ABSTRACT This study investigates the thermal vibration and buckling characteristics of smart sandwich nanoplates featuring a hexachiral auxetic core and piezoelectric/piezomagnetic face sheets. Utilizing the nonlocal strain gradient theory (NSGT) and a higher‐order shear deformation framework, the governing equations are derived to capture size‐dependent effects. The results demonstrate that while thermal loads induce a softening effect, the magnetic potential acts as a stiffening mechanism that enhances structural stability. Conversely, external electric potential and increased thickness‐to‐length ratios are found to reduce the natural frequencies. A key finding reveals that vibration resistance is significantly influenced by the core's auxetic topology, where minimizing ligament dimensions maximizes stiffness. These insights are critical for optimizing the design of high‐performance components in NEMS, aerospace sensors, and smart structural systems.
ABSTRACT This study offers a theoretical analysis of horizontally polarized shear wave within a composite structure composed of a functionally graded magneto electro elastic (FGMEE) plate, which is supported by an elastic foundation utilizing a microstructural coupled stress half space. To characterize the graded material properties of the FGMEE plate, the linear functional distribution along the ‐axis is applied. Using the WKB (Wentzel–Kramers–Brillouin) asymptotic technique, the displacement fields and the electric and magnetic potentials are approximated. The procedure begins by transforming the PDE into an ODE via variable separation, thereafter solving the ordinary differential equation using the WKB approximation method. Furthermore, dispersion equations have been formulated by applying appropriate boundary conditions for both EOMO and ESMS cases. In addition, some specific cases have been examined, illustrating consistency with earlier established results. To illustrate the outcomes quantitatively, a comparative study has been carried out for two examples of upper layer namely, ‐ and PZT‐5A‐ is regarded for the FGMEE layer. Numerical computations along with graphical representations reveal the effect of different parameters in the dispersion relations on the SH‐wave velocity profile. As a consequence, FGMEE materials combine magnetic, electrical, and elastic properties, making them ideal for multi‐functional sensors; also coupled stress half space adds microstructural effects, which improve sensitivity for nano/micro scale sensors.
ABSTRACT Adhesive contact involving magneto‐electro‐elastic materials is affected by surface energy, indenter geometry, and the coupled material response. This article presents a unified treatment of axisymmetric adhesive contact problems between an arbitrary shaped rigid indenter and an insulated magneto‐electro‐elastic (MEE) substrate. Employing the harmonic potential function method combined with the principle of superposition, the proposed approach yields closed‐form solutions for adhesive interactions between axisymmetric indenters with arbitrary power‐law profiles and diverse adhesive force distributions. Explicit formulations are established for surface displacements, stress fields, applied force, and stress intensity factor. Furthermore, the numerical results are presented for power‐law punches including conical, spherical, diamond and nearly flat punches, indicating that the pull‐off response is comparatively sensitive to variations in indenter geometry when the indenter is relatively sharp.
ABSTRACT Building upon applications of Littlewood–Paley theory to fluid mechanics, we extend known regularity criteria in terms of pressure and the gradient of the magnetization orientation to the three‐dimensional Navier–Stokes–Landau–Lifshitz system. Using Littlewood–Paley decomposition and techniques from homogeneous Besov spaces, we establish new regularity criteria in Lebesgue spaces in terms of the pressure (or its gradient) along with the gradient of the magnetization orientation field.
ABSTRACT The free vibration and buckling of partially cracked isotropic and functionally graded material rectangular microplates in a thermal environment are investigated. Thermal effects are introduced into the control equations as thermal bending moments and in‐plane forces, while the linear spring model represents cracks. Furthermore, analytical solutions in series form are obtained for the free vibration and buckling of fully clamped microplates using the symplectic superposition method. The obtained analytical solutions are uniformly applicable to solving the dynamic problems of isotropic and functionally graded material microplates. Using this solution, the effects of different parameters on the natural frequency and critical buckling temperature of the plate are systematically analyzed through specific numerical examples. The results indicate that cracks significantly reduce the natural frequency and critical buckling temperature of the plate, whereas the internal material length scale parameter increases the natural frequency and enhances the plate's resistance to thermal buckling. The accuracy of the analytical solutions obtained in this work is verified against existing results in the literature. Furthermore, convergence analysis demonstrates that the symplectic superposition solutions converge rapidly.
