Journal of Applied Mathematics

Volume 2019, Article ID 3084394, 19 pages

https://doi.org/10.1155/2019/3084394

## A New Approximate Analytical Solutions for Two- and Three-Dimensional Unsteady Viscous Incompressible Flows by Using the Kinetically Reduced Local Navier-Stokes Equations

Department of Mathematics, College of Education for Pure Science, University of Basrah, Basrah, Iraq

Correspondence should be addressed to Assma J. Harfash; moc.oohay@4791jamssa

Received 13 October 2018; Accepted 9 December 2018; Published 1 January 2019

Academic Editor: Oluwole D. Makinde

Copyright © 2019 Abdul-Sattar J. Al-Saif and Assma J. Harfash. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this work, the kinetically reduced local Navier-Stokes equations are applied to the simulation of two- and three-dimensional unsteady viscous incompressible flow problems. The reduced differential transform method is used to find the new approximate analytical solutions of these flow problems. The new technique has been tested by using four selected multidimensional unsteady flow problems: two- and three-dimensional Taylor decaying vortices flow, Kovasznay flow, and three-dimensional Beltrami flow. The convergence analysis was discussed for this approach. The numerical results obtained by this approach are compared with other results that are available in previous works. Our results show that this method is efficient to provide new approximate analytic solutions. Moreover, we found that it has highly precise solutions with good convergence, less time consuming, being easily implemented for high Reynolds numbers, and low Mach numbers.

#### 1. Introduction

Many of the physical phenomena in fluid mechanics are formulated according to the unsteady viscous incompressible Navier-Stokes (INS) equations, which has the non-dimensional formula consisting of the momentum equations and the continuity equation [1–8]where is the physical time, is the velocity field, is the pressure, and is the Reynolds number (, where is the scale velocity field, is the characteristic length, and is the kinematic viscosity of the fluid).

Analytical and numerical solutions of INS equations are known difficulty because they are non-linear equations, and they do not find the time evolution equation for the pressure that must be determined by solving the Passion equation at each time step, which requires effort and time. Therefore, there are a lot of studies that have developed an alternative formula description of incompressible fluid flows. One of these alternative formulas is the kinetically reduced local Navier-Stokes (KRLNS) equations which was suggested in [1] for the thermodynamic description of incompressible fluid flows at low Mach numbers. The system of KRLNS equations issuch thatwhere is the Mach number ( is the ratio of the characteristic flow speed to the isentropic sound speed ), and is the grand potential. The time scale in INS equations related to that of KRLNS equations; .

All studies which have presented the KRLNS equations for simulation of unsteady incompressible viscous flow problems, used the numerical schemes for solving these equations. The KRLNS equations are proposed for the simulation of low Mach number flows in [2], and used the spectral element method to find the numerical solution of the three-dimensional Taylor Green vortex flow. In [3], two-dimensional KRLNS system is simplified and compared with a Chorin’s artificial compressibility method for steady state computation of the flow in a lid-driven cavity at various Reynolds numbers, the Taylor Green vortex flow is demonstrated that the KRLNS equations correctly describe the time evolution of the velocity and of the pressure, for this purpose, the explicit Mac Cormack scheme is used. In [5] the KRLNS equations are applied to two-dimensional simulation of doubly periodic shear layers and decaying homogeneous isotropic turbulence, to solve these equations have been used the central difference scheme for the spatial discretization in both advection and diffusion terms and four stages Runge-Kutta method for the time integration, the numerical results are compared with those obtained by the artificial compressibility method, the lattice Boltzmann method, and the pseudospectral method. Higher order difference approximations are used in [6] to find the solutions of the KRLNS equations which are applied for two-dimensional simulations of Womersley problem and doubly periodic shear layers.

The main purpose of this paper is to find new approximate analytical solutions for two- and three-dimensional unsteady viscous incompressible flow problems. To achieve this objective, the flow problems that are described by alternative formulas of Navier-stokes equations, which are named KRLNS (3) and (4), the reduced differential transform method (RDTM) is proposed. The reasons that encourage us to propose RDTM to solve the present problems are being an effective and efficient method to find approximate analytical solutions for nonlinear equations and we believe that it has been achieved for the first time in its study. Moreover, we extend the application of RDTM and compare its reliability and efficiency with other methods. New approximate analytical solutions for two- and three-dimensional unsteady viscous incompressible flows were found using RDTM. The results that we obtained are better than others, refer to the results in [4, 8] in accuracy, convergence, and CPU time.

The structure of this paper is organized as follows: In Section 2, we begin with some basic definitions and the use of the RDTM on the KRLNS equations. Section 3 explains the manner we adopted to discuss the convergence of the solutions. In Section 4, we apply this method to solve four flow problems of different dimensions in order to show its ability and efficiency in finding new approximate solutions. Section 5 introduces conclusions of the present work.

#### 2. Reduced Differential Transform Method

The RDTM is an iterative procedure for obtaining a Taylor series solution of differential equations. This method is similar to the differential transform method which was first introduced by Zhou [9]. RDTM has been successfully used to many nonlinear problems [10–19] since it does not require any parameter, discretization, linearization, or small perturbations; thus it reduces the size of computational work and is easily applicable.

The main idea of this method depends on the representation the function of two variables as a product of single-variable function, i.e., , then the function can be represented as

*Definition 1. *If function is analytic and differentiated continuously with respect to time and space , thenis called t-dimensional spectrum function of , and is the transformed this function.

*Definition 2. *The reduced differential inverse transform of is defined as

Then, the inverse transformation of the set of gives the terms approximation solution as follows:and the exact solution is

To show some basic properties of dimensional RDTM [18], we have to consider to be a vector of variables, and function is analytic and continuously differentiable with respect to time and space in the domain of interest, then the fundamental mathematical operations performed by RDTM are readily obtained and listed in Table 1.