This paper presents some new exact solutions corresponding to the oscillating flows of a MHD Oldroyd-B fluid with fractional derivatives. The fractional calculus approach in the governing equations is used. The exact solutions for the oscillating motions of a fractional MHD Oldroyd-B fluid due to sine and cosine oscillations of an infinite plate are established with the help of discrete Laplace transform. The expressions for velocity field and the associated shear stress that have been obtained, presented in series form in terms of Fox H functions, satisfy all imposed initial and boundary conditions. Similar solutions for ordinary MHD Oldroyd-B, fractional and ordinary MHD Maxwell, fractional and ordinary MHD Second grade and MHD Newtonian fluid as well as those for hydrodynamic fluids are obtained as special cases of general solutions. Finally, the obtained solutions are graphically analyzed through various parameters of interest.

Exact expressions of velocity, temperature and mass concentration have been calculated for free convective flow of fractional MHD viscous fluid over an oscillating plate. Expressions of velocity have been obtained both for sine and cosine oscillations of plate. Corresponding fractional differential equations have been solved by using Laplace transform and inverse Laplace transform. The expression of temperature and mass concentration have been presented in the form of Fox-H function and in the form of general Wright function, respectively and velocity is presented in the form of integral solutions using Generalized function. Some limiting cases of fluid and fractional parameters have been discussed to retrieve some solutions present in literature. The influence of thermal radiation, mass diffusion and fractional parameters on fluid flow has been analyzed through graphical illustrations.

The unsteady motion of an Oldroyd-B fluid over an infinite flat plate is studied by means of the Laplace and Fourier transforms. After time t = 0, the plate applies cosine/sine oscillating shear stress to the fluid. The solutions that have been obtained are presented as a sum of steady-state and transient solutions and can be easily reduced to the similar solutions corresponding to Newtonian or Maxwell fluids. They describe the motion of the fluid some time after its initiation. After that time when the transients disappear, the motion is described by the steady-state solutions that are periodic in time and independent of the initial conditions. Finally, the required time to reach the steady-state is established by graphical illustrations. It is lower for cosine oscillations in comparison with sine oscillations of the shear, decreases with respect to ω and l and increases with regard to l r .

The influence of simultaneous variation of slip and temperature has been inquired for the case of free-convected flow of an MHD (magnetohydrodynamic), elastoviscous fluid past an unbounded upright plate. How the course of velocity is revamped in response to temperature alterations on boundary has been studied by considering two cases of constant temperature and variable temperature. The inverse and direct role of elastoviscous parameter (K), thermal Grashof number (G r ) and Hartmann number (M) in determining the pattern of flow has been discussed through exact expressions and graphical illustrations. Interestingly, a G r -regime has been identified corresponding to elastoviscous velocity variation. The Newtonian fluid velocity past an unbounded plate entailing slip factor has also been retrieved and compared with elastoviscous fluid velocity. This comparative analysis reflects on magnitude and profile adaptations in response to numerical changes.

An investigation of how the velocity of elasto-viscous fluid past an infinite plate, with slip and variable temperature, is influenced by combined thermal-radiative diffusion effects has been carried out. The study of dynamics of a flow model leads to the generation of characteristic fluid parameters ( G r , G m , M, F, S c and P r ). The interaction of these parameters with elasto-viscous parameter K ′ is probed to describe how certain parametric range and conditions could be pre-decided to enhance the flow speed past a channel. In particular, the flow dynamics’ alteration in correspondence to the slip parameter’s choice, along with temperature provision to the boundary in temporal pattern, is determined through uniquely calculated exact expressions of velocity, temperature and mass concentration of the fluid. The complex multi-parametric model has been analytically solved using the Laplace and Inverse Laplace transform. Through study of calculated exact expressions, an identification of variables, adversely (M, F, S c and P r ) and favourably ( G r and G m ) affecting the flow speed and temperature has been made. The accuracy of our results have also been tested by computing matching numerical solutions and by graphical reasoning. The verification of existing results of Newtonian fluid with varying boundary condition of velocity and temperature has also been completed, affirming the veracity of present results.