This research article presents the magnetohydrodynamic Casson fluid flow through an extending surface embedded in a porous medium. Furthermore, the Casson fluid flow is investigated under the effects of thermal radiation, Joule heating, viscous dissipation, and chemical reaction. The analytical solution of the modeled problem is utilized with the help of homotopy analysis method (HAM). The convergence region of the applied technique is portrayed graphically. The impacts of the embedded factors on the flow profiles are exhibited with the help of figures. Furthermore, numerical values of the surface drag force, heat, and mass transfer rates are highlighted via table. The results show that the augmented Darcy number, Casson and magnetic parameters have declined the velocity profile of the Casson fluid flow. Growth in Brownian motion augments the chaotic motion amongst the particles due to which the kinetic energy of the particles transforms to heat energy which consequently augmented the thermal profile, while reduced the concentration profile. The mass and energy profiles are positively effects with the increment of thermophoresis term. And the growing values of chemical reaction and Lewis number cause a reduction in the diffusivity of mass of fluid due to which less transfer of mass takes place that weakens the concentration layer thickness and declines the concentration profiles.

Details on the hydrothermal characteristics of turbulent flows in a solar channel heat exchanger (CHE) are highlighted. The device has transverse T-shaped vortex generators (VGs). Two staggered VGs (baffles) are inserted on the lower and upper walls of the CHE. The working fluid is Newtonian and incompressible, with constant physical properties. The ANSYS Fluent 17.0 is utilized in this survey. The second-order upwind and QUICK schemes were utilized to perform the discretization of pressure and convective terms, respectively. The SIMPLE algorithm was employed to achieve the speed-pressure coupling. The residual target 10⁻⁹ was selected as a convergence criterion. The effects of the T-VGs’ geometrical shape and Reynolds numbers were inspected. At the baffle level, the wall effect was augmented due to the reduction of the passage area of flows, which is estimated here to be 55%, resulting thus in a considerable resistance to the movement of fluid particles. The thermal distribution is highly dependent on the flow structures within the CHE. Since the fluid agitation yields an enhanced mixing, it allows thus an excellent heat transfer. The most considerable rates of thermal transfer were obtained with high Re, which resulted from the intensified mixing of fluid particles through the formation of recirculation cells and the interaction with the walls of the T-VGs and the CHE. The T-baffles with intense flow rates yielded negative turbulent speeds and intensify the fluid agitation, which improves the thermal exchange rates.

Nanofluids are engineered colloidal suspensions of nanoparticles in the base fluids. At very low particle concentration, nanofluids have a much higher and strongly temperature-dependent thermal conductivity, which enables them to enhance the performance of machining applications such as the cooling and lubrication of the cutting zone during any machining process, the vehicle’s braking system, enhanced oil recovery (EOR), engine oil, and the drilling process of crude oil. In the current work, the density is assumed as an exponential function of temperature due to larger temperature differences. The main focus of this mechanism is the variable density effects on heat and mass characteristics of nanoparticles across the stretching porous sheet with thermophoresis and Brownian motion to reduce excessive heating in high-temperature systems. This is the first temperature-dependent density problem of nanofluid across the stretching surface. The coupled partial differential equations (PDEs) of the present nanofluid mechanism are changed into nonlinear coupled ordinary differential equations (ODEs) with defined stream functions and similarity variables for smooth algorithm and integration. The changed ODEs are again converted in a similar form for numerical outcomes by applying the Keller Box approach. The numerical outcomes are deduced in graphs and tabular form with the help of the MATLAB (R2013a created by MathWorks, Natick, MA, USA) program. In this phenomenon, the velocity, temperature, and concentration profile, along with their slopes, have been plotted for various parameters pertaining to the current issue. The range of parameters has been selected according to the Prandtl number $0.07\le Pr\le 70.0$ and buoyancy parameter $0<\lambda <\infty$, respectively. The novelty of the current work is its use of nanoparticle fraction along the porous stretching sheet with temperature-dependent density effects for the improvement of lubrication and cooling for any machining process and to reduce friction between tool and work piece in the cutting zone by using nanofluid. Moreover, nanoparticles can also be adsorbed on the oil/water surface, which alters the oil/water interfacial tension, resulting in the formation of emulsions.

