Abstract
A fractional model is developed to investigate the thermal onset of carbon nanotubes containing single-wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs). The blood and carboxymethyl cellulose (CMC) are utilized to report the characteristics of the base material. The thermal phenomenon is further supported with inclined magnetic force and mixed convection features. The vertical plate with an oscillatory nature induced the flow. After formulating the problem in view of flow assumptions, the fractional framework is carried out via the Prabhakar technique. The validation of the fractional model is ensured in view of previous studies. The comparative thermal aspect of carbon nanotubes and base materials by varying flow parameters is tested.
1. Introduction
The recent trend in thermal engineering proposes a cheaper source of energy based on the utilization of nanoparticles in the current century. The thermal mechanism of various base fluids is usually lower and less stable. The continued work in nanotechnology has experimentally proven that base materials’ thermal onset can be improved when nanomaterials are immersed in a proper way. The nanoparticles are low sized metallic particles which reports exclusively enhanced thermal impact. Recent applications of nanomaterials are commonly noticed in thermal management systems, chemical processes, as a heating source in industries, solar systems, extrusion processes, etc. The work on nanofluids was initiated by Choi [1], and it is being further extended in different directions by scientists. Turkyilmazoglu [2] discussed different thermal aspects of nanofluids with the implementation of a single-phase model and tested their stability properties. Ahmad et al. [3] incorporated the insight thermal onset of micropolar nanofluid by incorporating the modified heat flux relations. Thumma et al. [4] observed the optimized contribution of nanofluid with wall-heated properties. The binary chemical flow regarding the nanofluid flow was addressed in the continuation of Abbasi et al. [5]. Rasool et al. [6] intended the nanofluid flow to be subjected to the Darcy–Forchheimer phenomenon. The isothermal conducting flow of Maxwell nanofluid with the contribution of Lorentz forces has been depicted by Rasool et al. [7]. Shafiq et al. [8] investigated the Casson fluid properties due to nanofluids with magnetic force impact. Ali et al. [9] observed the elongated surface moving flow subject to hybrid nanofluids along with the rotation phenomenon. Mahesh et al. [10] approached the non-Fourier framework for nanofluid flow in view of entropy generation applications.
In contrast to simple materials, carbon nanotubes (CNTs) report more impressive thermal impact and stable properties like electrical and thermal conductivities, density, dimensions, and size. In fact, carbon nanotubes (CNTs) are taken as cylindrical molecules consisting of rolled-up sheets of carbon atoms with a single layer. The CNTs are classified as single-walled carbon nanotubes (SWCNT) and multiwalled carbon nanotubes (MWCNT). The diameter of SWCNT is usually less than one nm, while the diameter of (MWCNT) is approaching 100 nm. Different aspects of CNTs have been studied by researchers with diverse flow features. Reddy et al. [11] discussed the cavity flow due to the uniform distribution of CNTs along with optimized consequences. Noranuar et al. [12] evaluated the rotating orientation of CNTs with Casson fluid following the disc flow. Imtiaz et al. [13] reported the fluctuated thickness features for CNTs with bidirectional moving regime. The heating object on the enclosure with CNT distribution has been depicted in the continuation of Vishnu Ganesh et al. [14]. Shoaib et al. [15] worked out the neural computing investigation for a CNTs problem. Alzahrani and Ijaz Khan [16] focused on the coating simulation of CNTs’ flow with the contribution of Wu’s slip evaluation.
The fractional research provides modern tools of computations for performing the analytical and numerical simulations. The widely work in the area of fractional mathematics, different algorithms are defined by researchers. Such tools are important to define the solution to differential and integral problems. The motivations and valuable applications of such tools are associated with the computation of various problems in engineering, industrial development, thermal engineering, chemistry, physics, bio-engineering, and the computational sciences. After focusing on different fractional tools, it is observed that the Caputo and Fabrizio (CF) technique is the first interesting and novel fractional definition, which has been widely used by researchers for different problems [17–19]. Atangana and Baleanu [20] provided a new type of fractional algorithm with a stronger approach. The AB tools provide a modification of the CF approach and enable simulations for nonsingular kernels [21–24]. In different fractional studies, the Prabhakar fractional approach is another analytical framework that is not focused in a comprehensive way. The distinct aspect of this fractional approach is the assessment of the odd behavior of a fluctuating person with a nonlocal kernel. Following this definition, the nonsingular and nonlocal kernel problems can be effectively treated [25–27].
