Analysis of a Two-Phase MHD Free Convection Generalized Water–Ethylene Glycol (50 : 50) Dusty Brinkman-Type Nanofluid Pass through Microchannel

Khan, Dolat; Almusawa, Musawa Yahya; Hamali, Waleed; Akbar, M. Ali

doi:https://doi.org/10.1155/2023/3099858

Journal of Mathematics

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 3099858 | https://doi.org/10.1155/2023/3099858

Analysis of a Two-Phase MHD Free Convection Generalized Water–Ethylene Glycol (50 : 50) Dusty Brinkman-Type Nanofluid Pass through Microchannel

Dolat Khan,¹Musawa Yahya Almusawa,²Waleed Hamali,²and M. Ali Akbar³

Academic Editor: M. M. Bhatti

Received28 Oct 2022

Revised30 Nov 2022

Accepted18 Apr 2023

Published09 May 2023

Abstract

The water–ethylene glycol (50 : 50) nanofluid has applications in the manufacture of polyester as a raw agent, air conditioning systems, antifreeze formulation, dehydrating agents in the gas industry, a precursor in the plastic industry, and convective heat transfer. These developments in nanotechnology and nanoscience have caught the interest of several researchers. Because it keeps machines and engines cool by reducing friction between their different parts, grease is a vital part of many machinery and engines. Also, due to the extensive use of fractional derivatives, this work seeks to evaluate the combined impacts of free convection flow and heat transfer, magnetic field, and Brinkman-type water–ethylene glycol (50 : 50) dusty nanofluid among microchannel. The flow that the buoyant force provides helps to carry heat naturally via convection. While the left plate moves at a consistent velocity and the right plate stays stationary, the fluid is also evenly dispersed with all dust particles that have a spherical form. Partial differential equations (PDE) are used to present the mathematical modeling. The resultant PDEs are generalized by utilizing the Caputo–Fabrizio fractional derivative. The problem’s closed-form solution is produced by combining a Laplace transformation with a finite sine Fourier transformation. It has also been studied that temperature, Brinkman nanofluid, and dust particle velocity relate to a variety of other factors, such as the magnetic parameter, Grashof number, dusty fluid parameter, and volume friction parameter. The graphical outcomes for the dusty fluid, Brinkman nanofluid, and temperature profiles are plotted using Mathcad-15. The Brinkman nanofluid and dusty fluid behave similarly for a variety of embedded factors. It is found that compared to the traditional one, the fractional dusty nanofluid model displays more realistic characteristics. The addition of nanoparticles in water–ethylene glycol (50 : 50) dusty nanofluid enhances the rate of heat transfer up to 41.04478% by increasing their volume fractional.

1. Introduction

Materials with a size of 100 nanometers or less are known as nanomaterials, and the technology used to create them is referred to as nanotechnology. When classifying nanomaterials into one of four classes, attention is given to both the structure and the properties of the materials [1]. The first person's to look at the terminology used with nanofluids were Choi and Eastman [2]. To describe fluids containing particles with a diameter of less than 100 nanometers, he coined the term “nanofluid.” Outline the justification for why nanoscale particles are preferred over microscale particles in many applications [3]. When comparing nanoparticles to microparticles, significant increases in thermophysical properties have been seen. Nanofluids may be used in several ways, such as improving the efficiency of diesel generators, cooling air conditioning systems, and cooling power plants [4]. Rehman et al. [5] analyzed the MHD flow of carbon in micropolar nanofluid in rotating frame geometry in addition to convective heat transfer. Additionally, Ijaz Khan et al. [6] reported the relevant issues in optimizing entropy formation in nanomaterial flows with mixed convection. While the mixed convection flow of a tangent hyperbolic nanofluid was reported by Khan et al. [7]. To get a more realistic study Khan et al. [8] investigated the fractional model in the drilling process based on hybrid nanofluids with heat generation and porous medium. Furthermore, the growth of nanomaterials has made it possible to produce nanoparticles (also known as nanocomposites). Many researchers have lately focused on the properties of hybrid nanofluids, as mentioned by Leong et al. [9]. Khan et al. [10] present their analysis and evaluation of the joint Sumudu and Laplace transform for the MHD unsteady flow of hybrid nanofluid in a rotating frame. In order to represent and solve fractional differential equations for heat transportation in nanofluids, Saqib et al. [11] investigated the application of integral transforms.

