There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The present analysis represents the concept of the Caputo–Fabrizio derivatives of fractional order to MHD flow of a second-grade fluid together with radiative heat transfer. The fluid flow is subjected to an infinite oscillating vertical plate embedded in a saturated porous media. The fluid starts motion due to an oscillating boundary and temperature difference between the plate and the fluid. The problem is modeled in terms of partial differential equations, which consist of momentum equation and heat equation. The Laplace transform method is used to obtain the closed-form solutions for velocity and temperature profiles. In order to understand the physics of the problem under consideration, numerical results are obtained using Mathcad software and brought into light through graphical representations. The influence of various physical parameters is studied and displayed in various figures. The corresponding skin friction coefficient and Nusselt number are provided in tables. A graphical comparison is provided showing a strong agreement with the published results in the open literature.

A modern approach of Caputo–Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium

In this paper, we consider an extended nonlinear Schrodinger equation that includes fifth-order dispersion with matching higher-order nonlinear terms. Via the modified Darboux transformation and Joukowsky transform, we present the superregular breather (SRB), multipeak soliton and hybrid solutions. The latter two modes appear as a result of the higher-order effects and are converted from a SRB one, which cannot exist for the standard NLS equation. These solutions reduce to a small localized perturbation of the background at time zero, which is different from the previous analytical solutions. The corresponding state transition conditions are given analytically. The relationship between modulation instability and state transition is unveiled. Our results will enrich the dynamics of nonlinear waves in a higher-order wave system.

Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation

In this work, we have carried out the influence of temperature dependent viscosity on thin film flow of a magnetohydrodynamic (MHD) third grade fluid past a vertical belt. The governing coupled non-linear differential equations with appropriate boundary conditions are solved analytically by using Adomian Decomposition Method (ADM). In order to make comparison, the governing problem has also been solved by using Optimal Homotopy Asymptotic Method (OHAM). The physical characteristics of the problem have been well discussed in graphs for several parameter of interest.

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Thin film flow in MHD third grade fluid on a vertical belt with temperature dependent viscosity.

The present analysis represents the MHD flow of micropolar fluid past an oscillating infinite vertical plate embedded in porous media. At the plate, free convections are caused due to the differences in temperature and concentration. Therefore, the combined effect of radiative heat and mass transfer is taken into account. Partial differential equations are used in the mathematical formulation of a micropolar fluid. The system of dimensional governing equations is reduced to dimensionless form by means of dimensional analysis. The Laplace transform technique is applied to obtain the exact solutions for velocity, temperature, and concentration. In order to highlight the flow behavior, numerical computation and graphical illustration are carried out. Furthermore, the corresponding skin friction and wall couple stress are calculated.

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MHD Flow of Micropolar Fluid over an Oscillating Vertical Plate Embedded in Porous Media with Constant Temperature and Concentration

This study investigates the effects of an arbitrary wall shear stress on unsteady magnetohydrodynamic (MHD) flow of a Newtonian fluid with conjugate effects of heat and mass transfer. The fluid is considered in a porous medium over a vertical plate with ramped temperature. The influence of thermal radiation in the energy equations is also considered. The coupled partial differential equations governing the flow are solved by using the Laplace transform technique. Exact solutions for velocity and temperature in case of both ramped and constant wall temperature as well as for concentration are obtained. It is found that velocity solutions are more general and can produce a huge number of exact solutions correlative to various fluid motions. Graphical results are provided for various embedded flow parameters and discussed in details.

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Effects of wall shear stress on unsteady MHD conjugate flow in a porous medium with ramped wall temperature.

The unsteady MHD free convection flow near an exponentially accelerated infinite vertical plate through porous medium with uniform heat flux in the presence of thermal radiation has been considered. The mathematical model, under the usual Boussinesq approximation, was reduced to a system of coupled linear partial differential equations for velocity and temperature. Exact solutions are obtained by the Laplace transform method. The influence of pertinent parameters such as the radiation parameter, Grashof number, Prandtl number, and time on velocity, temperature, and skin friction is shown by graphs.

/pdf/radiation-and-magnetohydrodynamics-effects-on-unsteady-free-5cx8y94zr7.pdf

Radiation and magnetohydrodynamics effects on unsteady free convection flow in a porous medium

The purpose of this study is to provide a mathematical model of the process of entropy formation that occurs during the MHD convective transport of Carreau fluid across a curved stretched surface with soret and dufour effects. The energy equation is explored with the combined characteristics of joule heating, viscous dissipation and heat absorption/generation. The modeled governing equations are converted into nonlinear ordinary differential equations using similarity variables, and the results are acquired through the utilization of the built-in command NDSolve. It is revealed that entropy generation and Bejan number increase for higher values of Brinkman number. The impact of the Hartman number increase the entropy generation and reduced the Bejan behavior. The curvature parameter reduced the flow and temperature of fluid but increased the concentration profile. A small change in the heat absorption parameter strengthens the temperature profile. Graphical representations are used to discuss the behavior of Bejan number, entropy generation, velocity, temperature, and concentration profiles for the different values of physical parameters. The results of computations of drag force, the local nusselt number, and the Sherwood number are shown in tabular form for numerous values of the parameters. 

Sami Ul-Haq

Papers

Effects of wall shear stress on unsteady MHD conjugate flow in a porous medium with ramped wall temperature.

Radiation and magnetohydrodynamics effects on unsteady free convection flow in a porous medium

Entropy analysis in MHD convective flow of Carreau fluid over a curved stretching surface with soret and dufour effects