New approximate analytical technique for the solution of time fractional fluid flow models

AbstractIn this note, we broaden the utilization of an efficient computational scheme called the approximate analytical method to obtain the solutions of fractional-order Navier–Stokes model. The approximate analytical solution is obtained within Liouville–Caputo operator. The analytical strategy generates the series form solution, with less computational work and fast convergence rate to the exact solutions. The obtained results have shown a simple and useful procedure to analyze complex problems in related areas of science and technology.

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A Decomposition Method for a Fractional-Order Multi-Dimensional Telegraph Equation via the Elzaki Transform

Symmetry ◽

10.3390/sym13010008 ◽

2020 ◽

Vol 13 (1) ◽

pp. 8

Author(s):

Nehad Ali Shah ◽

Ioannis Dassios ◽

Jae Dong Chung

Keyword(s):

Fractional Order ◽

Decomposition Method ◽

Approximate Analytical Solution ◽

Telegraph Equation ◽

Form Solution ◽

Derivative Operator ◽

Analytical Strategy ◽

Fast Convergence Rate ◽

Computational Work ◽

Telegraph Equations

In this article, the Elzaki decomposition method is used to evaluate the solution of fractional-order telegraph equations. The approximate analytical solution is obtained within the Caputo derivative operator. The examples are provided as a solution to illustrate the feasibility of the proposed methodology. The result of the proposed method and the exact solution is shown and analyzed with figures help. The analytical strategy generates the series form solution, with less computational work and a fast convergence rate to the exact solutions. The obtained results have shown a useful and straightforward procedure to analyze the problems in related areas of science and technology.

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Approximate Analytical Solution for Laminar Flow Over a Backward Step

Volume 1A: 35th Computers and Information in Engineering Conference ◽

10.1115/detc2015-46177 ◽

2015 ◽

Author(s):

P. Venkataraman

Keyword(s):

Analytical Solution ◽

Stokes Equations ◽

Approximate Analytical Solution ◽

Continuous Functions ◽

Navier Stokes ◽

Navier Stokes Equation ◽

Navier Stokes Equations ◽

Mesh Free ◽

Numeric Calculation ◽

Domain Discretization

Analytical solution of Navier-Stokes equations are extremely difficult and rare. It is one of the unsolved Clay Millennium problems in mathematics. Many solutions that exist are examples of degenerate cases where the nonlinearity is controlled. In this paper we explore the application of Bézier functions to solve the two-dimensional laminar fluid flow over a backward step. The Bézier functions provide a mesh free alternative to domain discretization methods that are currently used to solve such problems. The Navier-Stokes equation are handled directly without transformation and the setup is direct, simple, and involves minimizing the error in the residuals of the differential equations along with the error on the boundary conditions over the domain. The solutions for the velocity and pressure are available in polynomial form. They are single continuous functions over the entire domain. The procedure employs a combination of symbolic and numeric calculation in MATLAB. Two problems are explored. The first is the flow in a 2D channel to illustrate the technique. The second is the flow over the backward step. The solutions are compared to the corresponding finite element solutions from COMSOL Multiphysics software.

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An approximate analytical solution of fractional 2D Navier-Stokes equation using homotopy-perturbation method

10.1063/1.4932437 ◽

2015 ◽

Author(s):

N. Kumaresan ◽

Kuru Ratnavelu

Keyword(s):

Analytical Solution ◽

Perturbation Method ◽

Stokes Equation ◽

Homotopy Perturbation Method ◽

Approximate Analytical Solution ◽

Navier Stokes ◽

Navier Stokes Equation ◽

Homotopy Perturbation

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Modeling of Grouting Penetration in Porous Medium with Influence of Grain Distribution and Grout–Water Interaction

Processes ◽

10.3390/pr10010077 ◽

2021 ◽

Vol 10 (1) ◽

pp. 77

Author(s):