ABSTRACT Understanding the surface response to moving loads poses a fascinating mechanical challenge that is crucial to determining structural strength. Motivated by this issue, this work analyzes the dynamic behavior generated by a uniformly moving line load on the rough upper surface of an irregular functionally graded piezo‐electric‐magnetic fiber‐reinforced composite (FG‐PEMFRC) substrate made of fibers embedded in a matrix. Through exact analytical treatment, explicit expressions are obtained for the incremental normal and shear stresses, horizontal and vertical electric displacements, and horizontal and vertical magnetic inductions under four boundary conditions such as electrically open magnetically short (EOMS), electrically open magnetically open (EOMO), electrically short magnetically short (ESMS), and electrically short magnetically open (ESMO). Parametric effects, such as including maximum irregularity depth, an irregularity factor (covering rectangular, parabolic, and smooth profiles), the functionally graded parameter, surface‐roughness friction coefficient, and constituent volume fractions, are illustrated graphically. Moreover, three types of mechanical coupling are defined, namely electro‐mechanical (EMC), magneto‐mechanical (MMC), and electro‐magneto‐mechanical (EMMC), and are also graphically illustrated. Taken together, these findings may be useful in numerous domains, including road and runway construction, advanced sensor and energy‐harvester development, and so on.
ABSTRACT In a two‐dimensional magnetic field, a Williamson‐nanofluid (WNF), an incompressible non‐Newtonian MHD fluid, flows past a stretched cylinder. It has several uses in a variety of domains, including material processing, heat transfer, and physiological fluids. The temperature‐dependent viscosity is modeled using the Reynolds viscosity formulation, with the cylinder surface subjected to a Newtonian heating boundary condition. Previous studies have already solved the problem using numerical methods (shooting method and Runge–Kutta method) to attain a solution. The governing physical model is formulated using the boundary‐layer approximation, which transforms the fundamental conservation laws into a system of nonlinear PDEs. Appropriate similarity variables are subsequently used to transform the dimensional PDEs into nonlinear ODEs. The resulting equations are solved numerically in Mathematica using the NDSolve algorithm. The flow, temperature, and concentration profiles are then calculated over a wide range of governing physical factors to examine their influence on the transport characteristics of the flow system. To further improve the predictive capability of the analysis, an AI‐based framework employing the NN‐LMBPA is implemented in MATLAB. The numerical solutions generated by the conventional solver are used to construct the training dataset for the NN. The performance of the proposed model is evaluated through function fitting, training‐state analysis, error histograms, regression analysis, and performance curves, providing a comprehensive assessment of its predictive accuracy and convergence characteristics. The results demonstrate that the governing flow parameters exert a significant influence on the flow, temperature, and concentration fields. In particular, the temperature distribution increases with increasing values of the Brownian‐motion parameter and the Nusselt number (Nt), whereas the velocity field is improved by increasing the melting parameter and Casson fluid parameter. Conversely, an increase in the thermal‐slip parameter suppresses the temperature distribution within the‐boundary‐layer. These findings highlight the value of the proposed AI‐assisted computational framework for accurately predicting the nonlinear transport behavior of complex nanofluid flow systems. As the Prandtl (Pr) number, Solutal slip parameter, and Nt grow, the concentration of flow drops. By comparing the results with previous research, the numerical code is validated and shown to be in good agreement.
ABSTRACT The finite element method (FEM) provides a general framework for analyzing virtually all types of structures under various loading conditions. Nevertheless, in certain situations, designers may identify that the structural system exhibits predominantly beam‐like behavior, allowing the use of analytical theories. This is often the case when evaluating the performance of rectangular floor systems where bending stiffness is significantly lower in one direction compared to the other. Additionally, beam theory offers highly accurate structural response predictions for footbridges. The theory of layered beams is extensively utilized in engineering, particularly in the design of composite beams, cross‐laminated timber beams, and sandwich structures. When it comes to long‐span lightweight floors, addressing human‐induced vibrations is a crucial aspect of the design process. This paper derives and solves the exact equation of motion for a simply supported layered beam. The formulation is based on the exact differential equation governing layered beam behavior, ensuring applicability to a wide range of cross‐sectional configurations. The solution is given, including both steady‐state and transient responses. These combined features distinguish the present study from previous works, where such completeness is rarely achieved within a single paper. This study is structured into two main sections: the first section derives the equation of motion and offers a comprehensive solution for simply supported layered beams using a validated walking function. The second section illustrates the practical application of this solution in the vibration design of lightweight floors. The proposed method clarifies the theoretical foundations of the guidelines and provides a straightforward approach to designing layered beams. A case study is presented, focusing on a timber–steel–timber composite floor beam, where timber plates are bolted to a steel beam. The results show that the proposed method effectively explains the underlying theories and offers a practical design approach for layered beams subjected to human‐induced vibrations.
