scholarly journals Exact Solution of Fractional Convective Casson Fluid Through an Accelerated Plate

CFD letters ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 15-25
Author(s):  
Muhammad Nazirul Shahrim ◽  
Ahmad Qushairi Mohamad ◽  
Lim Yeou Jiann ◽  
Muhamad Najib Zakaria ◽  
Sharidan Shafie ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Newtonian Fluid ◽  
Casson Fluid ◽  
Electrical Networks ◽  
Moving Plate ◽  
Transform Method ◽  
The Laplace Transform ◽  
Physical Phenomena ◽  
The Impact ◽  
Fractional Parameter

Fractional derivative has perfectly adopted to model few physical phenomena such as viscoelasticity of coiling polymers, traffic construction, fluid dynamics and electrical networks. However, the application of the fractional derivatives for describing the physical characteristics of non-Newtonian fluid over a moving plate is still rare. In the present study, the effect of the Caputo fractional derivative on the Casson fluid flow which is induced by an accelerated plate is analytically analysed. The governing equations are initially transformed into dimensionless expressions by using suitable dimensionless variables. Then the Laplace transform method is utilized to calculate the exact solutions for the fractional governing partial differential equations. The obtained solutions are validated by comparing the results for specific case with the existing solutions in the literature. The impact of fractional parameter, Prandtl number, and time on the velocity and temperature profiles are graphically showed and discussed. The results depict that the temperature and velocity increase with the increment of fractional parameter and time. Interestingly, the velocity decreases at region near the plate but is enhanced at the area far away from the plate when the Casson fluid parameter is increased. This study is essential in understanding the factional non-Newtonian fluid flows which is more realistic in nature.

Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Yury Rossikhin ◽  
Marina Shitikova
Keyword(s):  
Fractional Derivative ◽  
Human Head ◽  
Contact Spot ◽  
Spherical Shells ◽  
Spherical Object ◽  
Frontal Impacts ◽  
Age Related ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
The Impact

AbstractThe collision of two elastic or viscoelastic spherical shells is investigated as a model for the dynamic response of a human head impacted by another head or by some spherical object. Determination of the impact force that is actually being transmitted to bone will require the model for the shock interaction of the impactor and human head. This model is indended to be used in simulating crash scenarios in frontal impacts, and provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. It is assumed that the viscoelastic features of the shells are exhibited only in the contact domain, while the remaining parts retain their elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shells are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. It is shown that the fractional parameter of the fractional derivative model plays very important role, since its variation allows one to take into account the age-related changes in the mechanical properties of bone.

scholarly journals A generalized kinetic model of the advection-dispersion process in a sorbing medium

2021 ◽  
Author(s):  
Dumitru Vieru ◽  
Constantin Fetecau ◽  
Najma Ahmed ◽  
Nehad Ali Shah
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Transform Method ◽  
Dispersion Process ◽  
Kinetic Adsorption ◽  
Advection Dispersion ◽  
The Laplace Transform ◽  
Time Fractional Derivative ◽  
Generalized Kinetic Model ◽  
Generalized Time

A new time-fractional derivative with Mittag-Leffler memory kernel, called the generalized Atangana-Baleanu time-fractional derivative is defined along with the associated integral operator. Some properties of the new operators are proved. The new operator is suitable to generate by particularization the known Atangana-Baleanu, Caputo-Fabrizio and Caputo time-fractional derivatives. A generalized mathematical model of the advection-dispersion process with kinetic adsorption is formulated by considering the constitutive equation of the diffusive flux with the new generalized time-fractional derivative. Analytical solutions of the generalized advection-dispersion equation with kinetic adsorption are determined using the Laplace transform method. The solution corresponding to the ordinary model is compared with solutions corresponding to the four models with fractional derivatives.

scholarly journals Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects

2022 ◽  
Vol 6 (1) ◽  
pp. 38
Author(s):  
Ridhwan Reyaz ◽  
Ahmad Qushairi Mohamad ◽  
Yeou Jiann Lim ◽  
Muhammad Saqib ◽  
Sharidan Shafie
Keyword(s):  
Fluid Flow ◽  
Analytical Solution ◽  
Thermal Radiation ◽  
Fractional Derivative ◽  
Chemical Reaction ◽  
Fractional Derivatives ◽  
Experimental Studies ◽  
Fluid Flows ◽  
Casson Fluid ◽  
The Impact

Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations for fractional derivatives in the mechanics of fluid flows are yet to be discovered. Nonetheless, theoretical studies will be useful in facilitating future experimental studies. Therefore, the aim of this study is to showcase an analytical solution on the impact of the Caputo-Fabrizio fractional derivative for a magnethohydrodynamic (MHD) Casson fluid flow with thermal radiation and chemical reaction. Analytical solutions are obtained via Laplace transform through compound functions. The obtained solutions are first verified, then analysed. It is observed from the study that variations in the fractional derivative parameter, α, exhibits a transitional behaviour of fluid between unsteady state and steady state. Numerical analyses on skin friction, Nusselt number, and Sherwood number were also analysed. Behaviour of these three properties were in agreement of that from past literature.