The current investigation provides fractional simulations for oscillating flows of carbon nanotubes to improve the thermal properties of blood and carboxymethyl cellulose (CMC)-based fluids. The thermal classification of CNTs is observed by using single-wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs). The mixed convection features are attributed to natural convection flow. The fractional computations are performed via the Prabhakar fractional model. This model provides answers to the following research questions:(i)How carbon nanotubes (CNTs) with SWCNTs and MWCNTs are effective to enhance the thermal measurement of blood and carboxymethyl cellulose (CMC)?(ii)Referring to blood and CMC base materials, which material reports more impressive thermal performances with the interaction of CNTs?(iii)How a mathematical model based on the basic definition of Prabhakar fractional technique is developed?(iv)Which nanomaterial reports a more stable thermal impact associated with the CMC-SWCNTs and CMC-MWCNTs interactions?(v)What is the role of magnetic force and mixed convection phenomenon in enhancing the thermal transportation process.(vi)For accelerating flow, how are slip effects important to control the flow?
2. Flow Model with Governing Equations
The thermal impact of CNTs with SWCNTs and MWCNTs is focused. The uniform suspension of SWCNTs and MWCNTs along with blood and carboxymethyl cellulose (CMC) is considered. The oscillating surface flow caused the vertically accelerating flow. The inclined direction along angle is considered to incorporate the magnetic force impact. The flow pattern is based on time-dependent flow. The base fluid properties are notified via viscoelastic flow model. The vertical plate oscillates with velocity with uniform frequency The stream temperature and concentrations are and respectively. The flow model using such assumptions is developed in the following set of equations:
Momentum equation:
Thermal equation:
Fourier law of thermal flux:
Diffusion balance equation:
Fick’s law:
The oscillating boundary constraints for slip flow are as follows:
The mathematical formulae for distinct flow characteristics is notified via Table 1.
Following below dimensionless variables:with Prandtl number , heat Grashof number , mass Grashof number , magnetic constant , and viscoelastic parameter . The dimensionless system in view of defined variables is as follows:with:
The thermal results reported in Table 2 justify the numerical reflection of materials like CMS, blood, SWCNTs, and MWCNTs.
3. Prabhakar Model
Let us utilize the definition of Prabhakar model as follows:
Definition of is as follows:where
Using the Prabhakar’s Laplace technique,
4. Solution Methodology via a Fractional Approach
4.1. Solution of the Heat Equation
The use of the Laplace transform on equations (9) and (12) yields
The solution is as follows:
The implementation framework of the Laplace transform is given in Tables 3 and 4.
4.2. Simulations for the Concentration Equation
Incorporating the Laplace algorithms to equations (10) and (13), giveshaving solution:
4.3. Solution for Velocity Profile
Implementing the definition of Laplace in equation (8):
The analytical solution is as follows:
Using the Stehfest and Tzou’s algorithms, we obtain as follows:where
5. Validation of the Fractional Model
First, the accuracy of implemented Stehfest and Tzou’s is ensured in for temperature profile and concentration profile in Figures 1(a) and 1(b), respectively. The reported results convey good agreement between both techniques. The solution based on Prabhakar’s approach is validated in Figure 2 by making the comparison with work of Ahmad et al. [28]. A fine accuracy of results is noticed.