ABSTRACT This note is a short version of arXiv preprint arXiv:2508.12786 (2025), explicitly tailored to explain why the solutions of a stochastic Maccari system as found in ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 103, no. 5 (2023): e202100199. are wrong. Furthermore, I explain that the figures do not show the (wrong) solutions. It was written on request of ZAMM's Editor in Chief Holm Altenbach.
ABSTRACT This paper develops a matrix‐collocation method based on the Frobenius–Euler polynomial family for nonlinear Duffing‐type oscillators. The Appell structure of the Frobenius–Euler basis yields a parameter‐independent subdiagonal operational derivative matrix, while the parameter controls the conditioning of the resulting linear system through the basis evaluation. The method uses Chebyshev–Gauss–Lobatto collocation nodes on the unit interval, a Newton–Raphson scheme with analytical Jacobian and homotopy continuation in the cubic coefficient, and a piecewise extension for long‐time integration over multiple oscillation periods. A structural conditioning analysis identifies a stable parameter regime in which the condition number of the collocation system is smaller than that of the Bernoulli operational matrix scheme by a factor of – at truncation orders , while the achievable accuracy matches that of the Bernoulli basis to floating‐point precision in the benchmark problems considered. The mechanism behind the conditioning advantage is explained through the proximity of a removable but numerically dangerous singularity in the Frobenius–Euler generating function at . Numerical experiments on three standard Duffing benchmark problems establish accuracy comparable to that of the improved Taylor matrix method (ITMM), the Laplace decomposition algorithm, the quasilinearized Bessel polynomial collocation method (QBPCM), and the modified variational iteration method; the piecewise scheme with segments of degree achieves final‐time errors below over four oscillation periods of the unforced cubic Duffing oscillator.
ABSTRACT Quadratic air drag is a common dissipative mechanism at moderate to high velocities, yet in nonlinear one‐degree‐of‐freedom systems, it typically blocks closed‐form solutions and energy‐based phase‐plane partitions. We study the autonomous class and use a constructive alternative: on trajectory segments with fixed velocity sign, the squared‐velocity substitution converts the dynamics into a first‐order linear equation for as a function of . This yields explicit phase‐plane branches that can be continued and matched across turning points, providing separatrix‐type boundaries even when an explicit time parametrization is unavailable. For the cosine potential, we derive analytic saddle‐separatrix branches and provide the critical‐velocity thresholds, as well as the winding‐number classification of rotations corresponding to each separatrix. Adding dry (Coulomb) friction introduces set‐valued dynamics at , producing sticking intervals and ribbon‐shaped regions of the initial conditions' plane leading to sticking; their boundaries follow from the same branchwise construction for constant, spatially varying, and normal‐force‐dependent friction laws. For the tilted periodic (washboard) potential , explicit separatrix formulae yield an escape‐velocity criterion and a critical damping threshold separating unbounded transport from capture for . Analytical boundaries and thresholds are validated against direct time‐domain simulations, and the resulting closed and semi‐closed expressions provide benchmark cases for numerical basin‐boundary and dynamical integrity computations in periodic settings.
