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In this innovative study, a unique approach was engaged to simulate the flow characteristics of nanofluid inside a tank featuring a surface subjected to uniform flux. The testing fluid for this investigation was fabricated by incorporating alumina powders with varying shapes into water. The derivation of the final equations involved the application of Darcy’s law and the formulation of the stream function. The container experienced the combined efficacy of both the Lorentz force and gravity forces. The incorporation of additives resulted in a significant enhancement of the Nusselt number (Nu), demonstrating an increase of 19.8% and 40.28%, contingent on the magnitude of the Hartmann number (Ha). Moreover, an elevation in the shape factor led to a notable rise in Nu by 14%. Remarkably, as the Ha increased, there was a substantial reduction in the cooling rate by 51.33%. Furthermore, in the absence of the Ha, an escalation in the Rayleigh number (Ra) caused Nu to surge by 65.8%. This study holds paramount importance as it introduces a novel technique for simulating nanofluid flow with a sinusoidal surface, providing valuable insights into the complex interplay of forces within the container. The utilization of varying shapes of alumina powders adds a layer of sophistication to the experimentation, making this investigation a noteworthy contribution to the existing body of knowledge. The findings not only enhance our understanding of heat transfer dynamics but also offer practical implications for applications involving nanofluids in containers with nonuniform surfaces subjected to heat flux.

This paper presents a numerical solution to the problem of time-dependent blood flow via a w-shaped stenotic conduit, driven by pulsatile pressure gradient. The problem is formulated in cylindrical coordinates by employing the theoretical model of tangent hyperbolic fluid. The electro-osmotic effects are also taken into consideration. To simplify the non-dimensional governing equations of the flow problem, a mild stenosis assumption is utilized and the impact of the blood vessel wall is mitigated by employing a radial coordinate transformation. An explicit finite difference method is used to solve the resulting nonlinear system of differential equations, considering the auxiliary conditions specified at the boundary of the blood channel. After obtaining the numerical solution to the problem, an examination is carried out for various flow variables, such as axial velocity, temperature field, mass concentration, skin friction, Nusselt number, and Sherwood number. These results are presented graphically, and a concise explanation is provided using physical facts. An increase in flow rate and blood velocity leads to a rise in response, while an increase in stenosis height, Weissenberg number, and power-law exponent leads to a reverse response. Furthermore, the temperature field is significantly affected by the Brinkman number and the Prandtl number.

Transport properties of hydrocarbon liquid-based nanofluids in non-Darcy media have key significance in chemical, thermal and mechanical engineering. Therefore, the key focus of this research is to investigate the transport mechanism in nanofluid using Koo–Kleinstreuer–Li (KKL) thermal conductivity model in non-Darcy media under squeezing and permeable effects. The functional fluid is a homogenous mixture of Cu and kerosene. The problem formation is carried out via nanofluid-enhanced properties and similarity rules. Then numerical scheme was endorsed for the results analysis under increasing physical ranges. It is observed that the velocity $F(\eta )$ increased when the values of ${\alpha }_1$ vary from 1.0 to 4.0. However, quick particles movement is noticed for ${\gamma }_1$ for 1.0–4.0 and $-$1.0 to $-$4.0. Further, the thermal process in Cu/kerosene depreciates for ${\alpha }_1=0.5$, 1.0, 1.5, 2.0, ${\gamma }_1=2$, 4, 6, 8 and ${\gamma }_1=-2.0$, $-$4.0, $-$6.0, $-$8.0, respectively. The stronger permeability of the lower plate highly reduced the fluid movement and depreciation in the movement can be optimized when the fluid sucks from the channel through the lower plate.

