Authors: Abdullah Dawar, Rawan Bossly, Fuad S. Alduais, Afrah Al-Bossly, Jihad Younis, Anwar Saeed
Categories: Research Article, Activation energy, Brownian motion, Chemical reaction, Curved surface, Joule heating, MHD, Nanofluid, Thermophoresis
Source: Heliyon
Authors: Abdullah Dawar, Rawan Bossly, Fuad S. Alduais, Afrah Al-Bossly, Jihad Younis, Anwar Saeed
In this work, a comparative analysis on magnetohydrodynamic nanofluid flow containing spherical and cylindrical-shaped alumina nanoparticles over a stretching curved surface is mainly focused. The effects of Brownian motion, Joule heating, thermophoresis, chemical reaction and activation energy are taken into consideration in this work. It is important to mention that this analysis considers a 4 % volume fraction of the alumina nanoparticles. A thermal convective and zero-mass flux conditions are imposed to scrutinize the heat transfer analysis. The leading equations are transformed into dimensionless form by using suitable similarity variables. A numerical solution is obtained using the shooting technique. The acquired outcomes depict that greater magnetic factor enhances the skin friction coefficient. The greater magnetic factor, thermal Biot number, Eckert number, and interfacial layer thickness augment the heat transfer rate, while a greater nanoparticle diameter diminishes the thermal transfer rate. A higher magnetic factor has an increasing impact on thermal distribution and a reducing impact on velocity distribution. A greater curvature factor enhances both the velocity and thermal distributions. The concentration distribution is enhanced by the higher interfacial layer thickness and activation energy factor, while it is reduced by a higher nanoparticle diameter, Brownian motion factor, and chemical reaction factor. From the comparative analysis, it is found that the velocity, thermal, and concentration distributions are higher for the cylindrical-shaped nanoparticles compared to spherical-shaped nanoparticles.
NomenclatureSymbol****NameaConstant [−]B0Magnetic field strength [kg.s−2.A−1]BiTThermal Biot number [−]CConcentration [mol.m−3]CpSpecific heat [J.K−1.kg−1]DBBrownian diffusion coefficient [m−2.s−1]DTThermophoretic coefficient [m−2.s−1]EActivation energy [−]EaCoefficient of activation energy [J]EcEckert number [−]hfHeat transfer coefficient [W.m−2.s−1]kThermal conductivity [W.K−1.kg−1]KCurvature factor [−]KrCoefficient of chemical reaction [s−1]MMagnetic factor [−]NtThermophoretic factor [−]NbBrownian motion factor [−]PrPrandtl number [−]RRadius [m]ScSchmidt number [−]TTemperature [K]ur,usVelocity components [m.s−1]r,sCurvilinear coordinates [m]ρDensity [kg.m−3]μDynamic viscosity [kg.m−1.s−1]σElectrical conductivity [Ω−1.m−1]νfKinematic viscosity [m2.s−1]ΩChemical reaction factor [−]δTTemperature difference factor [−]
Nanofluids are obtained after mixing the nano-sized particles in pure fluids that exhibit remarkable enhancements in thermal properties, making them highly effective for heat transfer capabilities. Nanofluids depicts a promising advancement in heat transfer technology, offering significant performance improvements while requiring continued research to overcome existing limitations and achieve practical, widespread application. The mixing of nanoparticles suspension in pure fluid was initiated first by Choi and Eastman [1]. Anjum et al. [2] found that the primary benefit of nanofluids lies in their significantly greater thermal conductance in comparison of conventional fluids, for high conductance of the nanoparticles and their ability to create efficient heat transfer pathways within the fluid. Ashraf et al. [3] discovered that the suspension of nanoparticles in pure fluid improve the heat transfer coefficients, with up to a 40 % increase over traditional fluids. Lone et al. [4] discussed a semi-numerical solution for MHD nanofluid flow on a heated sheet. Sheikholeslami et al. [5] examined that the existence of nanoparticles also augments the thermal boundary layer thickness and enhances micro-mixing through Brownian motion, further boosting heat transfer rates. Nanofluids find applications in various fields, including cooling in electronic devices, automotive radiators, solar water heating systems, and nuclear reactors, offering substantial improvements in thermal management [6,7]. Mohite et al. [8] discovered that the inclusion of nanoparticles, creates conductive pathways that elevate the thermal conductance of nanofluids, often modeled through the Maxwell theory and other advanced frameworks. Sarfraz and Khan [9] inspected the magnetized nanofluid flow containing SWCNTs and MWCNTs over a rotating disk with thermal radiation impact and noticed that with the growing magnetic factor there has been a reduction in fluid motion and augmentation in energy of the system. Khan et al. [10] examined the Darcy-Forchheimer nanofluid flow with CNTs over a curved elongated sheet. Ouyang et al. [11] discussed the MHD time-dependent trihybrid nanoparticles flow on a movable needle using viscous dissipative and Joule heating effect and have noticed that there has been an escalation in thermal distributions with upsurge in magnetic factor and Eckert number.