ABSTRACT In this work, the effects of a magnetohydrodynamics (MHD) field on the heat and mass transport properties of a chemically reactive, radiating Casson ternary nanofluid (TNF) across an exponentially stretching surface in a porous medium are examined. Water‐based silicon dioxide (SiO 2 ), copper (Cu), and silver (Ag) nanoparticles compose the nanofluid. The research examines how velocity, temperature, and concentration profiles are simultaneously affected by permeability, thermophoresis, radiation, Brownian motion, and chemical reactions. To explore the effects of important parameters, the governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) using a similarity transformation and solved numerically in MATLAB using the boundary value problem (BVP) fourth‐order code (bvp4c). The derived quantities are analyzed using the response surface methodology (RSM). The increased porosity, magnetic field intensity, and the Casson fluid parameter reduce velocity. Additionally, Brownian motion and radiation increase temperature dispersion. According to the results, the optimization study shows that the model is highly accurate in R 2 , with the reaching 96.07%, the reaching 99.68%, and the reaching a 99.65% level of significance. These findings provide novel nanofluid heat and mass transport technologies for industrial and technical applications.
ABSTRACT This study investigates the flow of a tetrahybrid nanofluid in both horizontal and vertical microchannels, marking a novel exploration in this area. Tetrahybrid nanofluids, composed of four different nanoparticles dispersed in a base fluid, are known for their superior thermal conductivity compared to nanofluids containing a single type of nanoparticle. The analysis considers the effects of thermal radiation, an exponential space‐dependent heat source, and a magnetic field within the thermal energy equation, along with slip and convective boundary conditions in the flow. The governing equations are first nondimensionalized and then solved semi‐analytically using a novel approach known as the Fibonacci wavelet method. The findings indicate that the Reynolds number exhibits a dual influence on the thermal profile in horizontal microchannels. The Eckert number and heat source have a more significant impact on the thermal field in horizontal microchannels. Additionally, the Hartmann number notably affects the skin friction coefficient with variations in volume fraction particularly in horizontal microchannels. Conversely, the influence of the Hartmann number on the Nusselt number is more evident in vertical microchannels as the Reynolds number varies. The outcomes of this study provide valuable insights for the design and optimization of advanced thermal management systems, including microelectronics cooling, micro heat exchangers, biomedical devices, energy systems, and microreactors, where enhanced heat transfer performance is of critical importance.
ABSTRACT Studies on the irreversibility optimization of advanced nanofluids, particularly ternary hybrid nanofluids (THNFs), have attracted significant attention from researchers and engineers due to their wide‐ranging industrial applications. Irreversibility, commonly quantified through entropy generation(EG), is a key factor in evaluating energy losses within thermal systems. Its impact is frequently observed in practical devices such as chillers, air coolers, refrigerators, and vehicle engines, where minimizing energy dissipation is essential for improving efficiency and performance. Motivated by these wide‐ranging applications, the present study investigates entropy generation in MHD Williamson THNF flow with heat transfer over a permeable stretching sheet. The momentum equation for the THNF is formulated by incorporating the effects of electro‐magnetohydrodynamic (EMHD) and surface porosity. The influences of viscous dissipation, radiation source, Dufour, and EMHD are considered in the energy transport relation. The mass concentration equation is formulated subject to chemical reaction, Soret effects, and activation energy. The conservation of microorganisms concept is used to regulate and reduce the agglomeration of the ternary nanoparticles in the flow regime. Aluminum oxide , silver , and copper nanoparticles are used for the formation of the THNF. The water‐based Williamson fluid model is considered for the analysis. The second thermodynamics law is used to model the irreversibilities. Through appropriate similarity transformations, the governing dimensional flow equations are converted into a nondimensional system and then solved numerically using the built‐in function (NDSolve) of Wolfram Mathematica. The flow behavior and EG analysis are performed graphically, while physical quantities are scrutinized numerically. The results indicate that the temperature profiles of both HNF and THNF increase with higher Hartmann number, Eckert number, Dufour, and radiation parameters, whereas an opposite trend is observed with increasing Prandtl number. The velocity field of both fluids decreases with rising Hartmann and Weissenberg numbers. The concentration profiles of HNF and THNF diminish with increasing chemical reaction parameter and Schmidt number. Furthermore, THNF exhibits an enhancement of up to 4% in both skin friction and Nusselt number compared to HNF. Additionally, entropy generation increases with higher values of the microorganisms diffusion parameter, Hartmann number, and Brinkman number.