Fractional calculus has useful applications in many areas of engineering and research, including viscoelasticity, including electromagnetics, fluid mechanics, biological population models, electrochemistry, signals processing, and optics. In order to solve various physical and natural issues that the conventional derivative cannot adequately represent, fractional calculus is applied, as shown in Ross [12]. Fractional derivatives have drawn considerable interest from scientists during the past 30 years. In response, numerous researchers have proposed different ways of defining fractional derivatives. In the 18^th century, Riemann–Liouville [13] became the most commonly used definition. The R-L concept of the fractional derivative has been faithfully reproduced in many physical systems, but it is only useable because of two key features. The Caputo fractional derivative has been extensively studied in the domains of chemistry, economics, physics, and other physical concerns. Exploration of diffusion, signal processing, image processing, damping, pharmacokinetics, material mechanics, and bioengineering processes are all conceivable using CFD [14–16]. To address concerns with the R-L formulation, CFD [17], a modified version of fractional calculus, was created. Yet because the CFD kernel has a singularity, it cannot be used to correctly describe materials with large heterogeneities [18]. To address this problem, a new definition with a nonsingular kernel has been proposed by Caputo–Fabrizio [19], and this novel idea has been applied in the research of several scholars [20–22]. Mohammadi [23] used fractional-time CF and Caputo derivatives to investigate the flow of couple stress fluids (CSF) between the two parallel plates. Research on the two-phase laminar flow of dusty fluid on elastic and stretching sheets was explored in detail by Mahanthesh [24].

Various branches of bioengineering, the pharmaceutical and chemical industries, mechanical engineering, nuclear power facilities, and so on all make use of fluids in some way. Scientists and researchers are increasingly focusing on fluids because of their widespread use, primarily discussing and debating their current and future capabilities, uses, and aims. Numerous mathematical models are put out in the literature to characterize the behavior of various fluids. The Brinkman model is among the most straightforward and widely used of these many options. A framework that may be used in huge, permeable regions was created by Brinkman’s groundbreaking study [25]. He examined the behavior of fluid flow when dynamic viscosity develops on the surface of a dense cluster of small spheres using the Navier-Stoke equation.

The numerical technique was used by Liu et al. [26] to examine the continuous dependency of the Brinkman–Forchheimer fluid interacting with a Darcy fluid in a bounded region. Ramzan et al. [27] looked at the impacts of heat radiation and the Brinkman fluid on chemical reaction. It has been shown that heat radiation and mass Grashof numbers have an increasing influence on fluid velocity, but chemical reaction and magnetic field have a diminishing impact. Furthermore, the research conducted by Kumar et al. [28] focused on investigating the influence of diffusion and thermodynamics on the flow of a Brinkman-type nanofluid, which is both heat-radiating and chemically reactive. The flow was unsteady and hydromagnetic and occurred near an exponentially accelerated vertical plate through a porous medium. The temperature of the plate varied in an arbitrary ramped manner. Heat transfer, thermal radiation, chemical reactions, and magnetohydrodynamics were studied by Khan et al. [29] in the context of Brinkman fluid across a perpendicular plate. Dubey and Murthy [30] investigated how double-diffusive disease first manifests itself. The dust fluid through Brinkman porous media was reported by Kumar and Mohan [31].

The fluid containing the dust particles is used in a wide variety of mechanical procedures, including transport, the production of cement and steel, the removal of flying ash from thermal power plants, and the reduction of the unfavorable impacts of air conditioners. Numerous commercial applications exist for two-phase flows that include solid particles dispersed in nanofluids or hybrid nanofluids. Many scientists have been studying fluid flow with embedded dust particles lately. Experts are interested in dusty fluid models because they are two-phase systems. These occurrences occur in fluid flows containing dispersed solid particles. Both the analysis and the solution to the problem of the movement of dusty particles have been sped up. Saffman [32] is credited with initiating the study of laminar flow in dusty fluids. Specifically, Chakrabarti [33] looked at the behavior of a dusty gas as it approached the boundary layer. The motions of the dust fluid in densely interconnected dusty plasmas were investigated by Shukla and Stenflo [34]. We show that dusty plasmas with strong couplings may display zonal winds or vortex-like dust fluid movements when subjected to shear waves of sufficiently large amplitude.

In view of the above literature, the main objective of the present research is to investigate the water–ethylene glycol (50 : 50) dusty nanofluid using a newly created C-F time fractional derivative to give an exact solution. For the enhancement of heat transfer, cobalt ferrite, and magnetite nanoparticles are dispersed in the base fluid. The fluid is taken into a microchannel. In a two-phase flow model, fluid and dust particle equations are calculated separately. C-F time fractional derivative used to generalize the approach for more realistic. The exact solution is obtained by coupling Laplace with the finite sine Fourier transform. Nusselt number and many embedded variables effects are graphically shown here using Mathcad-15.