This paper focuses on applying the Corcione model to the microchannel. The Corcione model is highly relevant because it provides accurate empirical relationships for forecasting the dynamic viscosity and effective thermal conductivity of nanofluids. These qualities are crucial for building and improving different thermal systems. The model presents and discusses two simple empirical correlating equations for forecasting the dynamic viscosity and effective thermal conductivity of nanofluids. Hence the aim of this work is to use Corcione’s model to demonstrate the fully developed laminar flow of an electrically conducting nanoliquid through an inclined microchannel. The energy equation takes into account the physical impacts of the heat source/sink, temperature jamp, and viscous dissipation. TiO₂ nanoparticles in water are taken into consideration in this work for enhanced cooling. Using the numerical program Maple, Runge–Kutta–Fehlberg 4th–5th-order method is utilized to solve the present research. Making use of graphs, all of the flow parameters are shown, and the physical consequences on the flow and temperature profiles are thoroughly examined. It is noted that a higher inclined angle enhances the velocity profile whereas a larger temperature jump declines the temperature profile. Furthermore, Corcione’s model often has greater velocities, temperatures, and reduced surface drag forces than the Tiwari–Das model.

In this research, a novel design stochastic numerical technique is presented to investigate the unsteady form magnetohydrodynamic (MHD) slip flow along the boundary layer to analyze the transportation and heat transfer in a solar collector through nano liquids which is a revolution in the field of neurocomputing. Thermal conductivity in variable form is dependent on temperature and wall slips are assumed over the boundary. For mathematical modeling, the solar collector is assumed in the form of a nonlinear stretching sheet and a quite new artificial neural networks (ANNs) based approach is used to solve the current problem in which inverse multiquadric radial basis (IMRB) kernel is sandwiched between a global search solver named genetic algorithms (GAs) and a highly effective local solver named sequential quadratic programming (SQP) i.e. IMRB-GASQP solver. The governing boundary value problem is altered in the form of a system of nonlinear ordinary differential equations (ODEs) through the utilization of similarity transformation and then the obtained system of ODEs is solved using IMRB-GASQP solver by altering the values of distinguished parameters involved in it to observe the fluctuation in the velocity and temperature profiles of nanofluid. The obtained results are effectively compared with the reference solutions using the Adams numerical technique in graphical and tabulated form. An exhaustive error analysis using performance operators is presented while the efficacy of the designed solver using various statistical operators is also part of this research.

This paper investigates the effects of radiation, internal heat source and magnetohydrodynamics (MHD) on the mixed convective boundary layer flow of a Casson nanofluid within a porous medium that is saturated and subject to an exponentially stretching sheet. The nanofluid model incorporates Brownian motion and thermophoresis, and the Darcy model is employed for the porous medium. By applying an appropriate similarity transformation, the nonlinear governing boundary layer equations are converted into a set of nonlinear coupled ordinary differential equations. These equations are then solved numerically using the Hermite wavelet method, with simulations conducted through the MATHEMATICA programming language. The analysis covers various aspects including temperature distribution, velocity, solute concentration and several engineering parameters such as skin friction coefficients, the Nusselt number (rate of heat transfer) and the Sherwood number (rate of mass transfer), all evaluated based on dimensionless physical parameters. The results indicate that elevated radiation intensifies temperatures and leads to thicker thermal boundary layers. As the Casson parameter increases, both the velocity and the momentum boundary layer become narrower. Additionally, a more pronounced chemical reaction rate reduces the thickness of the solutal boundary layer. The accuracy and reliability of the numerical Hermite wavelet method are validated through a comparative analysis with previous studies, demonstrating excellent concordance and confirming the robustness of the computational approach.

