1. Introduction
Nanotechnology has significantly advanced in heat transfer studies, which has enhanced the thermal characteristics of energy-transmitting fluids. Producing nanoparticles with great heat conductivity is one of the most trending uses of nanotechnology. To increase the thermal conductivity of fluids, nanofluids have great importance. They are prepared in laboratories by using nanoparticles with an average diameter of less than 100 nm which are suspended in typical heat transfer fluids such as oil, water, and ethylene glycol. First, Maxwell [
1] proposed nanofluids after an attempt to optimize the heat transfer rate of regular fluids by suspending micro-sized particles failed owing to sedimentation and clogging of the flow patterns. Based on these issues, Choi [
2] suggested in 1995 that the dispersion of nanoparticles into the host fluid might improve the thermal performance of the base fluid. Subsequently, a diverse range of devices have been developed for a variety of practical purposes and functions in various fields such as electrical engineering [
3], helping to improve the thermal efficiency of horizontal spiral coils used in solar ponds [
4], as a coolant in double pipe heat exchangers [
5], stenotic artery [
6] and drug agent [
7]. Later research by Imran Siddique et al. [
8], Maryam Aleem et al. [
9], as well as Anum Shafiq et al. [
10] broadened the literature on nanofluids. The discovery of nanofluids has achieved the major part of industry’s requirements, but the suspension of single nanoparticles is inadequate for the required thermal performance. Therefore, researchers have been attempting to develop a better and more efficient fluid. Yamada et al. [
11] defined an upgraded kind of nanofluid in 1989 by combining two or more nanoparticles of distinct characteristics with common fluids. This advanced categorization of nanofluid, known as a hybrid nanofluid, shows potential improvements in heat transfer rate, which can be applied in many domains such as biomedicine [
12,
13,
14], heat exchangers [
15], solar energy [
16] and so on. Some of the modern advances in the literature of hybrid nanofluids are observed in the studies carried by Hafeez et al. [
17]; their study provides a numerical modelling of MHD rotational flow of hybrid nanomaterial by applying a bvp4c technique between two parallel porous sheets. Iskandar Waini et al. [
18] examined the stable mixed convection flow along a vertical surface immersed in a porous medium using hybrid nanoparticles. Talha Anwar et al. [
19] established two independent fractional models, Caputo–Fabrizio and Atangana–Baleanu, to analyze the flow patterns and thermal characteristics of a NaAlg/SA-based hybrid nanofluid. Their study revealed that the CF fractional operator improves the thermal rate more efficiently than the AB fractional operator.
Heat transmission is crucial for temperature controls in many industrial applications. Even with increased demand for energy-efficient equipment, achieving good heat transmission of a fluid remains a challenge. As a result, nanoparticles, nanofluids, and hybrid nanofluids exploration are some of the most significant topics of research. Consequently, heat transfer becomes more robust. Nepal T. Balaji et al. [
20] investigated the micro channel heat sink, which is used to check the convective heat transfer properties of water-based hybrid nanofluids including graphene nanoplatelets and MWCNTs. Mumtaz Khan et al. [
21] examined FDM combined with L1-technique utilization to perform the heat transfer of fractional transient MHD flow of viscoelastic hybrid nanofluids through an inclined surface fixed in a Darcy porous medium. Unsteady natural convection and heat transmission of hybrid nanofluid for two upright parallel plates were analyzed by Chandra Roy and Ioan Pop [
22]. In the fields of biomechanics, aerospace, and chemical engineering, magnetohydrodynamics (MHD) free convection flow is extremely important. MHD primarily focuses on the study of the magnetic characteristics and behavior of electrically conducting fluids, including magneto fluids such as electrolytes, liquid metals, plasmas and salt water. Ndolane Sene [
23] examined the heat transmission analysis of Brinkmantype fluid with Caputo derivative. Zar Ali Khan et al. [
24] found the analytic solution of the transient flow of a generalized Brinkman-type fluid in a channel under the influences of MHD with Caputo–Fabrizio fractional derivative. Ridhwan Reyaz et al. [
25] explored the effects of heat radiation on the MHD Casson Fluid as well as the Caputo fractional derivative on an oscillating upright plate.
