In this modern era, nanofluids are considered one of the advanced kinds of heat transferring fluids due to their enhanced thermal features. The present study is conducted to investigate that how the suspension of molybdenum-disulfide $\left(Mo{S}_2\right)$ nanoparticles boosts the thermal performance of a Casson-type fluid. Sodium alginate (NaAlg) based nanofluid is contained inside a vertical channel of width d and it exhibits a flow due to the movement of the left wall. The walls are nested in a permeable medium, and a uniform magnetic field and radiation flux are also involved in determining flow patterns and thermal behavior of the nanofluid. Depending on velocity boundary conditions, the flow phenomenon is examined for three different situations. To evaluate the influence of shape factor, $Mo{S}_2$ nanoparticles of blade, cylinder, platelet, and brick shapes are considered. The mathematical modeling is performed in the form of non-integer order operators, and a double fractional analysis is carried out by separately solving Caputo-Fabrizio and Atangana-Baleanu operators based fractional models. The system of coupled PDEs is converted to ODEs by operating the Laplace transformation, and Zakian’s algorithm is applied to approximate the Laplace inversion numerically. The solutions of flow and energy equations are presented in terms of graphical illustrations and tables to discuss important physical aspects of the observed problem. Moreover, a detailed inspection on shear stress and Nusselt number is carried out to get a deep insight into skin friction and heat transfer mechanisms. It is analyzed that the suspension of $Mo{S}_2$ nanoparticles leads to ameliorating the heat transfer rate up to 9.5%. To serve the purpose of achieving maximum heat transfer rate and reduced skin friction, the Atangana-Baleanu operator based fractional model is more effective. Furthermore, it is perceived that velocity and energy functions of the nanofluid exhibit significant variations because of the different shapes of nanoparticles.

This article provides a comprehensive analysis regarding effects of ramped wall temperature and ramped wall velocity on incompressible time-dependent magnetohydrodynamic flow of Maxwell fluid. The flow is due to free convection and bounded to an infinite vertical plate embedded in porous medium. Solutions of mass, shear stress, and energy fields are computed symmetrically by introducing some suitable non-dimensional parameters along with the Laplace transformation technique. The expression for the Nusselt number is also calculated. A comparison between solutions incorporating isothermal temperature and ramped wall temperature conditions is also executed to examine the profile differences. A graphical study is performed to highlight the influence of parameters on mass flow and energy transfer.

The core purpose of this work is the formulation of a mathematical model by dint of a new fractional modeling approach to study the dynamics of flow and heat transfer phenomena. This approach involves the incorporation of the Prabhakar fractional operator in mathematical analysis to transform the governing system from a conventional framework to a generalized one. This generalized model evaluates the improvement in thermal efficacy of vacuum pump oil because of the inclusion of aluminum alloy nanoparticles. The flow of the under-observation nanofluid starts due to the combined effects of natural convection and the ramped velocity function at the boundary. Meanwhile, an analysis of the energy equation is conducted by taking the Newtonian heating mechanism into consideration. The characteristics of platelet-, brick-, cylinder-, and blade-shaped alloy nanoparticles are incorporated into the primary system using shape-dependent relations for thermal conductivity and viscosity. Both the classical and generalized models are solved to derive the exact solutions by first inserting some dimension-independent quantities and then operating the Laplace transform on the succeeding equations. These solutions are utilized for the development of graphical illustrations to serve the purpose of covering all features of the problem under consideration. Furthermore, changes in energy and flow functions due to the dominant influences of the relevant contributing factors are delineated with appropriate physical arguments. In addition, the numerical results of the skin friction coefficient and Nusselt number are displayed via multiple tables to analyze the disturbance in shear stress and discuss the contribution of the fractional parameters, the volume concentration of the considered nanoparticles, and the shape factor in the boost of the thermal potential of the considered nanofluid. The findings imply that aluminum alloy nanoparticles have the ability to produce a 44% enhancement in the thermal effectiveness of vacuum pump oil. Moreover, the flow velocity is reduced as the loading range of the nanoparticles rises.