# Accelerating Flow of Carbon Nanotubes with Carboxymethyl Cellulose and Blood Base Materials with Comparative Thermal Features: Prabhakar Fractional Model

Authors: Ali Raza; Kamel Al-Khaled; Taseer Muhammad; Sami Ullah Khan
Journal: Mathematical Problems in Engineering
 (2023)

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## 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.


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## Body

## 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 angleθ 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 HtCosωt with uniform frequency ω. The stream temperature and concentrations are T∞ and C∞, respectively. The flow model using such assumptions is developed in the following set of equations:Momentum equation:(1)∂uξ,t∂t−β1∗∂3uξ,t∂t∂y2=μnfρnf∂2uξ,t∂ξ2−σnfBo2ρnfsinθ1uξ,t+gβTnfTξ,t−T∞+gβcnfCξ,t−C∞.Thermal equation:(2)ρCpnf∂Tξ,t∂t=−∂qξ,t∂ξ.Fourier law of thermal flux:(3)qξ,t=−knf∂Tξ,t∂ξ.Diffusion balance equation:(4)∂Cξ,t∂t=−∂Jξ,t∂ξ.Fick’s law:(5)Jξ,t=−D∂Tξ,t∂ξ.The oscillating boundary constraints for slip flow are as follows:(6)uξ,0=0,Tξ,0=T∞,Cξ,0=C∞;∀ξ≥0,u0,t−h∂u∂ξξ=0=UoHtft,∂T∂ξξ=0=−hkT0,t,C0,t=Cw;t>0uξ,t⟶0,Tξ,t⟶T∞,Cξ,t⟶C∞;ξ⟶∞,t>0.The mathematical formulae for distinct flow characteristics is notified via Table1.Table 1
Flow characteristics with mathematical expressions.
Thermal featuresMathematical representationEffective densityρnf=1−φρf+φρsDynamic viscosityμnf=μf/1−φ2.5Electrical conductivityσnf=1+3σs/σf−1φ/σs/σf+2−σs/σf−1φσfThermal expansionρβTnf=1−φρβTf+φρβTsHeat capacitanceρCpnf=1−φρCpf+φρCpsThermal conductivitykf=kCNTs−2φkf−kCNTs+2kf/kCNTs+2φkf−kCNTs+2kfkfElectrical conductivityσnf=σf1+3σs/σf−1φ/σs/σf+2−σs/σf−1φFollowing below dimensionless variables:(7)ξ∗=ξhk,u∗=uνgk/h2,t∗=tνk/h2,T∗=Tξ,t−T∞Tw−T∞,q∗=qqo,C∗=Cξ,t−C∞Cw−C∞,Pr=μCpκ,Gr=gυβTfT∞ν2,M=σfυfBo2ρfUo2,Gm=gυβCfC∞ν2,β1=β1∗gν2ρkh,with Prandtl number Pr, heat Grashof number Gr, mass Grashof number Gm, magnetic constant M, and viscoelastic parameter β1. The dimensionless system in view of defined variables is as follows:(8)∂∂tuξ,t−β1∂2uξ,t∂ξ2=∂2uξ,t∂ξ2−Msinθ1uξ,t+GrTξ,t+GmCξ,t,(9)Pr∂Tξ,t∂t=−∂qξ,t∂ξ,(10)Sc∂Cξ,t∂t=−∂Jξ,t∂ξ,with:(11)uξ,0=0,Tξ,0=0,Cξ,0=0,u0,t−h∂u∂ξξ=0=Htft,∂T∂ξξ=0=−1+T0,t,C0,t=1uξ,t⟶0,Tξ,t⟶0,Cξ,t⟶0;ξ⟶∞.The thermal results reported in Table2 justify the numerical reflection of materials like CMS, blood, SWCNTs, and MWCNTs.Table 2
Thermal impact of CMS, blood, SWCNTs, and MWCNTs.
MaterialCMCBloodSWCNTsMWCNTsρkg/m3997105326001600CpJ/kgK41793594425796kW/mK0.6130.49266003000βT×105K−10.90.182744


## 3. Prabhakar Model

Let us utilize the definition of Prabhakar model as follows:(12)qξ,t=−kγDα,β,αγC∂Tξ,t∂ξ,(13)Jξ,t=−Dα,β,αγC∂Cξ,t∂ξ.Definition ofDα,β,αγC is as follows:(14)Dα,β,αγCht=Eα,n,−β,α−γhnt=eα,n,−β−γα;t∗hnt=∫0tt−τn−β−1Eα,n,−β−γαt−ταhnτdτEα,β,αγht=∫0tt−τβ−1Eα,β−γαt−ταhτdτ,where(15)Eα,βγz=∑m=0∞Γγ+mzmm!ΓγΓαm+β.Using the Prabhakar’s Laplace technique,(16)LDα,β,αγCht=Lhmt∗eα,m−β−γα;t=LhmtLeα,m−β−γα;t=Lhmtsβ−m1−αs−αγ.