Mengmeng Zhou ◽

Fengshuai Fan ◽

Zhuo Zheng ◽

Chenyang Ma

Keyword(s):

Grain Size ◽

Porous Medium ◽

Fluid Flow ◽

Analytical Solution ◽

Stokes Equations ◽

Linear Trend ◽

Injection Pressure ◽

Navier Stokes ◽

Two Phase ◽

Grain Distribution

In this study, a numerical model of grouting penetration in a porous medium is established. The fluid flow in the interstices of the porous medium is directly modeled by Navier–Stokes equations. The grouting process is considered as a two-phase flow problem, and the level set method is used to characterize the interaction between grout and groundwater. The proposed model has considered the nuances for each grain during grouting penetration, instead of representing the fluid flow as a continuum process. In the simulation, three kinds of porosity (0.3; 0.4; 0.5) and two kinds of grain size distribution (0.5~1 mm; 1~2 mm) are used. Results show that: the pressure drop along penetration distance is approximately in a linear trend. The variation of filling degree along grouting distance approximately obeys a quadratic polynomial function. The injection pressure is influenced by the porosity and grain size of the porous medium, especially by the former. A theoretical analysis is carried out to propose an analytical solution of the grouting penetration. The analytical solution gives a good estimation when the grain amounts in the porous medium are small, and the difference becomes larger as the grain amounts increase.

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A computational fluid dynamics investigation of the flow behavior near a wellbore using three-dimensional Navier–Stokes equations

Advances in Mechanical Engineering ◽

10.1177/1687814019873250 ◽

2019 ◽

Vol 11 (9) ◽

pp. 168781401987325

Author(s):

Mohammad Jalal Ahammad ◽

Mohammad Azizur Rahman ◽

Jahrul Alam ◽

Stephen Butt

Keyword(s):

Fluid Dynamics ◽

Computational Fluid Dynamics ◽

Fluid Flow ◽

Analytical Solution ◽

Stokes Equations ◽

Three Dimensional ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Hydrocarbon Reservoir ◽

Wellbore Pressure

The analysis of fluid flow near the wellbore region of a hydrocarbon reservoir is a complex phenomenon. The pressure drop and flow rates change in the near wellbore with time, and the understanding of this system is important. Besides existing theoretical and experimental approaches, computational fluid dynamics studies can help understanding the nature of fluid flow from a reservoir into the wellbore. In this research, a near wellbore model using three-dimensional Navier–Stokes equations is presented for analyzing the flow around the wellbore. Pressure and velocity are coupled into a single system which is solved by an algebraic multigrid method for the optimal computational cost. The computational fluid dynamics model is verified against the analytical solution of the Darcy model for reservoir flow, as well as against the analytical solution of pressure diffusivity equation. The streamlines indicate that the flow is radially symmetric with respect to the vertical plane as expected. The present computational fluid dynamics investigation observes that the motion of reservoir fluid becomes nonlinear at the region of near wellbore. Moreover, this nonlinear behavior has an influence on the hydrocarbon recovery. The flow performance through wellbore is analyzed using the inflow performance relations curve for the steady-state and time-dependent solution. Finally, the investigation suggests that the Navier–Stokes equations along with a near-optimal solver provide an efficient computational fluid dynamics framework for analyzing fluid flow in a wellbore and its surrounding region.

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Numerical Approximations to Solution of Biochemical Reaction Model

International Journal of Chemical Reactor Engineering ◽

10.2202/1542-6580.2606 ◽

2011 ◽

Vol 9 (1) ◽

Author(s):

Ahmet Yildirim ◽

Ahmet Gökdogan ◽

Mehmet Merdan

Keyword(s):

Analytical Solution ◽

Approximate Analytical Solution ◽

Transformation Method ◽

Reaction Model ◽

Biochemical Reaction ◽

Graphical Form ◽

Kutta Method ◽

Transform Method ◽

Runge Kutta Method ◽

Differential Transform

In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.

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