b9a12a3b9−a12b10b9−2a1a3b10b9−2a1b102b9+a3b102b9−b103b9b102b10+b11b10−m2,m7=b9a12a3b9+a12b11b9+2a1a3b11b9+2a1b112b9+a3b112b9+b113b9b112b10+b11b11+m2,m8=b9a12a3b9−a12m2b9−2a1a3m2b9+2a1m22b9+a3m22b9+m23b9m22b10−m2b11+m2.Inverting the LT of equation (26), we obtain(28)ψFSFTη,t=m4t+m5+m6exp−b10t+m7expb11t−m8exp−m2t.When we inverse the FSFT of equation (28), we get(29)ψy,t=2∑n=1∞m4t+m5+m6exp−b10t+m7expb11t−m8exp−m2tsinnπy.Equation (29) fulfills set BCs, illustrating the correctness of our reported overall solutions.


### 4.3. 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).(30)Nu=−∂θy,t∂y|y=0,Cf=11+λ11+λDtαCF∂ψy,t∂y|y=0.


## 4.1. Solution of the Energy Equation

Equation (12), after using the Laplace transform, gives(15)qa0Peq+a1θ¯y,q=∂2θ¯y,q∂y2.There is a comparable modified form of equation (17):(16)ψ¯y,q|q=0=0,ψ¯1y,q|q=0=0,ψ¯y,q=0,ψ¯1,q=0,θ¯y,q|q=0=0,θ¯0,q=1q2,θ¯1,q=0.Take∫0d andSinnπ multiply both sides of equation (15). Then, using the beginning boundary conditions, we apply the FSFT. We get(17)θ¯FTη,q=m1q2q+a1q+m2.Equation (17) is more appropriately represented as(18)θ¯FTη,q=m1a1q2m2−a1−m2m1m22q+a1−m2m1m2+qm22.In equation (18), the inverting LT is obtained as(19)θFTη,t=a1m1m2t+m3exp−t⋅m2−1.Here,(20)a0=11−α,a1=α1−α,m1=nπn⋅π2+a0Pe,m2=a1nπ2nπ2+Pea0,m3=a1−m2m22m1.We take inverse FSFT of equation (19). We obtain the following temperature profile solution:(21)θy,t=t1−y+2∑n=1∞sinnπym31+exp−t⋅m2.Using equation (16) and the LT of equation (11), we obtain(22)qa0q+a1ψ¯1y,q=Pmψ¯y,q−ψ¯1y,q.After some simplification, we have(23)a2q+a3ψ¯1y,q=q+a1ψy,q.Here,a2=Pm+a0/Pm, a3=Pma1/a0+Pm.


## 4.2. Solution of the Momentum Equation

After applying the LT to equations (10) and using (16) and (23), we get(24)Rea0qq+a1ψ¯y,q=11+λ1d2ψ¯y,qdy2+λa0qq+a1d2ψ¯y,qdy2+Ka4q+a5q+a3ψ¯y,q+Grθ¯y,q.Here,a4=1−a2/a2, a5=a1+a1a3/1−a2.Using the FSFT of equation (24) along with the ICs and BCs, we arrived at the following solution:(25)ψ¯FTη,q=b9m1q+a1q2q+m2⋅q+a1q+a3q2+b7q+b8.Equation (25) may be represented more clearly and precisely as follows:(26)ψ¯FTη,q=m4q2+m5q+m6q+b10+m7q−b11−m8q+m2,where(27)b0=1+λ1Rea0,b1=λa0,b2=K1+λ1a4,b3=Gr1+λ1,b4=b0+nπ2−b1nπ2+b2,b5=b0a3+nπ2a1+a3−nπ2b1a3+b2a1+a5,b6=nπ2a1a3+b2a1a5,b7=b5b4,b8=b6b4,b9=b3m1b4,b10=−b7+b72−4b82,b11=−b7−b72−4b82,m4=a12a3b9b10b11m2,m5=b9a12a3b10b11−a12a3b10m2+a12a3b11m2−a12m2b10b11−2a1a3b10b11m2b102b112m22,m6=b9a12a3b9−a12b10b9−2a1a3b10b9−2a1b102b9+a3b102b9−b103b9b102b10+b11b10−m2,m7=b9a12a3b9+a12b11b9+2a1a3b11b9+2a1b112b9+a3b112b9+b113b9b112b10+b11b11+m2,m8=b9a12a3b9−a12m2b9−2a1a3m2b9+2a1m22b9+a3m22b9+m23b9m22b10−m2b11+m2.Inverting the LT of equation (26), we obtain(28)ψFSFTη,t=m4t+m5+m6exp−b10t+m7expb11t−m8exp−m2t.When we inverse the FSFT of equation (28), we get(29)ψy,t=2∑n=1∞m4t+m5+m6exp−b10t+m7expb11t−m8exp−m2tsinnπy.Equation (29) fulfills set BCs, illustrating the correctness of our reported overall solutions.


