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Nanofluids are gaining extensive attention due to their thermo-physical properties in technological and industrial fields for controlling the effects of heat transfer. Classical nanofluid studies are generally confined to models described by partial differential equations of an integer order where the memory effect and hereditary properties of materials are neglected. In order to overcome these downsides, the present work focuses on studying nanofluids with fractional derivative formed by differential equations with Caputo time derivatives that provides memory effect on nanofluid characteristics. Also, investigation on natural convective flow, heat, and mass transfer of nanofluids formed by different base fluids with different shapes of copper nanoparticles past an infinite vertical plate with radiation effect is carried out. The governing fractional differential equations are solved by employing Laplace transform technique with suitable boundary conditions. The different base fluids-water (H 2 O), SA:sodium alginate (C 6 H 9 Na O 7 ), and EG:ethylene glycol (C 2 H 6 O 2 ) and different shapes of nanoparticles-blade, brick, platelet, and cylinder are considered for the study.The exact solutions are obtained for the temperature, concentration, and velocity distributions and the respective Nusselt number, Sherwood number, and skin-friction coefficient. The influence of non-dimensional parameters provides physical interpretations of temperature, concentration, and velocity fields, Nusselt number, Sherwood number, and skin friction in detail with the help of graphical representations. From the results, it is found that nanofluid with water-based blade-shaped nanoparticle exhibits more velocity and temperature distributions. Also, strengthen of fluid flow, temperature, and concentration of nanofluids are inversely correlate with fractional order derivatives.

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The present numerical study reports the thermal performance of the straight porous fin with temperature-dependent thermal conductivity, radiation, and magnetic field effects. The heat transfer model comprising the Darcy's law for simulating flow with solid-fluid interactions in porous medium, Rosseland approximation for heat transfer through radiation, Maxwell equations for magnetic field effect and linearly varying temperature dependent thermal conductivity, results into highly nonlinear ordinary differential equation. The governing equation is solved using a finite difference scheme with suitable boundary conditions. The obtained solutions are physically interpreted by considering the impact of different nondimensional parameters on thermal performance, efficiency, and effectiveness of the system through plotted graphs. A detailed result with regard to the Nusselt number at the fin base is calculated. The results obtained are observed to be in excellent agreement with previous studies. From the study, it is observed that there is a significant effect on the thermal performance of the fin in the presence of porous constraints; also, results reveal that the nonlinear thermal conductivity parameter strengthens the thermal performance, efficiency, and effectiveness of the fin. Furthermore, the results of the study reveal that the rate of heat transfer of the fin increases with the increase in the magnetic parameter and radiation parameter.

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dusty nanofluids, Marangoni convection, space-dependent heat source, two-phase flow, wavy surface | INTRODUCTIONThe Marangoni convection is a phenomenon, which is induced by thermocapillarity or solutocapillarity, that is, the variation of surface tension of the fluid with temperature or concentration. Surface tension is one of the fluid characteristics, which remain stable basically. However, the primary conditions affecting the surface tension of the fluid are variation in temperature and addition of chemicals in it. When temperature rises, the molecules in the fluid become more active, which results into the reduction in the surface tension of the fluid. Also, it changes according to the chemical reactions that occur when the chemicals are added to the fluid. Further, the surface tension of the fluid also gets affected by impurities present in it. For instance, the salt which is a highly solvable substance, increases the surface tension of water, whereas the soap, which is a sparingly soluble substance, decreases it. Hence, it is worthy to investigate the Marangoni effect on dusty nanofluid flow. The study on Marangoni convection is essential in view of numerous industrial applications including crystal growth, thin liquid films, semiconductor processing, coating flow technology, microfluidics, etc. After a promising research, this mechanism has been adopted in the field of nanotechnology to get an artificial rain by a process called cloud seeding. Here, substances like dry ice (silver iodide aerosol is commonly used, which acts as a cloud condenser) is dispersed into the upper parts of clouds, which alters the surface tension and then causes microphysical changes within the clouds. Therefore, condensation of water vapor in the atmosphere occurs. One of the widely practiced applications using this effect is suppressing the fog in airports during harsh weather. The concept of Marangoni boundary layer was introduced by Napolitano, 1 and, later, several authors 2-7 studied the flow under the Marangoni effect. It is evident from Mahanthesh et al 8 that Marangoni effect enhances the temperature and concentration fields of Casson liquid flow due to an infinite disk. The buoyancy and thermocapillary effect on convective fluid flow and transfer of heat inside a vertical cylindrical cavity under the impact of evaporation was evaluated numerically by Kozhevnikov and Sheremet. 9 Siddiqa et al 10 have assessed the effect of thermal radiation on Marangoni boundary layer flow along a vertical irregular surface and their mathematical results disclosed that water-particle mixture, that is, dusty fluid, reduces the heat transfer rate.The mechanism of heat transfer plays a significant role in hypersonic, space vehicles, gas turbines, etc. In those applications, the thermal performance of commonly employed traditional fluids like water, oils, glycol, etc, is deficient. With the intention of boosting the thermal conductivity of these base fluids, several processes, such as adding surfactant, salts, alcohol to the base fluids, inc...

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Recent developments in fluid dynamics have been focusing on nanofluids, which preserve significant thermal conductivity properties and magnify heat transport in fluids. Classical nanofluid studies are generally confined to models described by partial differential equations of an integer order, where the memory effect and hereditary properties of materials are neglected. To overcome these downsides, the present work focuses on studying nanofluids with fractional derivatives formed by differential equations with Caputo time derivatives that provide memory effect on nanofluid characteristics. Further, heat transfer enhancement and boundary layer flow of fractional Maxwell nanofluid with single‐wall and multiple walls carbon nanotubes are investigated. The Maxwell nanofluid saturates the porous medium. Also, buoyancy, magnetic, electric, and heating effects are considered. Governing continuity, momentum, and energy equations involving Caputo time‐fractional derivatives reduced nondimensional forms using suitable dimensionless quantities. Numerical solutions for arising nonlinear problems are developed using finite difference approximation combined with L1 algorithm. The influence of involved physical parameters on flow and heat transfer characteristics is analyzed and depicted graphically. Our simulations found out that surface drag of Maxwell nanofluid with single‐walled carbon nanotubes dominates nanofluids with multiple walls carbon nanotubes, but the reverse trend is noticed for larger Grashof number values.