Values Of Fractional Parameter Research Articles

New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics

The main objective of this paper is to introduce an analytical study for the water wave solutions of coupled fractional variant Boussinesq equation, which is modelled to investigate the waves in fluid dynamics. Wave transformation in fractional form is applied to convert the original fractional-order nonlinear partial differential equation into another nonlinear ordinary differential equation. The strategy here is to use the unified method to obtain a variety of exact solutions. The unified method works well and reveals distinct exact solutions which are classified into two different types, namely polynomial function and rational function solutions. The results are also depicted graphically for different values of fractional parameter. These findings are highly encouraging and have significant importance for some special physical phenomena in fluid dynamics

The effect of slip on electro-osmotic flow of a second-grade fluid between two plates with Caputo–Fabrizio time fractional derivatives

This work aims to probe the slip flow of second-grade fluid. The impetus of the flow is taken to be the electro-osmosis and the pressure gradient. The flow is considered to be in a thin channel-like passage formed by two parallel plates. The potential difference existing between the surface of the solid and fluid is taken to be non-symmetric. The governing equations are formed for the second-grade fluid with the Caputo–Fabrizio fractional derivative. The Laplace transform is used for transforming the problem into space parameters after introducing the dimensionless variables. Instead of developing an analytical expression for inverse Laplacian, the numerical Stehfest algorithm is used. A tabular comparison of the obtained results by two different methods (Stehfest and Tzou) is given and the conformity of the two ensures the validity of our obtained results. The results are also pictured in terms of graphs and carry the information of the slip flow effect. Furthermore, the effect of the fractional parameter on velocity has also been tabulated using different values of fractional parameter.

Wall slip and non-integer order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives

This article is focused on natural convection of unsteady flow of generalized Maxwell fluid over an oscillating vertical flat plate with constant temperature at the boundary. The Maxwell fluid with classical derivatives, describing one dimensional flow has been generalized to non-integer order derivatives known as fractional derivative with term of buoyancy. A modern definition of fractional derivative, recently introduced by Caputo and Fabrizio has been used to formulate the considered problem. Semi analytical solutions of the dimensionless problem have been obtained by using the Laplace transform. The solutions for temperature, velocity and shear stress are obtained with numerical inversion techniques of Laplace transform namely, Stehfest’s and Tzou’s algorithms. At the end, graphical illustrations for temperature, velocity, Nusselt number and shear stress are plotted. We have studied especially the influence of fractional parameter on temperature, velocity and shear stress respectively. We have observed that temperature can be enhanced for increasing the fractional parameter α while velocity and shear stress can be increased by decreasing the value of fractional parameter α.

Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model

The present article applies the idea of Caputo-Fabrizio time fractional derivatives to magnetohydrodynamics (MHD) free convection flow of generalized Walters’-B fluid over a static vertical plate. Free convection is caused due to combined gradients of temperature and concentration. Hence, heat and mass transfers are considered together. The fractional model of Walters’-B fluid is used in the mathematical formulation of the problem. The problem is solved via the Laplace transform method. Exact solutions for velocity, temperature and concentration are obtained. The physical quantities of interest are examined through plots for various values of fractional parameter: $\alpha$ , Walters’-B parameter $\Gamma$ , magnetic parameter M , Prandtl number Pr, Schmidt number Sc, thermal Grashof number Gr and mass Grashof number Gm. As a special case, the published results from open literature are recovered.

Mathematica numerical simulation of peristaltic biophysical transport of a fractional viscoelastic fluid through an inclined cylindrical tube

This paper studies the peristaltic transport of a viscoelastic fluid (with the fractional second-grade model) through an inclined cylindrical tube. The wall of the tube is modelled as a sinusoidal wave. The flow analysis is presented under the assumptions of long wave length and low Reynolds number. Caputo's definition of fractional derivative is used to formulate the fractional differentiation. Analytical solutions are developed for the normalized momentum equations. Expressions are also derived for the pressure, frictional force, and the relationship between the flow rate and pressure gradient. Mathematica numerical computations are then performed. The results are plotted and analysed for different values of fractional parameter, material constant, inclination angle, Reynolds number, Froude number and peristaltic wave amplitude. It is found that fractional parameter and Froude number resist the flow pattern while material constant, Reynolds number, inclination of angle and amplitude aid the peristaltic flow. Furthermore, frictional force and pressure demonstrate the opposite behaviour under the influence of the relevant parameters emerging in the equations of motion. The study has applications in uretral biophysics, and also potential use in peristaltic pumping of petroleum viscoelastic bio-surfactants in chemical engineering and astronautical applications involving conveyance of non-Newtonian fluids (e.g. lubricants) against gravity and in conduits with deformable walls.

Surface wave modes in PEMC backed chiral slab

The characteristics of surface wave modes in a PEMC backed chiral slab are studied theoretically. First, the analytical solution of electromagnetic fields and dispersion relations are carried out. Then, the fractional field solutions are found using the fractional curl operator. The numerical results are given by assuming that wave numbers k and k ± are either real or imaginary. These results are also evaluated at real and imaginary values of fractional parameter describing the order of curl operator. The discussion contains fractional dispersion curves at various cut-off frequencies and the fractional surface waves in chiral-PEMC and achiral-PMC slabs respectively. For numerical analysis it is assumed that the fractional order of the curl operator is related to chiral admittance, thickness of the slab, and PEMC admittance. For the values of the fractional order equal to 0, 1, and 2 geometry corresponds to PMC backed ordinary dielectric slab, PEMC backed chiral slab, and PEC backed chiral slab respectively. Consequently TE, HE (even), and HE (odd) modes are produced in the respective geometries.

Waves in Fractional Dual Planar Waveguides Containing Chiral Nihility Metamaterial

Field expressions for guided waves in three-layered fractional dual planar waveguides are derived by virtue of fractional curl operator. Fractional dual waveguides may be regarded as intermediate steps of the two waveguides which are related through principle of duality. All layers are parallel to the walls of the guide. Middle layer of the waveguide is filled with air however other two layers are filled with chiral nihility metamaterial. Parameter α describes the order of the fractional curl operator. No electric field exists in the chiral nihility layers of the waveguide for α = 0, whereas no magnetic field exists in the chiral nihility layers for α = 1. That is, power flow is confined to only air region of the waveguide for integer values of fractional parameter and corresponding situations have been previously discussed in a published work. In present work, it is noted that, for 0 < α < 1, neither electric field and nor magnetic field is zero inside chiral nihility layers of the fractional dual waveguides. By varying the values of order of the fractional curl operator, one can observe how situation dealing with no electric field changes to situation dealing with no magnetic field. Behavior of fields and power flow in different regions of the fractional waveguides is discussed. It is concluded that, even for fractional dual waveguides, real part of power flow is zero in chiral nihility regions of the waveguides.

Complex and higher order fractional curl operator in electromagnetics

Fractional curl operator may be used to obtain the fractional solutions to the Maxwell’s equations. For values of fractional parameter lying between zero and one, fractional solutions may be regarded as intermediate solutions between the original solutions and dual to the original solutions. The fractional parameter describes the order of the fractional curl operator. Behavior of the fractional solutions has been studied for real values of fractional parameter which are greater than unity and for complex values of the fractional parameter.