Magnetohydrodynamic (MHD) is the study of the dynamics of electrically conducting fluids in association of magnetic field. In MHD, the motion of the fluid and the magnetic field are interdependent, governed by the coupled equations. In engineering systems like electromagnetic pumps and generators, controlling fluid flow and heat transfer via magnetic field is essential for efficient operation. Therefore, MHD fluid flow is a key to optimizing performance and achieving desired outcomes in these and other advanced technological applications. Reddy et al. [12] revealed that MHD generates the Lorentz force, which significantly affects the fluid's motion and thermal flow. Kumar et al. [13] established that the Lorentz force either accelerates or decelerates the fluid flow, based on the direction of magnetic field relative to the fluid motion. This results in modifications to the velocity profile, often damping turbulence and promoting a more stable and uniform flow. Algehyne et al. [14] studied the thermal transportations for convective MHD fluid flow over an elongated surface. Ahmad et al. [15] inspected the significance of multi-slip constraints for the MHD nanofluid flow with influence of thermal radiation and chemical reaction. In terms of thermal distribution, the magnetic field alters convective heat transfer mechanisms as shown by Jalili et al. [16]. The suppression of turbulence due to the Lorentz force leads to reduced thermal mixing, which creates a more stratified temperature distribution within the fluid. Mahesh et al. [17] deliberated on the impact of thermal radiation on couple stress MHD fluid flow over a permeable dissipative elongated sheet. Kheder et al. [18] shown that MHD induces Joule heating, where electrical currents within the fluid generate heat, impacting the overall thermal distribution of the fluid flow. Rafique et al. [19] studied the MHD flow with varying viscous forces and slip constraints over an elongated surface. Obalalu et al. [20] discussed the implications of MHD and thermal transportation analysis for nanofluid flow over an elongated sheet and have observed that with escalation in magnetic factor there is augmentation in thermal profile while a reduction in velocity characteristic. Abbas et al. [21] inspected the effect of radiative thermal transportation on MHD fluid flow on a stationary no–rotating sphere that is immersed by permeable medium.
Fluid flow involving thermophoresis and Brownian motion plays a vital role in the behavior of particulate suspensions in various engineering and environmental processes. Thermophoresis is the motion of fluid's particles caused by the temperature difference as established by Thabet et al. [22]. This phenomenon is particularly significant in liquid-solid suspensions where the temperature gradient induces a particle flux, impacting the heat distribution in the fluid. The combined effects of thermophoresis and Brownian motion significantly alter the temperature and concentration profiles in fluid flows as observed by Shahzad et al. [23]. Prakasha et al. [24] proved that the thermophoresis effect tends to create a non-uniform particle distribution, with higher concentration in cooler regions and leads to localized changes in thermal conductance and specific heat capacity, influencing the overall temperature distribution. Waqas et al. [25] studied computationally the Brownian motion as well as thermophoresis effects on rotational fluid flow with the effect of activation energy. The random diffusion due to Brownian motion helps in mitigating sharp concentration gradient and contributes to more uniform particle dispersion. This interaction between thermophoretic and Brownian diffusion results in complex behavior, where the overall temperature and concentration distributions depend on the relative strengths of these two mechanisms as noticed by Kumar et al. [26]. In practical applications, these effects are fundamental for processes like air filtration, combustion, and nanoparticle synthesis. For example, in cooling systems, thermophoresis aids in removing nanoparticles from hot regions, while Brownian motion ensures effective distribution and mixing as showed by Rehman et al. [27]. Hani et al. [28] studied a novel analytical technique for nanofluid flow using the influences Brownian motion as well as thermophoresis effects. Almaida et al. [29] discussed these two effects for nanofluid flow in association with magnetic effect. Ashraf et al. [30] examined the slip flow with mixed convection over the surface of sphere with effects of viscous dissipation and thermophoresis.
Fluid flow with Joule heating contains the generation of heat within a conducting fluid due to the passage of an electric current in the fluid that creates an electrical resistance. This resistive heating adds an internal heat source, significantly influencing the thermal behavior of the fluid as showed by Nazeer [31]. Abo-Dahab et al. [32] established that the primary impression of Joule heating on heat transfer is the alteration of the temperature distribution within the fluid, leading to changes in thermal gradient. As the fluid's temperature increases, its viscosity and thermal conductance changes, affecting convective heat transfer rates. In many applications, like electro-thermal microfluidics and cooling systems in nuclear reactors, managing the effects of Joule heating is vital in maintaining system stability and efficiency [33]. Irfan et al. [34] discussed the significance of Joule heating in a nonlinear radiative and convective fluid flow. Abbas et al. [35] proved that the Joule heating enhances the heat transfer by increasing the fluid temperature, thus improving thermal energy exchange. Hayat et al. [36] showed that Joule heating induces thermoelectric convection, where temperature gradients create density variations, leading to buoyancy-driven flow that enhances mixing and overall heat transfer. Moreover, this convection can be beneficial in promoting uniform temperature distribution and reducing thermal resistance, but excessive heating might lead to thermal instability and undesirable temperature rises. Joule heating effects are essential for optimizing heat transfer processes, ensuring efficient operation, and preventing damage in sensitive applications [37]. The interaction between electric currents and fluid dynamics allows for the precise design and management of systems where Joule heating is a significant factor, enabling improved performance and reliability in technological and engineering applications. Sarfraz et al. [38] discussed the flow of fluid over a stretching cylinder with impact of Cattaneo-Christov model. Sarfraz et al. [39] discussed the various solutions for mixed convective non-linear radiative nanofluid flow on an exponentially shrinking surface and have noticed that with the growth in heat absorption and generation factor there is an augmentation in thermal distribution.