## 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 (17)PrsT¯ξ,s=−∂q¯ξ,s∂ξ,q¯ξ,s=−sβ1−αs−αγ∂T¯ξ,s∂ξ,∂T¯∂ξξ=0=−1s+T¯0,s;T¯ξ,s⟶0,ξ⟶∞.The solution is as follows:(18)T¯ξ,s=1Prs1−β/1−αs−αγ−1e−ξPrs1−β/1−αs−αγs.The implementation framework of the Laplace transform is given in Tables3 and 4.Table 3
Numerical variation of the temperature profile at different times for different numerical algorithms.
ξTξ,t by StehfestTξ,t by StehfestTξ,t by TzousTξ,t by Tzoust=0.5t=1.0t=0.5t=1.00.11.46830.72951.46740.68500.31.06990.47341.06930.43750.50.77960.30310.77900.27450.70.56790.19090.56750.16820.90.41370.11760.41340.09981.10.30140.07030.30120.05641.30.21950.04020.21940.02941.50.15990.02130.15980.01301.70.11640.00990.11640.00351.90.08480.00310.08470.0021Table 4
Numerical variation of the concentration field at different times for different numerical schemes.
ξCξ,t by StehfestCξ,t by StehfestCξ,t by Tzou’sCξ,t by Tzou’st=0.5t=1.0t=0.5t=1.00.10.89640.87650.89630.87570.30.72000.67250.72000.67070.50.57820.51510.57820.51250.70.46430.38390.46420.39090.90.37270.30040.37260.29751.10.29910.22870.29900.22581.30.24000.17360.24000.17101.50.19250.13150.19250.12901.70.15450.09930.15450.09711.90.12400.07470.12390.0728


### 4.2. Simulations for the Concentration Equation

Incorporating the Laplace algorithms to equations (10) and (13), gives(19)ScsC¯ξ,s=−∂J¯ξ,s∂ξ,J¯ξ,s=−sβ1−αs−αγ∂J¯ξ,s∂ξ,C¯0,s=1s;C¯ξ,s⟶0,ξ⟶∞,having solution:(20)C¯ξ,s=1se−ξScs1−β/1−αs−αγ.


### 4.3. Solution for Velocity Profile

Implementing the definition of Laplace in equation (8):(21)∂2u¯ξ,s∂ξ2−11+β1sMsinθ1+su¯ξ,s=−Gr1+β1sT¯ξ,s−Gm1+β1sC¯ξ,su¯0,t−h∂u¯∂ξξ=0=Fs;u¯ξ,s⟶0asξ⟶∞.The analytical solution is as follows:(22)u¯ξ,s=11+h1/1+β1sMsinθ+sGr1+β1s1sPrs1−β/1−αs−αγ−11+hPrs1−β/1−αs−αγPrs1−β/1−αs−αγ−1/1+β1sMsinθ+s+Gm1+β1ss1+hScs1−β/1−αs−αγScs1−β/1−αs−αγ−1/1+β1sMsinθ+s+Fse−ξ1/1+β1sMsinθ+s−Gr1+β1s1sPrs1−β/1−αs−αγ−11+hPrs1−β/1−αs−αγPrs1−β/1−αs−αγ−1/1+β1sMsinθ+s−Gm1+β1ss1+hScs1−β/1−αs−αγScs1−β/1−αs−αγ−1/1+β1sMsinθ+s.Using the Stehfest and Tzou’s algorithms, we obtain as follows:(23)wy,t=ln2t∑n=1Nvnw¯y,nln2t,where(24)vn=−1n+N/2∑r=q+1/2minq,N/2rN/22r!N/2−r!r!r−1!q−r!2r−q!,wy,t=e4.7t12w¯r,4.7t+Re∑j=1N−1kw¯r,4.7+kπit.