Some Solutions of the Timoshenko Beam Equation for Short Pulse-Type Loading

1958 ◽  
Vol 25 (3) ◽  
pp. 379-385
Author(s):  
H. J. Plass
Keyword(s):  
Timoshenko Beam ◽  
Short Pulse ◽  
Sine Wave ◽  
Beam Equation ◽  
Transform Method ◽  
The Laplace Transform ◽  
Support Conditions ◽  
Time Required ◽  
The Impact ◽  
Timoshenko Beam Equation

Abstract A collection of solutions to the Timoshenko beam equation is presented. Various types of support conditions and impact conditions are included. In every case the impact is assumed to be a pulse in the form of a half-sine wave. The results were found numerically, using the method of characteristics, except for one case, which was done in addition by the Laplace transform method, for check purposes. Agreement with experiment is good except for a pulse of duration comparable to the time required for the bending-type wave to travel a distance of one diameter. Discussion is included of the differences among the various cases studied.

scholarly journals The solution of local fractional diffusion equation involving Hilfer fractional derivative

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 809-814
Author(s):  
Yun Qiao ◽  
Quan-Xi Qiao
Keyword(s):  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Operator ◽  
Laplace Transform Method ◽  
Hilfer Fractional Derivative ◽  
Transform Method ◽  
Fractional Diffusion Equations ◽  
The Laplace Transform

In this present work the Yang-Fourier transform method incorporating the Laplace transform method is used to solve fractional diffusion equations involving the Hilfer fractional derivative and local fractional operator. The exact solution is obtained.

scholarly journals MATHEMATICAL AND STATISTICAL ANALYSIS OF RL AND RC FRACTIONAL-ORDER CIRCUITS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040030 ◽  
Author(s):  
NADEEM AHMAD SHEIKH ◽  
DENNIS LING CHUAN CHING ◽  
SAMI ULLAH ◽  
ILYAS KHAN
Keyword(s):  
Statistical Analysis ◽  
Fractional Derivative ◽  
Classical Model ◽  
Modern Concept ◽  
Inverse Transformation ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Fractional Order Circuits ◽  
Fractional Parameter

The RL and RC circuits are analyzed in this research paper. The classical model of these circuits is generalized using the modern concept of fractional derivative with Mittag-Leffler function in its kernel. The fractional differential equations are solved for exact solutions using the Laplace transform technique and the inverse transformation. The obtained solutions are plotted and presented in tables to show the effect of resistance, inductance and fractional parameter on current and voltage. Furthermore, the statistical analysis is presented to predict the seasonal of time and other parameters on the current flowing in the circuit. The statistical analysis shows that the variation in current is insignificant with respect to time and is more significant with respect to other parameters.

scholarly journals Study of Fuzzy Fractional Nonlinear Equal Width Equation in the Sense of Novel Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Muhammad Naeem ◽  
Ahmed A. Khammash ◽  
Ibrahim Mahariq ◽  
Ghaylen Laouini ◽  
Jeevan Kafle
Keyword(s):  
Fractional Derivative ◽  
Fuzzy Theory ◽  
Approximate Solutions ◽  
Transform Method ◽  
Numerical Examples ◽  
Suggested Technique ◽  
Wide Range ◽  
The Laplace Transform ◽  
Uncertainty Parameters ◽  
The Given

In this paper, we designed an algorithm by applying the Laplace transform to calculate some approximate solutions for fuzzy fractional-order nonlinear equal width equations in the sense of Atangana-Baleanu-Caputo derivatives. By analyzing the fuzzy theory, the suggested technique helps the solution of the fuzzy nonlinear equal width equations be investigated as a series of expressions in which the components can be effectively recognised and produce a pair of numerical results with the uncertainty parameters. Several numerical examples are analyzed to validate convergence outcomes for the given problem to show the proposed method’s utility and capability. The simulation outcomes reveal that the fuzzy iterative transform method is an effective method for accurately and precisely studying the behavior of suggested problems. We test the developed algorithm by three different problems. The analytical analysis provided that the results of the models converge to their actual solutions at the integer-order. Furthermore, we find that the fractional derivative produces a wide range of fuzzy results.

scholarly journals Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1154
Author(s):  
Temirkhan S. Aleroev ◽  
Asmaa M. Elsayed
Keyword(s):  
Fractional Derivative ◽  
Homotopy Perturbation Method ◽  
Analytic Solution ◽  
Separation Of Variables ◽  
Fourier Method ◽  
String Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
The Laplace Transform

This paper is proposed for solving a partial differential equation of second order with a fractional derivative with respect to time (the vibration string equation), where the fractional derivative order is in the range from zero to two. We propose a numerical solution that is based on the Laplace transform method with the homotopy perturbation method. The method of the separation of variables (the Fourier method) is constructed for the analytic solution. The derived solutions are represented by Mittag–LefLeffler type functions. Orthogonality and convergence of the solution are discussed. Finally, we present an example to illustrate the methods.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

An improved de Laval nozzle experiment

2021 ◽  
pp. 030641902110341
Author(s):  
Nicholas Goodman ◽  
Brian J Leege ◽  
Peter E Johnson
Keyword(s):  
Data Acquisition ◽  
Laval Nozzle ◽  
Flow Characteristics ◽  
Data Acquisition System ◽  
Isentropic Flow ◽  
Hands On ◽  
De Laval Nozzle ◽  
Physical Phenomena ◽  
The Impact ◽  
The Relationship

Exposing students to hands-on experiments has been a common approach to illustrating complex physical phenomena that have been otherwise modelled solely mathematically. Compressible, isentropic flow in a duct is an example of such a phenomenon, and it is often demonstrated via a de Laval nozzle experiment. We have improved an existing converging/diverging nozzle experiment so that students can modify the location of the normal shock that develops in the diverging portion to better understand the relationship between the shock and the pressure. We have also improved the data acquisition system for this experiment and explained how visualisation of the standing shock is now possible. The results of the updated system demonstrate that the accuracy of the isentropic flow characteristics has not been lost. Through pre- and post-laboratory quizzes, we show the impact on student learning as well.

Sign in / Sign up

Export Citation Format

Share Document