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6. Discussion of Results
The physical onset due to variations in parameters is presented in this section. In order to report the more beneficiary aspect of thermal problem, the comparative results are deduced for CMC-SWCNTs, CMC-MWCNTs decomposition. Figures 3(a) and 3(b)) present the change in thermal rate due to two fractional constants and . The investigated analysis is inspected for CMC-SWCNTs and CMC-MWCNTs suspensions. The declining trend in temperature against is claimed for SWCNTs and MWCNTs. However, the improvement in thermal profiles for SWCNTs and MWCNTs is noted when varied. The thermal report due to is more stable when compared with The observations for base and in view of fractional constant and Prandtl number has been addressed in Figures 4(a) and 4(b). A lower temperature range for and is noted. However, the stable heating rate for is achieved. The controls of thermal rate due to is based on the fact of low thermal diffusivity. Figure 5(a) displays the results for volume fraction on temperature profile in view of and nanofluid suspensions. A decreasing impact on the temperature profile with larger is noted. The declining change in thermal phenomenon is lower for The graphical onset reported in Figure 5(b) claims a comparative thermal influence of different suspensions like , and The thermal influence of has been noted as a more stable and enhancing. The lower thermal impact of is reported. Therefore, it is concluded that the thermal impact of blood is improved by utilizing the These novel observations may present many applications in health sciences, biosciences, and various engineering processes. The significance of and for concentration profile with different values of and has been featured in Figures 6(a) and 6(b)). Two times instants are used to compute the simulations. The enhancing and lower concentration rates are noted for and respectively. A comparatively stable impact for is achieved. The results depicted via Figures 7(a) and 7(b) report the behavior of concentration profile for and Lower information for concentration profile is achieved for both parameters. The control of the concentration profile associated to is due to less mass diffusivity.
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Figures 8(a) and 8(b) pronounce the role of and on velocity profile . Assigning variation of and slows down the velocity. Figure 9(a) exhibited the change for and due to Here, the declining results are noted. Similar kind of lower observations are noted in Figure 9(b) where role of is justified. Figures 10(a) and 10(b) denote the contribution of and on profile of . A significant enhanced in for and is claimed. Physically, the heat Grashof number causes the natural convection due to buoyancy and viscous forces, and subsequently, an increment in 𝐺𝑟 Gr enhances the buoyancy forces, which results in a velocity increment. Figures 11(a) and 11(b) are prepared for assessing the contribution of magnetic parameter and inclination angel for velocity profile. Both and declined the velocity. The association of magnetic forces is referred to as the Lorentz force, which resists the velocity flow. Furthermore, the maximum effect of the Lorentz forces is when the angle of inclination of the applied magnetics is perpendicular to the oscillating plate.
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Tables 3–6 show the results for numerical achievement based on Stehfest and Tzou’s schemes. Various time instants are used to compute the simulations for accelerating phenomenon. A declining change in thermal and concentration profile for larger is noted. For larger time instant, the heat and mass transfer rate is lower. From Table 6, it is observed that the Nusselt number, Sherwood number, and wall shear force declined with increasing
7. Conclusions
The fractional investigation is reported for the carbon nanotubes flow for suggesting the enactment in blood and carboxymethyl cellulose base liquids. For CNTs, the impacts of SWCNTs and MWCNTs are incorporated. The computational simulations are facilitated with Prabhakar’s fractional scheme. The comparative thermal onset is presented. The novel findings are as follows:(i)The suspension of and is declined for the fractional parameter.(ii)A control of thermal decomposition of and is noted for the Prandtl number.(iii)More stable and improved thermal impact of suspension is observed as compared to decomposition.(iv)The concentration change enhanced was due to a fractional constant or material.(v)The increasing trend in velocity flow is observed for mass and thermal Grashof constants.(vi)The implemented Prabhakar fractional scheme seems to be more effective for performing the computation of different complicated problems.
Nomenclature
| : | Fluid velocity |
| : | Times |
| : | Gravity acceleration |
| : | Thermal conductivity of the nanofluid |
| : | Skin friction |
| : | Mean absorption parameter |
| : | Nanofluid density |
| : | Characteristic velocity |
| : | Angle of magnetic inclination |
| : | Prandtl number |
| : | Heat Grashof number |
| : | Mass Grashof number |
| : | Schmidth number |
| : | Magnetic field |
| : | Laplace transform variable |
| : | Prabhakar fractional parameters |
| : | Magnetic field strength |
| : | Specific heat at constant pressure |
| : | Dynamic viscosity |
| : | Thermal expansion coefficient |
| : | Electrical conductivity |
| : | Ambient temperature |
| : | Nusselt number |
| : | Sherwood number |
| : | Generalized thermal conductivity |
| : | Prabhakar fractional . |
Data Availability
All data used to support the findings of this study are available in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.