## 4.1. Solution of the Heat Equation

The use of the Laplace transform on equations (9) and (12) yields (17)PrsT¯ξ,s=−∂q¯ξ,s∂ξ,q¯ξ,s=−sβ1−αs−αγ∂T¯ξ,s∂ξ,∂T¯∂ξξ=0=−1s+T¯0,s;T¯ξ,s⟶0,ξ⟶∞.The solution is as follows:(18)T¯ξ,s=1Prs1−β/1−αs−αγ−1e−ξPrs1−β/1−αs−αγs.The implementation framework of the Laplace transform is given in Tables3 and 4.Table 3
Numerical variation of the temperature profile at different times for different numerical algorithms.
ξTξ,t by StehfestTξ,t by StehfestTξ,t by TzousTξ,t by Tzoust=0.5t=1.0t=0.5t=1.00.11.46830.72951.46740.68500.31.06990.47341.06930.43750.50.77960.30310.77900.27450.70.56790.19090.56750.16820.90.41370.11760.41340.09981.10.30140.07030.30120.05641.30.21950.04020.21940.02941.50.15990.02130.15980.01301.70.11640.00990.11640.00351.90.08480.00310.08470.0021Table 4
Numerical variation of the concentration field at different times for different numerical schemes.
ξCξ,t by StehfestCξ,t by StehfestCξ,t by Tzou’sCξ,t by Tzou’st=0.5t=1.0t=0.5t=1.00.10.89640.87650.89630.87570.30.72000.67250.72000.67070.50.57820.51510.57820.51250.70.46430.38390.46420.39090.90.37270.30040.37260.29751.10.29910.22870.29900.22581.30.24000.17360.24000.17101.50.19250.13150.19250.12901.70.15450.09930.15450.09711.90.12400.07470.12390.0728


## 4.2. Simulations for the Concentration Equation

Incorporating the Laplace algorithms to equations (10) and (13), gives(19)ScsC¯ξ,s=−∂J¯ξ,s∂ξ,J¯ξ,s=−sβ1−αs−αγ∂J¯ξ,s∂ξ,C¯0,s=1s;C¯ξ,s⟶0,ξ⟶∞,having solution:(20)C¯ξ,s=1se−ξScs1−β/1−αs−αγ.


## 4.3. Solution for Velocity Profile

Implementing the definition of Laplace in equation (8):(21)∂2u¯ξ,s∂ξ2−11+β1sMsinθ1+su¯ξ,s=−Gr1+β1sT¯ξ,s−Gm1+β1sC¯ξ,su¯0,t−h∂u¯∂ξξ=0=Fs;u¯ξ,s⟶0asξ⟶∞.The analytical solution is as follows:(22)u¯ξ,s=11+h1/1+β1sMsinθ+sGr1+β1s1sPrs1−β/1−αs−αγ−11+hPrs1−β/1−αs−αγPrs1−β/1−αs−αγ−1/1+β1sMsinθ+s+Gm1+β1ss1+hScs1−β/1−αs−αγScs1−β/1−αs−αγ−1/1+β1sMsinθ+s+Fse−ξ1/1+β1sMsinθ+s−Gr1+β1s1sPrs1−β/1−αs−αγ−11+hPrs1−β/1−αs−αγPrs1−β/1−αs−αγ−1/1+β1sMsinθ+s−Gm1+β1ss1+hScs1−β/1−αs−αγScs1−β/1−αs−αγ−1/1+β1sMsinθ+s.Using the Stehfest and Tzou’s algorithms, we obtain as follows:(23)wy,t=ln2t∑n=1Nvnw¯y,nln2t,where(24)vn=−1n+N/2∑r=q+1/2minq,N/2rN/22r!N/2−r!r!r−1!q−r!2r−q!,wy,t=e4.7t12w¯r,4.7t+Re∑j=1N−1kw¯r,4.7+kπit.


## 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 Figure2 by making the comparison with work of Ahmad et al. [28]. A fine accuracy of results is noticed.Figure 1
Graphical comparison between Stehfest and Tzou’s techniques.
(a)(b)Figure 2
The comparative framework of the computed model by Ahmad et al. [28].