Fluid flow with chemical reactions involves the relationship between fluid dynamics and reactive processes within the flow, significantly affecting the concentration distribution of reactants and products. When a chemical reaction occurs within the fluid, the reactant concentration changes over time and space, leading to complex patterns in its concentration distribution. The reaction rate depends on factors like temperature, pressure, and the presence of catalysts, as well as the specific kinetics of the chemical reaction involved. These reactions can be exothermic or endothermic, thereby influencing the thermal profile of the fluid and subsequently affecting the flow characteristics through changes in density and viscosity. Ragunath et al. [40] discussed the impacts of chemical reaction on a time dependent squeezing fluid flow with Darcy-Forchheimer effects. The concentration distribution in a reacting flow is impacted by several mechanisms, including advection, diffusion, and the reaction itself. The chemical reaction consumes reactants and produces products, altering local concentration. Kumar et al. [41] simulated computationally the radiative Maxwell fluid flow over an elongated sheet with the impact of chemical reaction. Assiri et al. [42] inspected numerically the impacts of chemical reaction and magnetic field on radiative nanofluid flow. In practical applications, like combustion, environmental engineering, and chemical reactors, chemical reactions play a significant role in optimizing performance and efficiency of the systems [43,44]. For example, in combustion processes, the distribution of fuel and oxidizer affects the flame structure and efficiency of energy conversion. In environmental engineering, reactive transport modeling helps predict pollutant dispersion and degradation in natural waters [45]. Patil et al. [46] studied fluid flow through a porous elongating sheet with the impacts chemical reaction and magnetic field. Rath and Nayak [47] examined the naturally convective and thermally radiative fluid flow over a permeable surface with the impact of chemical reaction.
The motivation of this research stems from the growing demand for more efficient thermal management systems in engineering and industries like electronics, manufacturing and energy. Those nanofluids which containing alumina nanoparticles have established superior heat transfer capabilities compared to the conventional fluids. However, the nanoparticles shapes play a critical role in influencing the flow of and heat transfer properties of nanofluids. Analyzing the effects of spherical and cylindrical-shaped alumina nanoparticles in a MHD nanofluid flow system over a curved surface addresses a gap in the existing studies. Therefore, this work investigates comparatively the magnetohydrodynamic nanofluid flow with spherical and cylindrical-shaped alumina nanoparticles on a stretching curved surface is mainly focused. The effects of Brownian motion, Joule heating, thermophoresis, chemical reaction and activation energy are used in this work. It is important to mention that this analysis considers a 4 % volume fraction of the alumina nanoparticles. A thermal convective and zero-mass flux conditions are imposed to scrutinize the heat transfer analysis. The PDEs are transformed into ODEs and are solved numerically by using bvp4c. The primary objectives of this analysis •To compare the thermal performance of cylindrical and spherical shaped alumina nanoparticles within the water-based nanofluid flow.•To determine the effects of Brownian motion, Joule heating, thermophoresis, chemical reaction and activation energy on their respective distributions.
The novelty of this research lies in its systematic analysis of thermal conductance differences between nanoparticle shapes, contributing valuable insights into the optimization of nanofluid application in engineering and industries. The following research questions will be addressed in this research •How does the magnetic factor effect the surface drag and why is this effect is more pronounced in cylindrical-shaped nanoparticles compared to spherical-shaped nanoparticles?•What are the mechanisms behind the greater rate of heat transfer observed due to the higher embedded factors and how does the cylindrical and spherical-shaped nanoparticles influence these factors?•Why do the cylindrical-shaped nanoparticles results in greater velocity, temperature and concentration distributions compared to spherical-shaped nanoparticles?
These research questions are designed to investigate deeper into the observed phenomena and could help in future studies or can expand on the insights provided by the current work.
Assume the flow of a nanofluid over an extending curved surface containing Al2O3 nanoparticles. The radius of the curved surface is denoted by R. The curvilinear coordinates system (r,s), where r represents the normal direction and s represents the flow direction, is chosen for the model formulation. The curved surface stretches along s− direction with velocity us=as with a>0 is constant. B0 is the strength of magnetic field that is used in normal direction. Let Tf, Tw and T∞ are fluid, surface and free stream temperatures such that Tf>Tw>T∞. Also, the zero-mass flux condition is adopted to eliminate the mass transfer rate at the curved surface (i.e. Cw=0) [48,49]. The Brownian motion, chemical reaction, thermophoresis, activation energy and Joule heating impacts are taken into consideration (See Fig.1). The leading equations can be written as [50,51]:(1)∂(r+R)ur∂r+R∂us∂s=0,(2)us2r+R=1ρnf∂p∂r,(3)ur∂us∂r+(Rr+R)us∂us∂s+(1r+R)usur=−1ρnf(Rr+R)∂p∂s+μnfρnf{∂2us∂r2+(1r+R)∂us∂r−(1(r+R)2)us}−σnfρnfB02us,(4)ur∂T∂r+(Rr+R)us∂T∂s=knf(ρCp)nf{∂2T∂r2+(1r+R)∂T∂r}+σnf(ρCp)nfB02us2+μnf(ρCp)nf{∂us∂r+(1r+R)us}2+(ρCp)np(ρCp)nf{DBδ∂C∂r∂T∂r+DTT∞(∂T∂r)2},(5)ur∂C∂r+(Rr+R)us∂C∂s=DB{∂2C∂r2+(1r+R)∂C∂r}+δDTT∞{∂2T∂r2+(1r+R)∂T∂r}−(C−C∞)Kr(TT∞)nexp(−EakBT),In the above equations, B0 is the strength of magnetic field, C is the concentration, Cp is the specific heat, DB is the Brownian diffusion coefficient, DT is the thermophoretic coefficient, Ea is the coefficient of activation energy, k is the thermal conductivity, Kr is the coefficient of chemical reaction, R is the radius of curvature, T is the temperature, ur and us are the velocity components along r and s directions respectively, ρ is the density, μ is the dynamic viscosity and σ is the electrical conductivity.Fig. 1Geometrical representation of the flow problem.Fig. 1
With boundary conditions [48,52,53]:(6){r=0:us=as,ur=0,−knf∂T∂r=hf(Tf−T),DBδ∂C∂r+DTT∞∂T∂r=0,r→∞:us→0,∂us∂r→0,T→T∞,C→C∞.
The thermal convective and zero-mass flux conditions are commonly used in fluid dynamics and heat transfer problems. The convective boundary condition is used to model the heat transfer from the surface to surrounding fluid. Furthermore, Neild and Kuznetsov [54] presented the idea of zero-mass flux condition to control the nanoparticle fraction at the boundary. By incorporating the zero-mass flux condition, the mass flow at the surface of the sheet is zero (see equation (10)).
