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This article provides the computational results of laminar, boundary layer flow of a dilute gas-particle mixture over a semi-infinite vertical wavy surface. The governing parabolic partial differential equations are switched into another frame of reference by using primitive variable formulations (PVF). Two-point finite difference scheme is applied to acquire the unknown quantities of the carrier and the particle phase. The results are obtained for the cases: (i) water–metal mixture and (ii) air–metal mixture and are displayed in the form of wall shear stress, wall heat transfer, velocity profile, temperature profile, streamlines and isotherms for different emerging physical parameters. The solutions are compared, as well, with the available data in the literature. Quantitative comparison shows good compatibility between the present and the previous results. For the dusty fluid model it is found that the rate of heat transfer reduces considerably when the amplitude of the sinusoidal waveform increases from 0 to 0.5.

With the help of symbolic computation, the Benjamin–Bona–Mahony (BBM) equation with variable coefficients is presented, which was proposed for the first time by Benjamin as the regularized long-wave equation and originally derived as approximation for surface water waves in a uniform channel. By employing the improved (G′/G)$(G^' /G)$-expansion method, the truncated Painlevé expansion method, we derive new auto-Bäcklund transformation, hyperbolic solutions, a variety of traveling wave solutions, soliton-type solutions and two solitary wave solutions of the BBM equation. These obtained solutions possess abundant structures. The figures corresponding to these solutions are illustrated to show the particular localized excitations and the interactions between two solitary waves.

A model for double-diffusive convection in anisotropic and inhomogeneous porous media has been analysed. In particular, the effects of variable permeability, thermal diffusivity and variable gravity with respect to the vertical direction, have been studied. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using three dimensional simulation. Our results show that the linear theory produce a good predicts on the onset of instability in the basic steady state. It is known that as Rc${R_c}$ increases the onset of convection is more likely to be via oscillatory convection as opposed to steady convection, and the three dimensional simulation results show that as Rc$Rc$ increases, the actual threshold moving toward the nonlinear stability threshold and the behaviour of the perturbation of the solutions becomes more oscillated.

The main purpose of this work is to investigate unsteady magnetohydrodynamic (MHD) boundary layer flow of Maxwell fluid over a stretching surface with nonlinear thermal radiation. Heat and mass transfer analysis is carried out in the presence of convective boundary conditions and first-order chemical reaction. A uniform magnetic field is applied normal to the direction of the fluid flow. The nonlinear coupled partial differential equations are solved numerically using an implicit finite difference method with quasi-linearization technique. Effects of the emerging parameters on the dimensionless velocity, temperature and concentration are investigated. The rate of heat transfer in terms of Nusselt number and rate of mass transfer in terms of Sherwood number are also computed and addressed.

An investigation of heat and mass transfer characteristics of unsteady free convective flow of viscous incompressible micropolar fluid between the vertical porous plates in the presence of thermal radiation is carried out in the present work. The fluid is considered to be grey, absorbing–emitting but non scattering medium and the Cogley–Vincent–Gilles formulation is adopted to simulate the radiation component of heat transfer. The resulting system of equations is solved numerically with Crank–Nicolson implicit finite difference method. The effects of various physical parameters such as transient, micropolar parameter, radiation parameter, Reynolds number, Schmidt number, heat and mass transfer Biot numbers on the velocity, temperature and concentration field are discussed graphically.

It has been found that the dynamical behavior of many complex physical systems can be properly described by nonlinear DDEs. However, in the related literature, research focusing on such equations with rational nonlinearity is rare. Hence, the present study makes an attempt to fill the existing gap. To this end, we consider two distinct DDEs with rational nonlinearity. We observed that the model equations assume three kinds of traveling wave solutions; hyperbolic, trigonometric and rational including kink-type solitary waves and singular periodic solutions. Our discussion is based on the auxiliary equation method.

An emerging concept of hybrid nanofluid with a new improved model of its thermophysical properties are introduced in the present work. Hybrid nanofluid is an advanced type of conventional heat transfer fluids, which has been employed for the enhancement of heat transfer rate. Two distinct fluids, namely hybrid nanofluid (Cu−Al2O3/water)$({\rm{Cu - A}}{{\rm{l}}_{\rm{2}}}{{\rm{O}}_{\rm{3}}}{\rm{/water}})$ and nanofluid (Cu/water) are used to investigate the parametric features of the flow and heat transfer phenomena over a permeable stretching sheet in the presence of magnetic field. The effects of various physical parameters and effecting physical quantities of interest are analyzed. From this study it is observed that the heat transfer rate of hybrid nanofluid (Cu−Al2O3/water)$({\rm{Cu - A}}{{\rm{l}}_{\rm{2}}}{{\rm{O}}_{\rm{3}}}{\rm{/water}})$ is higher than that of Nanofluid (Cu/water) under magnetic field environment. More combinations of different nanocomposites can be tried so that the desired heat transfer rate can be achieved.