## 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. Figures3(a) and 3(b)) present the change in thermal rate due to two fractional constants α and β. The investigated analysis is inspected for CMC-SWCNTs and CMC-MWCNTs suspensions. The declining trend in temperature against α is claimed for SWCNTs and MWCNTs. However, the improvement in thermal profiles for SWCNTs and MWCNTs is noted when β varied. The thermal report due to CMC−SWCNTs is more stable when compared with CMC−SWCNTs. The observations for CMC−SWCNTs base and CMC−SWCNTs in view of fractional constant γ and Prandtl number Pr has been addressed in Figures 4(a) and 4(b). A lower temperature range for γ and Pr is noted. However, the stable heating rate for MC−SWCNTs is achieved. The controls of thermal rate due to Pr is based on the fact of low thermal diffusivity. Figure 5(a) displays the results for volume fraction φ on temperature profile in view of CMC−SWCNTs and CMC−MWCNTs nanofluid suspensions. A decreasing impact on the temperature profile with larger φ is noted. The declining change in thermal phenomenon is lower for CMC−MWCNTs. The graphical onset reported in Figure 5(b) claims a comparative thermal influence of different suspensions like −SWCNTs , blood−SWCNTs,CMC−MWCNTs and blood−MWCNTs. The thermal influence of CMC−SWCNTs has been noted as a more stable and enhancing. The lower thermal impact of blood−MWCNTs is reported. Therefore, it is concluded that the thermal impact of blood is improved by utilizing the SWCNTs. These novel observations may present many applications in health sciences, biosciences, and various engineering processes. The significance of CMC−SWCNTs and CMC−MWCNTs for concentration profile with different values of α and β 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 β and α, respectively. A comparatively stable impact for CMC−SWCNTs is achieved. The results depicted via Figures 7(a) and 7(b) report the behavior of concentration profile for γ and Sc. Lower information for concentration profile is achieved for both parameters. The control of the concentration profile associated to Sc is due to less mass diffusivity.Figure 3
Temperature profile forα and β with γ=0.8,Pr=7.2,φ=0.02 at (a): t=0.8 and (b): t=1.9.
(a)(b)Figure 4
Variation for temperature profile with deviation inγ,Pr and α=β=0.8,φ=0.02.
(a)(b)Figure 5
Variation for temperature profile with deviation in (a) volume fraction and (b) comparison of different nanoparticles.
(a)(b)Figure 6
Variation for concentration field with deviation inα,β and γ=0.8,Sc=5.2,φ=0.02 at (a) t=0.8 and (b) t=1.9.
(a)(b)Figure 7
Variation for concentration field with deviation inγ,Sc and α=β=0.8,φ=0.02.
(a)(b)Figures8(a) and 8(b) pronounce the role of α and β on velocity profile uξ,t. Assigning variation of α and β slows down the velocity. Figure 9(a) exhibited the change uξ,t for CMC−SWCNTs and CMC−MWCNTs due to γ. Here, the declining results are noted. Similar kind of lower observations are noted in Figure 9(b) where role of β1 is justified. Figures 10(a) and 10(b) denote the contribution of Gr and Gm on profile of uξ,t. A significant enhanced in uξ,t for Gr and Gm is claimed. Physically, the heat Grashof number Gr 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 M and inclination angel θ for velocity profile. Both M and θ 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.Figure 8
Variation for momentum profile with deviation inα,β and γ=0.8,Sc=5.2,Gr=4.1,Gm=6.4,θ=π/6,Pr=7.2,φ=0.02,M=0.6,t=0.8.
(a)(b)Figure 9
Velocity profile forγ and β1 when α=β=0.8,Sc=5.2,Gr=4.1,Gm=6.4,θ=π/6,Pr=7.2,φ=0.02,M=0.6,t=0.8.
(a)(b)Figure 10
Velocity profile forGr,Gm when α=β=γ=0.8,Sc=5.2,θ=π/6,Pr=7.2,φ=0.02,M=0.6,t=0.8.
(a)(b)Figure 11
Velocity profile forM and θ1 when α=β=γ=0.8,Sc=5.2,Gr=4.1,Gm=6.4,Pr=7.2,φ=0.02,t=0.8.
(a)(b)Tables3–6 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 ξ 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 α.Table 5
Numerical variation of the velocity profile at different times for different numerical schemes.
ξuξ,t by Stehfestuξ,t by Stehfestuξ,t by Tzou’suξ,t by Tzou’st=0.5t=1.0t=0.5t=1.00.11.35772.85951.36372.87480.31.41433.25071.42173.26940.51.41253.43441.42073.45490.71.37063.46871.37393.49000.91.30253.39821.31103.41951.11.21843.25691.22663.27761.31.12563.07021.13343.09001.51.02962.85701.03692.87561.70.93412.63120.94092.64851.90.84182.40270.84802.4186Table 6
Numerical analysis of the local Nusselt number, Sherwood number, and skin friction.
αNuShCf0.11.14660.43383.25430.21.22380.46313.07310.31.33120.50372.98750.41.47350.55752.96290.51.65020.62442.97710.61.85330.70133.01540.72.06870.78283.06710.82.28050.86293.12560.92.47480.93643.1843


## 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 ofCMC−SWCNTs and CMC−MWCNTs is declined for the fractional parameter.(ii)
A control of thermal decomposition ofSWCNTs and MWCNTs is noted for the Prandtl number.(iii)
More stable and improved thermal impact ofSWCNTs−CMS suspension is observed as compared to MWCNTs−CMS decomposition.(iv)
The concentration change enhanced was due to a fractional constant orCMC−MWCNTs 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.



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