The characteristics of buoyancy-driven convection of nanofluid stream containing motile gyrotactic micro-organisms over a continuous heated surface are explored. The benefits of including micro-organisms to the suspension incorporate micro-scale mixing and foreseen enhanced stability of nanofluid. For heat transfer and mass transfer processes, non-Fourier’s heat flux theory and non-Fick’s mass flux theory are employed. This theory is actively under investigation to resolve some drawbacks of the famous Fourier’s Law and Fick’s Law. The modified parameters in conventional laws are thermal and solutal relaxation times, respectively. The governing equations are remodeled using appropriate similarity transformations into a system of coupled ordinary differential equations. Finite Element Methodology is used to obtain the solution of nonlinear coupled differential equations. The governing equations are associated with dimensionless parameters like $Ri,Nb,Nt,Le,{\beta }_e,{\beta }_c,Pe,\mathrm{\Omega }$. The influence of these parameters is analyzed graphically on velocity, temperature profile, concentration profile and density of micro-organisms. The computational results obtained reveal that the temperature profile and concentration profile have an inverse relationship with thermal relaxation and solutal relaxation time, respectively. Furthermore, the velocity increases with increasing values of the Richardson number, while a reverse pattern is observed for bioconvection Rayleigh number and Buoyancy ratio parameter.

We have modified the mineral oil used in transformers by dispersing 1-wt.% metal oxide nanostructures (commercially available ${\mathrm{\text{TiO}}}_2$, ZnO and ${\mathrm{\text{Fe}}}_3{\mathrm{\text{O}}}_4$ spherical structures and ZnO rod-shaped structures synthesized by arc discharge) into the oil through ball milling without surfactant. A good dispersion that lasted for at least 24 h was obtained for all nanofluids, however sedimentation was discovered by 72 h after ball milling. All nanofluids with different nanostructures exhibited enhanced thermal conductivity compared with the raw transformer oil. The nanofluid with ZnO nanoparticles showed better thermal conductivity than the nanofluids with ${\mathrm{\text{TiO}}}_2$ and ${\mathrm{\text{Fe}}}_3{\mathrm{\text{O}}}_4$ nanoparticles. The nanofluid with elongated ZnO nanoparticles (nanorods) synthesized by arc discharge showed the best thermal conduction among all the nanofluids studied in this work over the whole measurement period. The enhanced thermal conductivity of the nanofluid with elongated nanostructure is considered to be due to the rod-shaped nanostructure creating heat flow paths with lower thermal resistance. The arc discharge provides a cost-effective and scalable method to fabricate metal oxide nanostructures for potential nanofluid applications.

The water-based bioconvection of a nanofluid containing motile gyrotactic micro-organisms (moves under the effects of gravity) over a nonlinear inclined stretching sheet in the presence of a nonuniform magnetic field has been investigated. This regime is encountered in the bio-nanomaterial electroconductive polymeric processing systems currently being considered for third-generation organic solar coatings, anti-fouling marine coatings, etc. Oberbeck–Boussinesq approximation along with ohmic dissipation (Joule heating) is considered in the problem. The governing equations of the flow are nonlinear partial differential equations and are converted into ordinary differential equations via similarity transformations. These equations are then solved by the Finite Element Method. The effect of various important parameters on nondimensional velocity, temperature distribution, nanoparticle concentration, the density of motile micro-organisms is analyzed graphically in detail. It is observed from the obtained results that the flow velocity decreases with rising angle of inclination $\delta$ while temperature, nanoparticle’s concentration and density of motile micro-organisms increase. The local skin friction coefficient, Nusselt number, Sherwood number, motile micro-organism’s density number are calculated. It is noticed that increasing the Brownian motion and thermophoresis parameter leads to an increase in temperature of fluid which results in a reduction in Nusselt number. On the contrary, the Sherwood number rises with an increase in Brownian motion and thermophoresis parameter. Also, interesting features of the flow dynamics are elaborated and new future pathways for extension of the study identified in bio-magneto-nano polymers (BMNPs) for solar coatings.