The Maxwell [55], and Hamilton and Crosser [56] models are not sufficient to obtained the higher thermal conductivity. The primary drawback of these models is the absence of a relationship between nanoparticle diameter and nanolayer properties. These factors are more essential to augment the nanofluids thermal conductivity. Thus, Murshed et al. [57,58] defined the thermal conductivity model (7)knf=φklr(ks−klr)(1+2ψ13−ψ23)+ψ13(2klr+ks)(φψ23klr−φψ23kf+kf)ψ13(2klr+ks)+φ(klr−ks)(ψ23+ψ13−1),where ψ1=2dnp+h2dnp and ψ2=dnp+hdnp. Furthermore, h=2πσj where σj is the nanolayer diffusivity which varies from 0.2 nm to 0.8 nm. In these relations, h designates the interfacial thickness and dnp represents the nanoparticle diameter. By means of surface's electron density, Hashimoto et al. [59] used the identical approach to fix h. By considering σj=0.4nm, one can acquire h=1.0nm. Following the molecular and experimental simulations of Xue et al. [60] and Yu et al. [61], they established that h≈1.0nm. Thus for the spherical-shaped nanoparticle, h=1.0 [57,58,[62], [63], [64]]. For the cylindrical-shaped nanoparticle, h=2.0 [58,65]. Furthermore, klr=γkf is the interfacial layer where 0<γ≤kskf and γ(>1.0) is the empirical factor [57,58]. Thus, the modified thermal conductivity model is [59]:(8)knfkf=φγ(ks−γkf)(1+2ψ13−ψ23)+ψ13(2γkf+ks)(φψ23γ−φψ23γ+1)ψ13(2γkf+ks)+φ(γkf−ks)(ψ23+ψ13−1).
Other thermophysical relations are defined as [62]:(9)μnfμf=1(1−φ)2.5,ρnfρf=(1−φ)+φρsρf,(ρCp)nf(ρCp)f=(1−φ)+φ(ρCp)s(ρCp)f,σnfσf=1+3(σsσf−1)φ(σsσf+2)−(σsσf−1)φ.Here μ, ρ, Cp, σ and φ shows the dynamic viscosity, density, specific heat, electrical conductivity and nanoparticle volume fraction, respectively. The subscripts f, nf and s shows the base fluid which is water, the nanofluid and nanoparticles which is Al2O3.
The thermophysical properties of the water and Al2O3 nanoparticles are defined in Table 1.Table 1Thermophysical properties of water and Al2O3 nanoparticles [62].Table 1Thermophysical propertiesH2OAl2O3Cp[J.Kg−1.K−1]4180765ρ[Kg.m−3]997.13970k[W.m−1.K−1]0.607140σ[Ω−1.m−1]0.0053.5 × 10^7^
The similarity variables are defined as [51,66]:(10)ξ=aνfr,us=asf′(ξ),ur=−Rr+Raνff(ξ),p=ρfa2s2P(ξ),θ(ξ)=T−T∞Tf−T∞,Φ(ξ)=C−C∞C∞.
By applying these similarity variables, we (11)∂P∂ξ=ρδ(1ξ+K)(∂f∂ξ)2,(12)1ρδ(2Kξ+K)P(ξ)=μδρδ{∂3f∂ξ3+(1ξ+K)∂2f∂ξ2−(1(ξ+K)2)∂f∂ξ}+(Kξ+K){f(ξ)∂2f∂ξ2−(∂f∂ξ)2}−σδρδM∂f∂ξ+(K(ξ+K)2)f(ξ)∂f∂ξ,
Differentiate equation (12) with respect to ξ and then substitute in equation (11), we (13)μδρδ{∂4f∂ξ4+(2ξ+K)∂3f∂ξ3−(1(ξ+K)2)∂2f∂ξ2+(1(ξ+K)3)∂f∂ξ}−σδρδM{∂2f∂ξ2+(1ξ+K)∂f∂ξ}+(Kξ+K){f(ξ)∂3f∂ξ3−∂f∂ξ∂2f∂ξ2}+(K(ξ+K)2){f(ξ)∂2f∂ξ2−(∂f∂ξ)2}−(K(ξ+K)3)f(ξ)∂f∂ξ=0,
with boundary (14)f(ξ=0)=0,∂f∂ξ(ξ=0)=1,∂f∂ξ(ξ→∞)=0,∂2f∂ξ2(ξ→∞)=0.
Furthermore, the temperature and concentration equations are reduced (15)1Prkδ(ρCp)δ{∂2θ∂ξ2+(1ξ+K)∂θ∂ξ}+1(ρCp)δM{Nb∂Φ∂ξ∂θ∂ξ+Nt(∂θ∂ξ)2}+(Kξ+K)f(ξ)∂θ∂ξ+μδ(ρCp)δEc{∂2f∂ξ2−(1ξ+K)∂f∂ξ}2+σδ(ρCp)δMEc(∂f∂ξ)2=0,(16){∂2Φ∂ξ2+(1ξ+K)∂Φ∂ξ}+NtNb{∂2θ∂ξ2+(1ξ+K)∂θ∂ξ}+Sc(Kξ+K)×f(ξ)∂Φ∂ξ−ScΩ(1+δTθ(ξ))nexp(−E1+δTθ(ξ))Φ(ξ)=0,
with boundary (17){∂θ∂ξ(ξ=0)=BiTkδ(,θ(ξ=0)−1),θ(ξ→∞)=0,Nb∂Φ∂ξ(ξ=0)+Nt∂θ∂ξ(ξ=0)=0,Φ(ξ→∞)=0.In the above equations, the thermophysical relations are considered (18)μδ=μnfμf,ρδ=ρnfρf,σδ=σnfσf,(ρCp)δ=(ρCp)nf(ρCp)f,kδ=knfkf.