The investigation of non-Newtonian materials is another intriguing research issue due to its interdisciplinary character and interesting rheological dynamics. Non-Newtonian fluids are flexible due to their applicability in numerous sectors and production processes. The relevance of non-Newtonian fluids may be seen in the oil packing, cooling/heating processes, hydraulics, lubricant industry and opto-electronics. In the literature, scientists have researched many non-Newtonian models, among which is included the Casson Fluid model [
26], made known in 1959 by Casson, while inspecting the rheological data of pigment ink in a printer. Casson Fluid is a shear-thinning liquid with infinite viscosity at zero shear stress. When the yield stress is higher than the shear stress, the fluid acts like a solid. Toothpaste, slurries, blood, paint, molten polymers, honey, jelly, tomato sauce and chocolate are examples of Casson Fluid. This fluid model has been beneficial to polymer processing industries, food manufacturers, cosmetics, textiles, biomechanics, pharmaceuticals and many more. Ali Raza et al. [
27] investigated the flow of Casson nanoparticles by applying Laplace Transform across a vertical moving plate using the Atangana–Baleanu time-fractional derivative and studies have shown that the fractional, ordinary velocity fields of Casson Fluid decreases when compared to viscous fluid. Muhammad Nazirul Shahrim et al. [
28] were using the Laplace Transform to study the precise solution of fractional convective Casson Fluid over an accelerated plate. M. Veera Krishna et al. [
29] explored the radiative MHD flow of Casson hybrid nanofluid through an infinite exponentially accelerated vertical porous surface using the Laplace methodology, and the temperature of Casson hybrid nanofluid is considerably superior to that of Casson nanofluid.
In present times, fractional calculus [
30] is essential in engineering and applied scientific disciplines such as physics, electronics, mechanics, population modelling, biosciences, economics and signal processing. Fractional calculus contains two categories singular operators and nonsingular operators. (1) Caputo derivative (2) Riemann–Liouville derivative are singular operators. The Caputo–Fabrizio derivative and the Atangana–Baleanu derivative are non-singular operators. They arose as a result of the application of conventional differentiation to the concept of non-local derivatives. As per several subject specialists, the findings obtained through the use of fractional operators are more precise and realistic than those obtained using classic differentiation. When it comes to understanding fluid performance, fractional operators are extremely important because of their self-similar qualities and memory-capturing capabilities. The Caputo derivative is the most commonly encountered derivative in the fractional calculus literature. The rationale stems from the fact that this derivative is consistent with the initial conditions utilized in modelling real-world issues. Michele Caputo proposed the Caputo fractional derivative in his study in the year 1967 [
31]. Talha Anwar et al. [
32] analyzed different shape effects of fractal fractional model for thermal analysis of hybrid nanofluid with a power-law kernel and noticed that the heat transfer rate was most effective for blade-shaped nanoparticles when graphene nanoparticles and graphite oxide were equally dispersed. Muhammad Saqib et al. [
33] used the Atangana–Baleanu fractional derivative to examine the time fractional model of the convective flow of carboxy–methyl–cellulose (CMC)-based CNTs nanofluid through a porous media in a microchannel and observed that MWCNTs are more efficient than SWCNTs in improving the thermal conductivity of the nanofluids. Marjan Mohd Daud et al. [
34] implemented the Caputo fractional derivative principle to Casson Fluid convective flow in a microchannel with radiant heat flux. M Ahmad et al. [
35] described a generalization for natural convection flow of Maxwell nanofluid in two upright parallel plates adopting Caputo–Fabrizio utility of fractional order derivatives. Sidra Aman et al. [
36] derived precise estimates for MHD flow of Casson nanofluid with hybrid nanoparticles using the Caputo time fractional derivatives.
