The aim of the article is to study the unsteady magnetohydrodynamic-free convection flow of an electrically conducting incompressible viscous fluid over an infinite vertical plate with ramped temperature and constant concentration. The motion of the plate is a rectilinear translation with an arbitrary time-dependent velocity. Closed-form solutions for the temperature, concentration and velocity fields of the fluid are obtained. The influence of transverse magnetic field that is fixed relative either to fluid or plate is studied. Furthermore, the effects of system parameters on the fluid velocity are analyzed through numerical simulations and graphical illustrations.

Exact general solutions for the dynamics of an incompressible viscous fluid with non-integer order derivative without singular kernel are established using the integral transforms. These solutions, which are organized in simple forms in terms of exponential and trigonometric functions, can be conveniently engaged to obtain known solutions from the literature. The control of the new non-integer order derivative on the velocity of the fluid moreover a comparative study with an older model, is analyzed for some flows with practical applications. The non-integer order derivative with non-singular kernel is more appropriate for handling mathematical calculations of obtained solutions. It is also more reliable for numerical computations