2. Formulation and Solution

Assume that a water–ethylene glycol dusty nanofluid of the Brinkman type is flowing unsteadily and incompressibly between microchannels. The fluid as a whole evenly disperses a dust particle. The distance between two vertical, equivalent plates is represented by , as shown in geometrical Figure 1. The fluid motion is along the x-direction. The uniform magnetic field is transverse to the dusty nanofluid. The left plate moves with , while the right plate stays stationary. While denotes the temperature of the right plate by , the left plate’s temperature is .

As we have shown, there is a relationship between velocity and temperature [24]:

Utilizing the conditions:

Expressions for are given by [29]

We define the following dimensionless variables to create a dimensionless system of PDEs [24]:

When we plug equation (6) into equations (1), (2), (3), and (4), we get the following dimensionless form:

This includes physical conditions that have no dimensions, such as [24]

We now get the following the fractional parameter and CFD in equations (7) and (8), (9), respectively:where is the fractional parameter [35] and is the CF for the fractional operator:

Such that, ,

3. Solutions to the Problem

We get precise solutions to the system by combining the joint LT and FSFT.

3.1. Calculation of the Temperature Profile

Equation (14) Laplace transform yields the following results:

Likewise, the converted version of equation (10) is as follows:

Multiplying equation (16) by on both sides and then integrating it with limits from 0 to yields the following expression: Using starting and boundary conditions and then using the FSFT, we obtain:

Equation (18) may also be expressed as follows:

Given is the inverse LT of equation (19).here,

Equation using the inverse finite sine-Fourier transforms (20). We get the temperature profile’s solution:

Equation (13)’s Laplace transform is used, together with equation (17), to provide the following results:

As a consequence, following computation, we obtainhere,

3.2. The Velocity Profile Solution

Taking use of the following transformation [23]:

Equation (26) is applied to equation (12), and the solution is

Ics and BCs with transformations are as follows:

Equations (27) and (28), when applied with the Laplace transform, result in

Next, we apply the finite sine Fourier transforms to equation (29) to obtain

Equation (30) can be expressed as follows:where

Laplace inversion of equation (31), as

Equation (33) is inverse FSFT, as follows:

The following is obtained from (26) using equation (34):

4. Special Case: Without Particle Velocities, the Brinkman Nanofluid Model Shows

When we put into equation (8), we get a model for a fluid with a single phase of flow.

Putting equation (36) via its definition of fractional derivatives , we get

A Laplace transform applied to equation (37) ultimately yields:Transforming (38) into a function of its FSFT, we get

Using partial fraction w.r.t. s, we may decompose the right-hand side of equation (39), yielding:where

To get (26), we substitute the inverse transform of (40) using the joint LT and FSFT:

4.1. Nusselt Number

Nusselt number is obtained from equation (22) by using the connection from Khan et al. [29].

5. Results and Discussion

This article aims to investigate generalized Brinkman-type water–ethylene glycol (50 : 50) dusty nanofluids among microchannels that incorporate unsteady flow, heat transport, and MHD analysis. Combining Laplace and FSFT creates closed-form solutions. Figures and tables demonstrate how variables affect fluid and particle velocities and temperature.

Figures 2(a) and 2(b) show the study of the Brinkman parameter and contrast the velocities of Brinkman-type nanofluids and dust particles with those of Newtonian viscous fluids. When compared to a fluid of the Brinkman type , a Newtonian viscous fluid has a high velocity. This illustrates that Brinkman parameter values delay velocity profiles more than Newtonian viscous fluids. Higher Brinkman values increase drag, slowing fluid and dust motion.

(a)

(b)

The effect of a fractional parameter on the velocities of the Brinkman nanofluid and dusty fluid is examined in Figures 3(a) and 3(b). On both velocities, the memory effect is quite significant. When the other parameters are constant, changing the fractional parameter leads to different solutions for the fluid and dust particle velocities. Which is a suitable option for experimental research so that the findings can be more accurately compared. The many solutions demonstrate that the fractional model is more plausible than the classical one.

(a)

(b)

The investigation of on fluid and dust particle velocity is covered in Figures 4(a) and 4(b). Therefore, it is evident that as the rises, so do the buoyant forces. Physically, the buoyant forces in the fluid, form in reaction to the greater temperature and quicken the fluid’s velocity.

(a)

(b)

The impact of the volume fraction of fluid and dusty fluid on the velocities profile is seen in Figures 5(a), and 5(b). The velocity of the fluid decreases as its volume fraction increases, indicating that the nanoparticles are physically slowing the fluid’s flow. The resistive forces that cause the nanoparticles to slow down in the channel were created when the fluid’s nanoparticle concentration increased. The viscosity of the fluid rises with fluid concentration, decreasing both the fluid and dusty fluid velocities.