ABSTRACT Heat and mass transfer in flows of rotating disk under deceleration are significantly affected through internal reaction dynamics. In such context, exothermic/endothermic chemical reactions by heat generation/absorption arise as key components that govern thermal transport and concentration regulation. Their impact becomes even more significant in the existence of Stefan mechanism, where surface transpiration further alters correlation between reaction controlled transport phenomena and concentration gradient. Such coupling of reactive species and Stefan effect are critical in industrial procedures such as catalytic reactors, polymer fabrication, cooling of rotating machinery, and chemical vapor deposition processes. Therefore, the present study investigates the influence of Stefen mechanism on incompressible magnetized flow over a stretchable decelerating rotating disk together with electric field, Hall current, and exothermic/endothermic chemical reactions. The 3D, axi‐symmetric boundary layer flow governing equations resulting from unsteady Navier's Stokes, energy and mass equations are normalized with the help of time frame similarity transformations. The reduced dimensionalized system of differential equations are first discretized through first‐and second‐order central finite difference approximations and then solved iteratively using Successive over Relaxation (SOR) method. The graphs of the controlling parameters on velocity, temperature, and concentration distributions are shown graphically for different scenarios of Stefan blowing/suction. It is deduced that electric parameter modified the velocity fields along radial and tangential directions while the Hall parameter has opposite effects on such velocity profiles under the influence of Stefan phenomenon. The temperature field is enhanced withing the boundary layer region for temperature difference parameter. The species concentration is reduced by chemical reaction parameter. The numerical values of wall shear stresses, Nusselt number, and Sherwood number are calculated at the deceleration disk surface. A generalized additive model (GAM) framework was used to measure the nonlinear impact of various control parameters on near‐wall response gradients. The GAM fits disclose parameter‐dependent monotonic trends and subtle nonlinearities in the thermal and solutal boundary layer responses across the explored parameter ranges. The graphical results on radial, tangential, and axial velocity fields against the unsteadiness parameter are matched well with published work. To validate the numerical scheme, numerical values of shear stresses for unsteadiness parameter are also compared well with literature in limiting scenario.
ABSTRACT The current study compares the heat transfer characteristics of a dual offset jet (DOJ) consisting of two plane isothermal turbulent offset jets and a single offset jet (SOJ). The Reynolds Averaged Navier–Stokes (RANS) equations‐based incompressible flow solver, in addition to the two‐equation turbulence model standard , is utilized to generate the simulation data. In the case of DOJ configuration, the offset ratio (, ratio of the jet centerline–horizontal wall distance to the nozzle width) of the jet on the upper side is varied from 6 to 12; the same for the jet on the lower side is retained at 3. The Reynolds number () at the nozzle outlet for both the jet configurations is varied between 15,000 and 35,000. The bottom plate is heated and set at a constant temperature boundary condition. The analysis of heat transfer results indicates that the DOJ thermal boundary layer remains thicker than that for the SOJ in the developed zone. The local Nusselt number () along the heated bottom wall for the DOJ is found to be superior than the SOJ for a given value of and . Moreover, the average Nusselt number () along the bottom wall is found to be greater for the dual offset jet configuration compared to the single offset jet. A regression analysis which is performed to correlate with , and results in correlation functions for the DOJ and for the SOJ, applicable in the range and .
ABSTRACT This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non‐convex domains by analyzing the regularity of the solution in weighted Sobolev spaces of Kondratiev type from which we extract a result on the regularity in the specific scale of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our proofs are based on Schauder's fixed point theorem.
ABSTRACT The existing investigation organized a novel analysis of the 3D flow features of ternary and penta‐hybrid nanofluids flow across porous space, along with the corresponding heat‐transfer mechanisms across an extended porous channel incorporating gyrotactic microbial movement and internal heat generation. This study marks a substantial advancement over past investigations on ternary and Penta (HNF) models by incorporating enhanced and unique thermal properties via the inclusion of five distinctive types of tiny material particles in the formulation of the model equations. The Buongiorno model of nanofluid is introduced to account for the effects of convective heating, Brownian movement, and thermophoretic diffusion on the nanoparticles. The resulting complex system of nonlinear governing PDEs is changed into ODEs by employing proper transformation functions and subsequently computed through the RKF45 discretization method, yielding closed‐form approximate solutions for various flow profiles within the MATLAB programing platform. A comprehensive parametric analysis reveals that increases in porous‐media permeability, magnetic factor, and ratio parameter enhance the velocity profiles in both directions. Notably, the Penta (HNF) demonstrates superior thermal and mass‐transport performance relative to the ternary (HNF). To validate the computed results accuracy obtained through the RKF45 technique, tabulated comparisons with two independent algorithms were performed, showing strong agreement between the methods. These computational insights into nanofluid flow and microbial dynamics offer valuable insights for optimizing heat transfer and control mechanism in advanced energy systems.