The term P(ξ) can be reduced (19)P(ξ)=(ξ+K2K){{μδ∂3f∂ξ3+(1ξ+K)∂2f∂ξ2−(1(ξ+K)2)∂f∂ξ}−σδM∂f∂ξ+ρδ(Kξ+K){f(ξ)∂2f∂ξ2−(∂f∂ξ)2}+ρδ(K(K+ξ)2)f(ξ)∂f∂ξ},In the above transformed ordinary differential equations, M(=σfB02ρfa) is the magnetic parameter, K(=aνfR) is the curvature parameter, BiT(=hfkfνfa) is the thermal Biot number, Pr(=(ρCp)fνfkf) is the Prandtl number, Ec(=us2(Cp)f(Tf−T∞)) is the Eckert number, Ω(=Kra) is the chemical reaction factor, Nt(=(ρCp)np(ρCp)fDT(Tf−T∞)νfT∞) is the thermophoresis factor, Sc(=νfDB) is the Schmidt number, δT(=Tf−T∞T∞) is the temperature difference parameter, Nb(=(ρCp)np(ρCp)fDBC∞νfδ) is the Brownian motion factor and E(=EakBT∞) is the activation energy factor.
The quantities of interest, like skin friction, Nusselt number and Sherwood number are described (20)Cfs=τrsρfus2,Nus=sqwkf(Tf−T∞),Shs=sqmDB(Cw−C∞),where τrs=μnf{∂us∂r+(Rr+R)∂ur∂r−(1r+R)us}|r=0, qw=−knf∂T∂r|r=0 and qm=−DB∂C∂r|r=0.
Using the above similarity transformations, equation (20) is reduced as [49]:(21)Cs=ResCfs=−μδ(∂2f∂ξ2|ξ=0−1K∂f∂ξ|ξ=0),Nu=NusRes=−kδ∂θ∂ξ|ξ=0,where Res(=as2νf) is the local Reynolds number.
A numerical technique called bvp4c technique can handle many boundary value problems. The error tolerance of 10^−6^ is defined, ensuring that the computed solution meets the required accuracy. To apply this technique, higher-order ODEs are reduced to first-order ODEs. Hence, it is supposed (22)f=ϕ(1),f′=ϕ(2),f″=ϕ(3),f‴=ϕ(4),fiv=ϕ′(4),θ=ϕ(5),θ′=ϕ(6),θ″=ϕ′(6),Φ=ϕ(7),Φ′=ϕ(8),Φ″=ϕ′(8).
Thus we (23)ϕ′(4)=−{μδρδ{(2ξ+K)ϕ(4)−(1(ξ+K)2)ϕ(3)+(1(ξ+K)3)ϕ(2)}−σδρδM{ϕ(3)+(1ξ+K)ϕ(2)}+(Kξ+K){ϕ(1)ϕ(4)−ϕ(2)ϕ(3)}+(K(ξ+K)2){ϕ(1)ϕ(3)−(ϕ(2))2}−(K(ξ+K)3)ϕ(1)ϕ(2)}μδρδ,(24)ϕ′(6)=−{1Prkδ(ρCp)δ{(1ξ+K)ϕ(6)}+1(ρCp)δM{Nbϕ(8)ϕ(6)+Nt(ϕ(6))2}+(Kξ+K)ϕ(1)ϕ(6)+μδ(ρCp)δEc{ϕ(3)−(1ξ+K)ϕ(2)}2+σδ(ρCp)δMEc(ϕ(2))2}1Prkδ(ρCp)δ,(25)ϕ′(8)=−{(1ξ+K)ϕ(8)+NtNb{ϕ′(6)+(1ξ+K)ϕ(6)}+Sc(Kξ+K)×ϕ(1)ϕ(8)−ScΩ(1+δTϕ(5))nexp(−E(1+δT)ϕ(5))ϕ(7)},
with boundary (26){ϕInitial(1)−0,ϕInitial(2)−1,ϕFinal(2)−0,ϕFinal(3)−0,ϕInitial(6)−BiTkδ(ϕInitial(5)−1),ϕFinal(5)−0,NbϕInitial(8)+NtInitialϕ(6),ϕFinal(7)−0.