Cellulose nanofluids have a great application potential in the energy industry, and their thermal properties are substantially affected by the components and microstructures of nanofluids. Therefore, this study investigated the isobaric heat capacity and thermal conductivity of cellulose I$\beta$ nanofluids mixing with H₂O by molecular dynamics (MD). The results showed that the existence of water in cellulose increased the isobaric heat capacity of the system, especially for the random cellulose/H₂O nanofluids. Additionally, nonequilibrium molecular dynamics (NEMD) simulations based on the Fourier law of thermal conduction were conducted to examine the thermal conductivity of the simulated systems. As indicated by our results, the cellulose I$\beta$ crystal was advantageous in terms of its high directional thermal conductivity along the chain direction. Thus, the thermal conductivity of the cellulose/H₂O nanofluids along the chain direction used the high directional thermal conductivity of the cellulose I$\beta$ crystal. Consequently, the cellulose/H₂O nanofluids integrated the superiorities of high isobaric heat capacity of water and great directional thermal conductivity of cellulose I$\beta$ crystal, thereby improving the heat transfer efficiency in thermodynamic systems. In addition, the potential energy of the cellulose crystal system was mainly generated by intermolecular repulsion, while those of the cellulose/H₂O nanofluid systems were mainly produced through intermolecular attraction.

This paper analyzes the influence of mixed convective fourth grade nanofluid flow by a stretchable Riga device in the presence variable thermal conductivity and mass diffusivity. Heat and mass transportation are considered with Cattaneo–Christov (CC) model. Thermal radiation and dissipation are also taken in the energy expression. Suitable transformation is employed to reduce partial differential system into nonlinear ordinary system. The governing nonlinear expression is solved via optimal homotopy analysis method. Impact of different physical variables is discussed via graphs. Velocity profile is enhanced for higher values of cross viscous parameter and fourth grade fluid variable. Fluid temperature enhances for higher estimation of thermal relaxation parameter but reverse behavior is seen for solutal concentration variable on nanoparticle concentration.

In recent years, the research for enhanced thermal transportation is centered around the utilization of nanostructures to avail the prospective benefits in areas of biomedical, metallurgy, polymer processing, mechanical and electrical engineering applications, food processing, ventilation, heat storage devices, nuclear systems cooling, electronic devices, solar preoccupation, magnetic sticking, bioengineering applications, etc. The thermal aspects of nanoliquids and associated dynamics properties are still necessary to be explored. In this thermal contribution, the flow of Casson nanofluid configured by an infinite disk is analyzed. The significance of Marangoni flow with activation energy, thermal and exponential space-dependent heat source, nonlinear thermal radiation and Joule heating impacts is also incorporated. Similarly, variables are affianced to recast the governing flow expressions into highly coupled nonlinear ODEs. The numerical simulation for the prevailing model is elucidated by applying the bvp4c built-in function of computational commercial software MATLAB. Consequences of sundry parameters, namely, magnetic parameter, Prandtl number, radiation parameter, exponential space-dependent heat source parameter, thermal-dependent heat source parameter, Eckert number, Dufour parameter, Soret number, Schmidt number, Marangoni number and Marangoni ratio parameter, mixed convection parameter, buoyancy ratio parameter, bioconvection Rayleigh number, activation energy parameter, thermophoresis parameter, Brownian motion parameter, bioconvection Lewis number, Peclet number microorganisms difference variable versus involved flow profiles like velocity, temperature, concentration of nanoparticles and microorganism field are obtained and displayed through graphs and tabular data.

This article examines entropy production (EP) of magneto-hydrodynamics viscous fluid flow model (MHD-VFFM) subject to a variable thickness surface with heat sink/source effect by utilizing the intelligent computing paradigm via artificial Levenberg–Marquardt back propagated neural networks (ALM-BPNNs). The governing partial differential equations (PDEs) of MHD-VFFM are transformed into ODEs by applying suitable similarity transformations. The reference dataset is obtained from Adam numerical solver by the variation of Hartmann number (Ha), thickness parameter $(\alpha )$, power index ($n)$, thermophoresis parameter (Nt), Brinkman number (Br), Lewis number (Le) and Brownian diffusion parameter (Nb) for all scenarios of proposed ALM-BPNN. The reference data samples arbitrary selected for training/testing/validation are used to find and analyze the approximated solutions of proposed ALM-BPNNs as well as comparison with reference results. The excellent performance of ALM-BPNN is consistently endorsed by Mean Squared Error (MSE) convergence curves, regression index and error histogram analysis. Intelligent computing based investigation suggests that the rise in values of Ha declines the velocity of the fluid motion but converse trend is seen for growing values of $n$. The rising values of Ha, Nt and Br improve the heat transfer but converse trend is seen for growing values of $\alpha$. The inclining values of Nt incline the mass transfer but it shows reverse behavior for escalating values of Le. The inclining values of Br incline the EP.

