LAUSR

The aim of the present work is to apply the fractional derivative to the heat transformation of a nanofluid along with ramped wall temperature. The flow is analyzed under the effect of magnetohydrodynamic together with heat transfer. A nanofluid under the application of fractional-order differential equations by Caputo–Fabrizio derivatives with respect to time has the ability to explain the behavior of nanofluid under the influence of memory concept. For the same purpose, Caputo–Fabrizio time-fractional derivative is applied to investigate the behavior of nanoparticles on the thermal conductivity of a fluid. Appropriate nondimensional variables are engaged in the equation which governs the problem and guides us to obtain the exact solutions for the fields of velocity and temperature. These obtained solutions for the nondimensional set of governing equations are found by extracting them from the governing equations by applying Laplace transform techniques along with Caputo–Fabrizio time-fractional derivative. The influence of the fractional variable on the velocity, temperature, and Nusselt number is graphically exposed and discussed. The velocity for the state of wall temperature as ramped falls down with the enlarging values of the fractional parameter. Variation in Nusselt number is shown in the tabular form. Solutions are visualized graphically to make an analysis of how the variation is taking place in the physical behavior of the nanofluid flow with respect to the change in distinct physical parameters. The obtained results here will have useful industrial and engineering implementations. It is found that fluid velocity in the flow direction decreases with the increase in the magnetic parameter. The relationship of fractional parameter with the velocity and temperature of the nanofluid is found as direct proportional for a smaller time. However, this direct proportionality converts into inverse proportionality for larger values of time. It is observed that the increase in the nanoparticles volume fraction causes an increase in temperature distribution. It is due to lower specific heat of nanoparticles and its higher thermal conductivity than that of the base fluid.