Being motivated by Ndolane Sene [
37], who analyzed the exact solution for a class of fluids model with the Caputo derivative by using Laplace and Fourier Sine Transform method, it is noticed that there has been no attempt in the prior literature to investigate MHD and hybrid nanofluids with Caputo fractional derivatives by using Fourier Sine Transform and Laplace Transform. Hence, the current study proposes to expand on the work of Ndolane Sene by adopting MHD with different shapes of hybrid nano fluid model using graphene (Gr), multiwall carbon nanotubes (MWCNTs) as nanoparticles and water as host fluid to analyze the heat transmission rate. The implementation of the Caputo derivative and its approach to obtaining the analytical results by employing the Laplace and Fourier transforms will be novel. The Caputo fractional derivative is used to fractionalize the MHD free convection Casson hybrid Brinkman-type fluid model. The Fourier sine and Laplace Transformation is used to transform non-linear governing PDEs into ordinary differential equations. These exact solutions are shown for temperature and flow fields of hybrid nanofluid. Eventually, by making
the classic non-Newtonian solutions are recovered for velocity field. Further, the influence of several parameters on the fluid flow and thermal characteristics were discussed and shown in graphical and tabular form. The practical applications of employing these nanoparticles are in wastewater treatment, 3D printing, solar cell (dye-sensitized solar cells) industries.
The contents of the present paper are outlined as follows.
Section 2 describes the fractional mathematical model using Caputo fractional derivatives.
Section 3 gives the approaches to obtain analytical solutions using Fourier sine and Laplace Transform methods for temperature and velocity fields. Further discussed are the limiting cases, heat transmission rate and shearing stress. Discussion and the interpretations of the influences of the parameters utilized in the modelling have been provided in
Section 4. We conclude the paper with findings which are discussed in
Section 5.
2. Fractional Mathematical Model with Caputo Derivative
Consider an unsteady MHD free convective Casson hybrid flow of water with graphene and MWCNTs nanoparticles over an infinite upright plate. The system rectilinear coordinate is implemented for the analysis, and the fluid flow is taken in the y-direction, whereas the
x-axis is picked perpendicular to the plate. Magnetic field of strength
is applied normal to the fluid flow direction. The fluid is viscous, incompressible, conducting and not electrified. The fluid is assumed to be gray, absorbing and emitting radiation but as a non-scattered medium. Different forms of nanoparticles (cylinder, blade, brick, platelet and spherical) are disseminated into the host fluid to obtain hybrid nanofluid. At time
, the plate and hybrid nanofluid are both in equilibrium state with temperature
. As time progresses,
, the fluid is driven by the velocity U and at the same time, temperature of the fluid raised to
and then far away from the plate its ambient temperature is
, causing free convection to occur, as presented in
Figure 1. Body force emerges as buoyancy force in this circumstance because of the temperature difference. Further, for analyzing the flow phenomena of the hybrid nanofluid, the Brinkman-type fluid model is being used.
The following forms can be used to depict the rheological equation for an incompressible Casson Fluid (Nakamura et al. [
38]).
Here, where represents the component of the deformation rate, is the product of the component of deformation rate with itself, is the critical value of this product based on the non-Newtonian model, symbolizes the yields stress, and denotes the plastic dynamic viscosity of the non-Newtonian flow.
The mathematical structure of the corresponding conventional flow of Casson hybrid nanofluid (graphene–MWCNTs–H
2O) can be concise by Boussinesq’s approximation (Nehad Ali Shah and Ilyas Khan [
39]) with the following partial differential governing equations given under the aforementioned assumptions.
The dimensional initial and boundary conditions employed in this study are detailed below.
Table 1 lists the thermo-physical attributes of hybrid nanofluids and nanofluids.
Table 2 portrays the thermophysical properties of the host fluid (H
2O) and nanoparticles (graphene and MWCNT).
Table 3 displays the sphericity and shape factor for various shapes of nanoparticles.
The following non-dimensional parameters are constructed using Buckingham’s pi- theorem (W.D.Curtis [
41]).
When transforming the Equations (2)–(5) using the dimensionless variables specified in Equation (6) and further dropping * sign, a more simplified form of the non-dimensional fluid model is obtained as:
Fractional calculus is an effective tool for describing real-world phenomena with the so-called memory effect. The Caputo derivative is used because the memory effect and a constant function’s derivatives yield zero. Equations (9) and (10) are obtained by replacing the integer order derivative with the Caputo derivative in Equations (7) and (8) and generalizing the integer-order derivative to non-integer partial differential equations. They are:
We treat the following relationships as dimensional initial and boundary conditions that momentum and temperature satisfy.
where