ABSTRACT Waste discharge concentration in thermal performance is crucial for optimizing heat transfer efficiency in industrial and environmental processes, and the dynamic motion of the spinning sphere interacts with waste discharge to either enhance or hinder thermal transport. This analysis is important in industries where waste byproducts accumulate, and they help in heat removal processes. This article aims to study trihybrid magnetized cross nanofluid in the presence of waste discharge concentration. For thermal analysis, the space‐conjugate temperature‐based nonuniform heat sink/source around a spinning sphere is considered. The trihybrid nanofluid contains three nanoparticles of Cu, Fe 3 O 4 , and SiO 2 . The spinning sphere geometry has the capability to describe the dynamic behavior of the nanofluid in optimizing heat dissipation and waste discharge control. Partial differential equations (PDEs) are generated based on physical assumptions, and they are transformed into ordinary differential equations (ODEs) by similarity transformation. The Bvp4c scheme is used to get the initial solution and combined it with artificial neural network (ANN), considering a three hidden‐layer neural network to solve and predict the solution. For higher values of , the velocity increases and for , the velocity decreases. During the training process, it is seen that the best validation and test datasets are noted. The temperature profile decreases for (0.1–0.5), and the temperature profile increases for (0.3–0.6), (0.35–0.50), and (2.2–5.2). As increases from 0.01 to 0.04, the concentration profile decreases.
The scientists are fascinated by increasing the industrial and engineering progression, and this is why it is essential to examine the working of each device. This is promising by confirming that specific progressions are unalterable. The entropy generation rate is a degree of energy missing amount inside a structure and is worthwhile in falling wasted-ness or exploiting productivity. The dynamics of Nanofluid, an innovative field of fluid dynamics with usages in energetics, remedial study, and natural science, has risen in new spans. Nanoparticles overwhelming the appropriate capacities can be formed naturally or synthetically in research laboratories. Healthcare drug transportation is one of the uses of nanomaterials. The progress of nanoparticles to support yield chemotherapy treatments straight to malevolent developments and provide medications to damaged portions of blood vessels to treat cardiovascular infection. In the present paper, the thermophysical and rheological properties of hybrid nanofluids on flow patterns are studied. Mixed convective flow of blood-based hybrid nanofluids considering two distinct nanomaterials, such as () and . Porous medium effects are described by using the Darcy-Forchheimer relation. Nonlinear radiation is used to study heat transfer, and a stretching boundary is further subjected to convective conditions. The entropy generation equation for every type of nanofluid is determined by using the second law of thermodynamics. Additionally, in order to make the model more comprehensive, the cubic autocatalysis chemical reaction with irreversibility analysis is taken into consideration. Adopting the procedure of transformations, highly nonlinear ordinary differential equations (ODEs) are received from leading partial differential equations (PDEs). The coupled ordinary differential equations are handled by the utilization of the optimal homotopy analysis approach. Some remarkable special cases of Bejan number transport, entropy generation rate, velocity, concentration, and temperature are addressed to improve the novelty of the stated article. The study shows that fluid motion is retarded with an increase in volume fraction of both () and -water nanofluids. The higher values of the porosity parameter reduced the velocity of fluid in the y-direction. It is found that the Forchheimer number is discovered to have a contrasting impact on velocity-temperature curves. Furthermore, the temperature ratio factor exaggerates the temperature field, thermal Biot number, and volume fraction of () and nanoparticles. The concentration of nanofluids declines for both homogeneous reaction and heterogeneous reaction variables. Higher approximations of both homogeneous reaction variable and diffusion parameter associated with homogeneous reaction variable lead to improved entropy rate. Bejan number rises with but declines with . The heat development amount owing to water-based nanoparticles is quite remarkable related to blood-based nanomaterials. Also, local Nusselt number and skin friction coefficient are enhanced for higher volume fraction of ()- and -water nanofluids. The current work is useful in the latest technologies containing biosciences, pharmaceuticals, engineering, and chemical industries to improve their efficiency.