Fig. 2 represents the flow chart of the applied numerical technique.Fig. 2Flow chart of the applied numerical technique.Fig. 2
To validate our outcomes with published results, Table 2 is shown. Here, the current results are compared for the case when the flow is viscous and non-magnetized. From this analysis, we have seen that the growing values of K reduce the skin friction (Cs). From the present and previous results, it is confirmed that both the results are matching meticulously with each other, confirming the correctness and validation of the approach used for solution.Table 2Comparison of Cs values for different values of K when the flow is viscous and non-magnetized.Table 2KAhmad et al. [67]Dey et al. [68]Rosca and Pop [69]Abbas et al. [70]Afridi et al. [71]Present results5.01.157 6301.1576301.150761.157631.15763121.15762510.01.073 4901.0734901.071721.073491.07348861.07349520.0––1.035011.035611.03560981.03561530.01.0235301.02231.023151.023531.02353101.02353640.0––1.017291.017591.01758661.01759250.01.014050–1.013801.014051.01404921.014054100.0––1.006871.007041.00703841.007043200.01.003560–1.003421.003561.00356411.0035691000.01.000790–1.000681.000791.00079931.000804
The physical explanation of each parameter behavior against the velocity (∂f/∂ξ), temperature (θ(ξ)), and concentration (Φ(ξ)) distributions are depicted through Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15, Fig. 16, Fig. 17, Fig. 18. Also, the influences of those factors on (ResCfs) and (Res−1/2Nus) are displayed in Table 3, Table 4. In this work, a comparative study on the variations in nanofluid flow containing spherical and cylindrical-shaped alumina nanoparticles is mainly focused. It should be noted that 4 % volume fraction of the alumina nanoparticles (i.e., φ = 0.04) is considered in this work. Furthermore, the default values of the embedded factors are chosen as M=0.1, K=5.0, BiT=0.5, Pr=6.2, Ec=0.1, Ω=0.8, Nt=0.1, Sc=1.3, δT=1.0, dnp=5.0, γ=65.25, Nb=0.1 and E=0.1. From Tables 3 and it is found that the greater values of M enhances ResCfs for both h=1.0 and h=2.0. The greater M enhances the Lorentz force which plays a crucial role against the nanoparticles motion. This opposing force augments the friction at surface which causes retardation in the velocity profile as well. Therefore, the greater values of M enhances ResCfs for both h=1.0 and h=2.0. From the comparative analysis, it is found that the skin friction is more pronounced for cylindrical-shaped nanoparticles as compared to spherical-shaped nanoparticles. In comparison to spherical-shaped alumina nanoparticles, cylindrical-shaped alumina nanoparticles have a larger aspect ratio. As a result, when they align and engage with the fluid flow, they encounter a stronger drag force which increases the resistance, consequently the skin friction heightens. From Tables 4 and it is found that the greater values of M, BiT, Ec and γ enhances Res−1/2Nus while the greater dnp reduces Res−1/2Nus. The heat transfer rate is calculated for both the spherical and cylindrical-shaped alumina nanoparticles (i.e. h=1.0, h=2.0). The greater M enhances the opposing force which shows the particles motion and enhances the heat transfer rate. In comparison to spherical-shaped nanoparticles, cylindrical-shaped nanoparticles have a greater surface area to volume ratio. Heat transfer is improved because of the greater surface area's ability to interact with the surrounding fluid. The greater values of BiT enhances the thermal transference rate. Actually, a higher BiT denotes more effective surface heat transfer in relation to the internal heat resistance of the particle. Because of their larger characteristic length and surface area to volume ratio which enable more efficient heat transfer interactions with the surrounding fluid. Furthermore, the cylindrical-shaped nanoparticles have a higher rate of thermal transference than spherical-shaped nanoparticles. The greater values of Ec enhances the rate of heat transfer. Because of viscous dissipation, a greater Ec denotes a higher conversion of kinetic to thermal energy. This phenomenon enhances the thermal behavior of fluid and as a result, the heat transfer rate also augments. Furthermore, the cylindrical-shaped nanoparticles have a higher thermal transference rate than spherical-shaped nanoparticles. The higher values of dnp diminishes thermal transference rate. The surface area to volume ratio reduces with increasing nanoparticle diameter. This decreased ratio lessens the contact amid of nanoparticles and base fluid, which reduce the augmentation of heat flow convection and thermal conductance and as a result the rate of heat transference reduces. Similar results are reported in Refs. [62,65]. Furthermore, the cylindrical-shaped nanoparticles have a higher thermal transference rate than spherical-shaped nanoparticles. The higher γ augments thermal transference rate. The nanofluid thermal conductivity improves with higher interfacial layer. Compared to the bulk fluid, the molecules in this layer have different thermal characteristics, which frequently results in higher thermal transference rate. Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer than spherical-shaped nanoparticles. Fig. 3, Fig. 4 shows the effect of M on ∂f/∂ξ and θ(ξ). From Fig. 3, Fig. 4, we have seen that greater M reduces the velocity profile while enhances the thermal distribution. Because of the collaboration between magnetic field and the electrically conducting fluid, a greater magnetic factor in a nanofluid system with alumina nanoparticles dispersed in a water base fluid reduces the velocity profile. The water's electrical conductance is increased by the alumina nanoparticles, amplifying the impact of the Lorentz force generated by magnetic field. The Lorentz force reduces the nanofluid's velocity by acting as a drag force against the fluid motion. The resistive force intensifies with a rise in the magnetic factor, which causes the fluid's velocity to decrease more. The combined effects of reduced fluid motion and improved thermal characteristics of the alumina-water nanofluid can credited to the improvement of temperature profile with an increased magnetic factor. Higher temperatures are found near the surface due to a larger thermal layer and reduced convective heat transfer due to Lorentz force's reduced velocity profile. The total thermal conductance of nanofluid is increased by the higher thermal conductance of alumina nanoparticles as compared to the base fluid. The fluid's internal thermal energy transmission becomes more effective as a result, which raises the temperature profile even more. Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer and surface drag than spherical-shaped nanoparticles which evident to results observed in Fig. 3, Fig. 4. From these Figures, we have found that ∂f/∂ξ and θ(ξ) are higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 5, Fig. 6 shows the consequence of K on ∂f/∂ξ and θ(ξ). From Fig. 5, Fig. 6, we have seen that higher K enhances both ∂f/∂ξ and θ(ξ). As the curvature factor enhances, the curved surface become flat where temperature and velocity profiles show dominant behaviors. From Table 2, we can observe that the greater values of K diminishes the skin friction which results higher velocity profile. Since by definition, there is a reverse relation between the curvature factor and kinematic viscosity of water. According to this definition, the kinematic viscosity of base fluid enhances with growing curvature factor which results higher temperature profile. Therefore, the greater values of K enhances both ∂f/∂ξ and θ(ξ). Similar results are also reported in Refs. [49,72]. Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer and surface drag than spherical-shaped nanoparticles which evident to results observed in Fig. 5, Fig. 6. From these Figures, we have found that ∂f/∂ξ and θ(ξ) are higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 7, Fig. 8 shows the effect of Nt on θ(ξ) and Φ(ξ). From Fig. 7, Fig. 8, we have seen that the greater values of Nt enhances both the temperature and concentration distributions. The greater Nt enhances the thermal profile by promoting the nanoparticles movement from the higher temperature region to lower temperature region. The fluid dynamic is produced by this thermophoretic force, which more equally distributes heat energy throughout the flow field. The overall temperature profile rises due to thermal energy carried by the nanoparticles as they travel from hotter colder zone. An improved temperature profile results from this redistribution, which helps the nanofluid temperature distribution to be more consistent. Therefore, the greater Nt enhances θ(ξ). Similar to this, a larger Nt cause a more noticeable movement of nanoparticles from hot to cold regions, which improves the concentration profile of a nanofluid flow over a stretching curved surface. In the fluid's colder areas, this movement, caused by the temperature gradient, leads to a greater accumulation of nanoparticles. Clearly Φ(ξ) increases in the low-temperature zones and decreases in the high-temperature zones as they migrate. The concentration profile varies more significantly as a result of this differential mobility, with higher concentration of nanoparticles found farther from the heated surface. Therefore, the greater Nt enhances Φ(ξ). Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer and surface drag than spherical-shaped nanoparticles which evident to results observed in Fig. 7, Fig. 8. From these Figures, we have found that θ(ξ) and Φ(ξ) are higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 9 depicts the effect of BiT on θ(ξ). From Fig. 9, we have seen that the greater values of BiT enhances the temperature distribution. As we know by definition, that BiT is in direct relation with thermal transfer coefficient (hf). So, the higher values of BiT augments hf which as a result enhances the thermal boundary layer thickness and hence augment θ(ξ). Thus, the greater values of BiT enhances θ(ξ). Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer than spherical-shaped nanoparticles which evident to results observed in Fig. 9. From this Figure, we have found that the temperature distribution is higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 10, Fig. 11 shows the effect of γ on θ(ξ) and Φ(ξ). From Fig. 10, Fig. 11, we have seen higher of γ enhances both θ(ξ) and Φ(ξ). We have considered the alumina nanoparticles in the present analysis, so 1<γ<65.25. From the effective thermal conductance model, it is found that the greater values of γ enhances the heat transfer rate (as discussed in Table 4) which results an enhanced thermal and concentration distributions. Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer than spherical-shaped nanoparticles which evident to results observed in Fig. 10, Fig. 11. From these Figures, we have found that θ(ξ) and Φ(ξ) are higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 12, Fig. 13 shows the effect of dnp on θ(ξ) and Φ(ξ). From Fig. 12, Fig. 13, we have seen that the greater values of dnp reduces both the temperature and concentration distributions. As verified by Choi and Eastman [1] from the experimental analysis that the thermal layer reduces due to the accumulation of solid nanoparticles. Also, the higher values of nanoparticle diameter reduce the thermal transference rate which results reduction in the thermal boundary layer and as a result, θ(ξ) reduces. A similar impact of dnp on Φ(ξ) is also found. Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer than spherical-shaped nanoparticles which evident to results observed in Fig. 12, Fig. 13. From these Figures, we have found that θ(ξ) and Φ(ξ) are higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 14 depicts the influences of Ω on Φ(ξ). From Fig. 14, we have seen that higher Ω reduces Φ(ξ). The higher value of chemical reaction factor reduces the concentration boundary layer thickness which decreases Φ(ξ). Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer than spherical-shaped nanoparticles which evident to results observed in Fig. 14. From this Figure, we have found that Φ(ξ) is higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 15 shows the effect of E on Φ(ξ). From Fig. 15, we have seen that the greater values of E enhances Φ(ξ). A slower reaction rate is the result of fewer reactant molecules having the energy required to undergo a chemical reaction at a particular temperature when there is higher activation energy. This slower reaction rate permits the reactant concentration to stay higher at the surface and throughout the flow field when fluid flows over a curved stretching surface. The curved surface improves reactant mixing and distribution by exerting more control over the flow dynamics. Because of this, chemical reaction with a higher activation energy happen less frequently, which slows down the rate at which reactants diminish and keeps Φ(ξ) higher throughout the fluid flow. Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer than spherical-shaped nanoparticles which evident to results observed in Fig. 15. From this Figure, we have found that the concentration distribution is higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 16 shows the effect of Nb on Φ(ξ). From Fig. 16, we have seen that the greater values of Nb reduces Φ(ξ). Higher Brownian motion results in more frequent and powerful collisions, which spreads the nanoparticles more broadly throughout the fluid. This dispersion results in a reduced concentration profile close to the surface by lowering the concentration of nanoparticles. Therefore, the higher values of Nb reduces Φ(ξ). Furthermore, the cylindrical-shaped nanoparticles have a higher rate of heat transfer than spherical-shaped nanoparticles which evident to results observed in Fig. 16. From this Figure, we have found that Φ(ξ) is higher for the cylindrical-shaped nanoparticles as compared to the spherical-shaped nanoparticles. Fig. 17 shows the variation in Cs via K and M. Fig. 17(a) shows the variation in Cs via K and M when h=1.0. From this Figure, it is observed that the greater M enhances the Lorentz force which plays crucial role against the nanoparticles motion. This opposing force augments the friction at surface which causes retardation in the velocity profile as well. Furthermore, the greater K reduces Cs as shown in Table 2. When h=2.0, similar impacts of K and M on Cs are found in Fig. 17(b). From the comparative analysis, it is found that the skin friction is more pronounced for cylindrical-shaped nanoparticles as compared to spherical-shaped nanoparticles. In comparison to spherical-shaped alumina nanoparticles, cylindrical-shaped alumina nanoparticles have a larger aspect ratio. Fig. 18 shows the variation in Nu via BiT and dnp. Fig. 18(a) shows the variation in Nu via BiT and dnp when h=1.0. The greater values of BiT enhances the thermal transference rate. Actually, a higher BiT denotes more effective surface heat transfer in relation to the internal heat resistance of the particle. Because of their larger characteristic length and surface area to volume ratio which enable more efficient heat transfer interactions with the surrounding fluid. The higher values of dnp diminishes thermal transference rate. The surface area to volume ratio reduces with increasing nanoparticle diameter. This decreased ratio lessens the contact amid of nanoparticles and base fluid, which reduce the augmentation of heat flow convection and thermal conductance and as a result the rate of heat transference reduces. Similar results are reported in Refs. [62,65]. The higher γ augments thermal transference rate. The nanofluid thermal conductivity improves with higher interfacial layer. Compared to the bulk fluid, the molecules in this layer have different thermal characteristics, which frequently results in higher thermal transference rate. When h=2.0, similar impacts of Nu via BiT and dnp are found in Fig. 18(b). Furthermore, the cylindrical-shaped nanoparticles have a higher thermal transference rate than spherical-shaped nanoparticles.Fig. 3Outcome of M on ∂f/∂ξ.Fig. 3Fig. 4Outcome of M on θ(ξ).Fig. 4Fig. 5Outcome of K on ∂f/∂ξ.Fig. 5Fig. 6Outcome of K on θ(ξ).Fig. 6Fig. 7Outcome of Nt on θ(ξ).Fig. 7Fig. 8Outcome of Nt on Φ(ξ).Fig. 8Fig. 9Outcome of BiT on θ(ξ).Fig. 9Fig. 10Outcome of γ on θ(ξ).Fig. 10Fig. 11Outcome of γ on Φ(ξ).Fig. 11Fig. 12Outcome of dnp on θ(ξ).Fig. 12Fig. 13Outcome of dnp on Φ(ξ).Fig. 13Fig. 14Outcome of Ω on Φ(ξ).Fig. 14Fig. 15Outcome of E on Φ(ξ).Fig. 15Fig. 16Outcome of Nb on Φ(ξ).Fig. 16Fig. 17Variation in Cs via K and M; (a) when h=1.0, (b) h=2.0.Fig. 17Fig. 18Variation in Nu via BiT and dnp; (a) when h=1.0, (b) h=2.0.Fig. 18Table 3Variation in ResCfs via M for both spherical and cylindrical-shaped nanoparticles.Table 3MResCfsh=1.0h=2.00.11.6625919841.6625920130.21.7300410151.7300410070.31.7944431401.7944431320.41.8561443151.856144300Table 4Variation in Res−1/2Nus via M, BiT, Ec, dnp and γ for both spherical and cylindrical-shaped nanoparticles.Table 4MBiTEcdnpγRes−1/2Nush=1.0h=2.00.10.25816273090.28157875910.20.26896450500.29197541060.30.28009774410.30269013680.40.29160347010.31376410910.10.07685394560.07945047280.20.14241245460.14901462710.30.19898037850.21040644340.40.24827501980.26496828260.10.11600444210.14491878690.20.17495740020.20168445640.30.23345180540.25793744280.40.29160347010.313764109110.00.30146206340.328239891420.00.30043828170.327025180430.00.29839905620.324318903350.00.29160347010.31376410911.10.23286446200.24252608915.00.29160347010.313764109130.00.29839905620.324318903365.250.30146206340.3282398914
A comparative analysis on the variations in nanofluid flow containing spherical and cylindrical-shaped alumina nanoparticles is mainly focused in this article. The Brownian motion, thermophoresis, Joule heating, chemical reaction and activation energy impacts are also taken into account. It should be noted that 4 % volume fraction of the alumina nanoparticles is considered in this analysis. A numerical solution of the modeled equations is presented by using shooting technique. From this comparative analysis, the following key points are i.It is found that a greater magnetic factor enhanced the skin friction coefficient. Additionally, the skin friction is more pronounced for cylindrical-shaped nanoparticles compared to spherical-shaped nanoparticles.ii.It is observed that greater values of the magnetic factor, thermal Biot number, Eckert number, and interfacial layer thickness enhanced the heat transfer rate, while a greater nanoparticle diameter reduced the rate of heat transfer. Furthermore, cylindrical-shaped nanoparticles have a higher rate of heat transfer than spherical-shaped nanoparticles.iii.A higher magnetic factor has an increasing impact on thermal distribution and a reducing impact on velocity distribution. A greater curvature factor enhanced both the velocity and thermal distributions.iv.The temperature distribution is enhanced by a higher thermophoresis factor, thermal Biot number, and interfacial layer thickness, while it is reduced by a higher nanoparticle diameter.v.The concentration distribution is enhanced by a higher interfacial layer thickness and activation energy factor, while it is reduced by a higher nanoparticle diameter, Brownian motion factor, and chemical reaction factor.vi.From the results observed in this analysis, it is found that the velocity, thermal, and concentration distributions are higher for cylindrical-shaped nanoparticles compared to spherical-shaped nanoparticles.
Abdullah Dawar: Writing – original draft, Conceptualization. Rawan Bossly: Supervision, Formal analysis, Formal analysis, Data curation. Fuad S. Alduais: Writing – original draft, Validation. Afrah Al-Bossly: Investigation, Visualization. Jihad Younis: Software, Methodology. Anwar Saeed: Resources, Investigation.
The current analysis can be extended by exploring the effect of varying volume fraction of the spherical and cylindrical-shaped nanoparticles to optimize the heat transfer performance. Additionally, the effects of an inclined magnetic field and induced magnetic field on the flow profiles could provide further insights.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.