Nanofluid lubrication and machining are challenging and significant tasks in manufacturing industries that are used to control the removal of a material from a surface by using a cutting tool. The introduction of a nanofluid to the cutting zone provides cooling, lubricating, and chip-cleaning benefits that improve machining productivity. A nanofluid is a cutting fluid that is able to remove excessive friction and heat generation. Chemical reactions and temperature-dependent density are essential in the thermal behavior of a nanofluid. The present study presents a careful inspection of the chemical reactions, temperature-dependent density, viscous dissipation, and thermophoresis during the heat and mass transfer of a nanofluid along a magnetically driven sheet. The physical attitude of viscous dissipation and the chemical reaction improvement rate in magneto-nanofluid flow is the primary focus of the present research. By applying the proper transformation, nonlinear partial differential expressions are introduced to the structure of the ordinary differential framework. The flow equations are simplified into nonlinear differential equations, and these equations are then computationally resolved via an efficient computational technique known as the Keller box technique. Flow factors like the Eckert number, reaction rate, density parameter, magnetic force parameter, thermophoretic number, buoyancy number, and Prandtl parameter governing the velocity, temperature distribution, and concentration distribution are evaluated prominently via tables and graphs. The novelty of the current study is in computing a heat transfer assessment of the magneto-nanofluid flow with chemical reactions and temperature-dependent density to remove excessive friction and heating in cutting zones. Nanofluids play significant roles in minimum quantity lubrication (MQL), enhanced oil recovery (EOR), drilling, brake oil, engine oil, water-miscible cutting fluids, cryogenic cutting fluids, controlled friction between tools and chips and tools and work, and conventional flood cooling during machining processes.

Several primary mechanisms are less utilized in engineering and recent technologies due to unsustainable heating. The impact of viscous dissipation and Joule heating is very important to examine current density and heat rate across a magnetized cylinder. The key objective of this examination was to insulate excessive heat around the cylinder. The present effort investigated the impact of viscous dissipations, Joule heating, and magnetohydrodynamics (MHD) on the transitory motion of convective-heat transport and magnetic flux features of dissipative flows throughout a magnetized and warmed cylinder at suitable places. The suggested turbulent dynamical structure of mathematics is offered for an associated method of partial differentiation equations impacted by boundary values. The complex equations are translated via non-dimensional shapes by using relevant non-dimensional numbers. The non-dimensional representation has been improved to make it easier to conduct uniform computational calculations. The computational answers for these linked dimensionalized formulations have been achieved using the Prandtl coefficient Pr, Joule heating parameter $\zeta$, Eckert number $E_c$, the magneto-force number $\xi$, the buoyancy parameter $\lambda$, and multiple additional predefined factors. The important contribution of this work is based on non-fluctuating solutions that are utilized to examine the oscillating behavior of shearing stress, rate of fluctuating heat transport, and rate of fluctuating magnetic flux in the presence of viscous dissipation and Joule heating at prominent angles. It is shown that the velocity of a fluid increases as the buoyancy parameter increases. The maximum frequency of heat transmission is illustrated for each Eckert variable.

After the introduction of Alam–Imdad’s relation-theoretic contraction principle, the field of metric fixed point theory has attracted much attention. A number of fixed point theorems in the context of relational metric space employing various types of contractions has been appeared during the last seven years. In this manuscript, one proved a metrical fixed point theorem for $\varphi$-contraction involving (c)-comparison functions employing an amorphous relation. The result proved in this paper refines, modifies, unifies and sharpens several existing fixed point results. We also constructed an example in order to attest the credibility of our results. Finally, we applied our result to establish the existence and uniqueness of solution of certain periodic boundary value problem.