## 4.3. 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).(30)Nu=−∂θy,t∂y|y=0,Cf=11+λ11+λDtαCF∂ψy,t∂y|y=0.


## 5. Results and Discussion

The unsteady, incompressible, unidirectional flow of the Jeffery fluid along with dust particles and parallel plates and the time-fractional model are investigated in this article. It has also considered the effects of free convection and heat transmission. The exact solutions are obtained by using LT and FSFTs. Figures2–14 illustrate the effect of different physical parameters on the Jeffery fluid velocity distribution, dust particle velocity distribution, and temperature profile. The examination of several parameters for the base fluid 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.Figure 2
The impact of different values ofα on ψy,t.Figure 3
The impact of different values ofα on ψ1y,t.Figure 4
The impact of different values ofK on ψy,t.Figure 5
The impact of different values ofK on ψ1y,t.Figure 6
The impact of different values ofλ on ψy,t.Figure 7
The impact of different values ofλ on ψ1y,t.Figure 8
The impact of different values ofGr on ψy,t.Figure 9
The impact of different values ofGr on ψ1y,t.Figure 10
The impact of different values ofRe on ψy,t.Figure 11
The impact of different values ofRe on ψ1y,t.Figure 12
The impact of different values ofRe on ψy,t.Figure 13
The impact of different values ofRe on ψ1y,t.Figure 14
The impact of different values ofPe on temperature.In order to investigate the effects ofα on Jeffery fluid and dusty fluid 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 figures that we have several velocity outlines for different values. Although the other parameters are constant, the variation of fractional parameters α has a significant memory impact on both velocities, which shows that the fractional model is more realistic than the classical one.The effect of the dusty fluid parameterK on both velocities profiles is shown in Figures 4 and 5. It is clear from the figure that the velocity declines with the rising values of K. Physically this behavior is true, as, by Stocks drag formulaK=6πrμ, it is clear that increasingK increase viscous forces which resists the flow and consequently decreases both the velocities profile.Figures6 and 7 are drawn to study the impact of the Jeffrey fluid parameter λ on Jeffrey fluid and dusty fluid velocities. From these figures, it is clear that increasing the Jeffrey fluid parameter λ retards both velocity profiles. Physically, the Jeffrey fluid parameter λ increases non-Newtonian behavior or causes viscous forces, which raises the thickness of the momentum boundary layer. This results in a drop in both velocities.TheGr behavior of dust particles and Jeffrey fluid velocities is shown in Figures 8 and 9. It is clear from these figures that increasing the values of Gr enhances both velocity profiles. Physically, due to the increasing Gr, buoyant forces rise and viscous forces reduces, which enhance both velocities.Figures10 and 11 show the effect of the Jeffrey fluid and dust particle velocity profiles. By raising the values of Re, the profile of both the Jeffrey fluid and dusty fluid velocities is produced. As the Reynolds number rises, more inertial forces are generated, slowing flow behavior because Re is the ratio of inertial to viscous forces.Figures12 and 13 depict the impact of Pm on dust particle velocity and Jeffrey fluid 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 Jeffrey fluid velocity.The result ofPe on the temperature outline is plotted in Figure 14. By raising the values of Pe the temperature profile 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 fluid decrease, and therefore, temperature distribution decreases.Variations in the rate of heat transmission inPe 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 profiles.Table 1
Variation in the Nusselt number for different values ofPe.
PeαtNu10.510.2227320.510.8277230.511.4588440.512.94575Table 2
Variation in skin friction for different values ofα.
αtλKRePeGrPmCf0.10.63110250.12.060330.30.63110250.11.912120.60.63110250.10.940321.00.63110250.10.30632


## 6. Conclusions

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



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