This study describes a novel application of artificial intelligent-based computing paradigm via knacks of neural networks backpropagated through Levenberg–Marquardt scheme (NNs-BLMS) to investigate the mathematical model of Carreau nanofluid thin film flow over a stretching sheet under the magnetic field effect (CNTFFSSM). The system of PDEs of the designed model is transformed by using suitable transformations to the system of ODEs. The Adams deterministic numerical technique is utilized for generation of a dataset for the proposed technique for six varients to produce corresponding six scenarios of the designed model by varying magnetic parameter, Brownian motion parameter, unsteadiness parameter, Weissenberg number, thermophoresis number and Lewis number. The computational intelligent solver NNs-BLMS is applied to the CNTFFSSM model by performing processes on training, testing and validation samples. The efficiency of the proposed technique is validated by comparison of the standard solution and outcomes of the proposed solver for designed model through histograms, error analysis and regression analysis. The outcomes disclosed that velocity field is a decreasing function of magnetic parameter, while an increasing function of the Weissenberg number. The temperature and concentration profiles enhance with higher thermophoresis parameter, while diminish with higher Weissenberg number. Moreover, higher Lewis number causes lower concentration profile.

In recent years, nanotechnologies have been widely used in several fields regarding their rapid developments which create a lot of prospects for researchers and engineers. More specifically, replacement of conventional liquids with nanoliquids is considered as an innovative solution to heat transfer problems. This research work reports mixed convective phenomenon of viscoelastic fluids with variable stretched surface. This study prospects the applications in biotechnology and irrigation systems. Characteristics of thermophoresis, nonlinear thermal radiation and Brownian moment are examined. Significant influence of no mass flux relation and variable thicked surface on second grade is studied in this work. Moreover, heat-mass transportation phenomenon is explored through thermophoresis, nonlinear thermal radiation and Brownian diffusion. Transformation procedure is adopted in order to achieve required ODEs. HAM procedure is adopted to obtain the series solution. Outcomes of physical parameters for velocity, temperature and concentration fields are discussed. Here velocity profile of second-grade fluid is detected to be a declining function K (local second-grade parameter). Higher $N_b$ (Brownian moment parameter) intensifies the temperature profile while concentration of viscoelastic liquid dwindles against $N_b$.

The enhancement of the heat transfer of fluids is a very important task in engineering and technology which can be accomplished through the hybrid nanoparticles. In this theoretical investigation, the natural convection flow of nanoliquid through a square container is examined under the impacts of thermal boundary conditions. The considered nanoliquid is a combination of titanium alloy Ti6Al4V (class of titanium) and aluminum alloy AA7075 (class of aluminum) nanoparticles and water as a base fluid. Aluminum alloys are commonly used in household wiring and manufacturing of wheels, etc. The titanium alloy Ti6Al4V is commonly acknowledged as the “workhorse” of titanium which is extensively used in aerospace and biomedical engineering. The mathematical model is developed in the form of nonlinear partial differential equations and later transformed into dimensionless form. The resultant dimensionless set of equations is simulated numerically by using the eminent finite element method (FEM). The iterative Newton’s Raphson method is used to get the optimal solutions. The stream function and temperature contours are displayed against the Rayleigh number and nanoparticle volume friction. The variation of local and average Nusselt numbers is also plotted. The outcomes revealed that water-based titanium alloy is the best choice to enhance the heat transfer rate in the cavity. The results obtained in this investigation are very important in engineering research, academic, and discussion about the heat transfer analysis with these two types of alloy nanoparticles inside the cavity flows.