(a)

(b)

The results are shown as the impact of on the velocities of Brinkman-type nanofluids and dust particles in Figures 6(a) and 6(b). Increased dust particle mass caused by rising causes a slowdown in the flow of nanofluid and dust particle velocity. Physically, the fact is that, the fluid became more viscous with higher values of the mass particle parameter, which is why there is an apparent slowdown at both velocities.

(a)

(b)

Figures 7(a) and 7(b) illustrate how affects the velocity of a nanofluid and dust particle. The viscosity of the momentum boundary layer contracts as increases, increasing the Lorentz forces that are resistant to fluid flow and delaying the velocity of nanofluid and dust particles.

(a)

(b)

It is clear that a rise in would result in a delay in the viscous forces, which would increase fluid velocity, according to Stocks’ drag formula () for sphere-shaped dust particles. As shown in Figures 8(a) and 8(b), increasing the amount of dust particles increases the nanofluid and dust particle velocities (b).

(a)

(b)

Figure 9 reports the effect of the fractional perimeter on the temperature profile. This, by fixing the other parameters, increases the generalization of the heat profile relative to the traditional one. Physically, this shows that the temperature profile may be made more realistic by altering the fractional parameter, which provides additional solutions.

The impacts of volume friction on depicted in Figure 10. The range is between 0 and 0.04 when it reaches more than 0.04 segmentation occurs. A rise in the nanoparticle volume friction percentage will, in either scenario, result in a lower temperature as well as a change in the temperature distribution.

Table 1 displays the thermophysical characteristics of nanoparticles for your review. Table 2 shows the difference in skin friction caused by different parameter values. Skin friction is significant in several engineering areas, especially civil engineering. The viscous forces and, consequently, the surface friction increase as the raises. Skin friction is decreased by raising the . By raising the buoyancy forces to rise, the viscosity drops, and the surface friction goes down as a result. Table 2 clearly illustrates how skin friction reduces as volume friction increases.

Table 3 displays the Nusselt number against the volume fraction of nanofluid. It is clear that the enhancement of volume fractional makes the fluid more viscous, and as a result, the rate of heat transfer enhances up to 41.04478%.

The comparison between the present study and that of Narahari and Pendyala [36] is shown in Figure 11, which demonstrates a high degree of congruence. The solutions are shown to be sound and valid, as each one precisely overlaps with the other.

6. Conclusions

In this study, the investigation of the generalized Brinkman-type dusty nanofluid at free convection temperature is the major focus. While the dust scatters evenly across the whole fluid, the flow is between the microchannels. Physical boundary conditions and PDE are used to formulate the issue. By combining the usage of the Laplace transform with the finite sine Fourier transform, the issue is generalized without a single kernel using Caputo–Fabrizio’s fractional operator. The program Mathcad-15 is used to carry out parametric investigations. The validity and correctness of the present answers have been proven in several exceptional instances. The main findings from this study are as follows:(i)Compared to the classical model, the fractional model of dusty nanofluid has a more realistic feature.(ii)The variation of the fractional parameter gets more realistic results for skin friction and heat transfer rate.(iii)By combining Laplace transform and FSFT, we can reduce the computational time needed to obtain exact solutions for these types of problems.(iv)The increase in expands the nanofluid and particle velocities.(v)The water–ethylene glycol nanofluids improve the heat transfer rate up to 41.04478%.(vi)The findings of this study have practical applications in the engineering and product manufacturing industries. For example, dusty fluids are commonly used in gas cooling systems and nuclear reactors.

Nomenclature

:	Variable temperature
:	Ambient temperature
:	Time
:	Coefficient of thermal epansion
:	Applied magnetic field
:	Velocity of the nanofluid
:	Constant velocity
:	Specific heat capacity
:	Skin friction
:	Velocity of the particle
:	Caputo–Fabrizio
:	Nondimensional Brinkman fluid parameter
:	Density
:	Is the distance between parallel plates
:	Grashof number
:	Dynamic viscosity
:	Gravitational acceleration
:	Kinematic viscosity
:	Electrical conductivity
:	Heaviside step function
:	Fractional parameter
:	Stock’s resistance coefficient
:	Dusty fluid parameter
:	Nondimensional magnetic parameter
:	Thermal conductivity
:	Dimensionless temperature of the fluid
:	Normalization function
:	Number density of the dust particles
:	Nusselt number
:	Prandtl number
:	Particles mass parameter
:	Temperature of the fluid
:	Temperature of wall
:	Mass of the particle
:	Nanofluid
:	Nondimensional Brinkman fluid
:	Fluid
:	Nanoparticle.

Data Availability

All the data in this study are available in the manuscript.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

Dolat khan was responsible for supervision, methodology, and draft writing, Musawa Yahay Almusawa was the project administrator and methodology. M. Ali Akbar was responsible for funding and draft writing. Waleed Hamali was responsible for the investigation, methodology, and draft writing.

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