In this work, heat transfer enrichment using nanofluid in a lid-driven porous cavity having an isothermal solid block has been investigated numerically through three distinct cases based on the moving direction of horizontal wall(s). The modeled governing partial differential equations are numerically solved by SIMPLE algorithm and the resulting outcomes are validated with previous works both in qualitative and quantitative nature. Numerical results of various emerging parameters such as Richardson number ($0.01\le \mathrm{\text{Ri}}\le 100$), Darcy number (10${}^{-5}\le \mathrm{\text{Da}}\le 10^{-1}$), Block length ($0.25\le \mathrm{\text{BL}}\le 0.75$) and volume fraction of suspended nanoparticles ($0.0\le \varphi \le 0.05$) were discussed. The results clearly show that the direction of moving walls plays a crucial role on the flow and heat transfer, in particular, the opposite direction of moving walls yields the highest heat transfer rate. In addition to that, significant influence of isothermal block is found and determined that the block length of 0.75 causes the maximum rate of heat transfer in the entire system.

This study extracts heat transfer analysis for the axisymmetric flow of CuO-water nanofluid imposing normal to a spiraling disk uniformly rotating and radially stretching. The combined influence of uniformly rotating and linearly radial stretching surface engenders the logarithmic spirals in motion. The governing system is transformed into a system of the ordinary differential equations that represent the flow. The Tiwari–Das [Int. J. Heat Mass Trans. 50, 2002 (2007)] model is utilized to envisage the heat transfer and other physical properties of the nanofluid. A highly precise numerical technique named as Legendre wavelet-based spectral method is practiced to computing the impacts of the pertinent variables on the flow. The results of physical significance including the wall shear, skin frictions and heat transfer rate are attained for the multiple spiraling parameters and the nanoparticle volume fraction. The asymptotic results for the large spiraling parameter are also presented to analyze the flow behavior for the high spiraling parameter. The influence of the nanoparticles is also investigated and the enhancement in thermal conductivity is measured. The temperature and thermal layer thickness reduce for large spiraling parameter. Also addition of the nanoparticles enhances the heat transfer rate of the nanofluid.

The flow of air containing small particles past a pointed area of an aircraft, bullet and bonnet of vehicles exhibits the flow over paraboloid surface of revolution. Therefore, the study of Al₂O₃-H₂O nanofluid over upper paraboloid horizontal surface of revolution (UPHSR) is organized. The concerned model develops via thermal conductivity dealing with the particles shape factor and similarity transforms. Afterward, numerical analysis is performed and the influences of pertinent parameters on the velocity and temperature $𝜃(\eta )$ in Al₂O₃-H₂O are examined. The deep inspection of the results in the view of physics behind them revealed that Al₂O₃-H₂O drops for the stringer magnetic field. Further, nonlinear thermal radiations and internal heat generation made Al₂O₃-H₂O a better heat conductor which increased its applications in a broad zone.

Some of the fundamental properties of the nanofluids affect not only the transport phenomenon but also enhance the the heat transfer characteristics. The advancement in Rotatory machine technology can also be attributed mainly to a lot of research work that had been done on rotating disk flows with addition of nanoparticles in the base fluid. Taking cue of these developments, we examined the nanofluid (Cu+H₂O) flow over a rotating disk moving upward/ downward with viscous dissipation by reducing the governing Navier–Stokes equations into ordinary differential equations using appropriate transformations and then numerically evaluated them by the BVP Mid-rich scheme in Maple software. The study of velocity and thermal profiles is carried out and examined graphically. The results show that the upward movement of the disk escalates the radial and azimuthal velocity profiles along with the thermal gradient. In contrast, the addition of nanoparticles decreases the heat transfer rate at a constant disk movement.