Numerical solution of the three-dimensional Burger’s equation by using the DQ-FD combined method in the determination of the 3D velocity of the flow

In this paper, the differential quadrature and the finite difference combined method (DQ-FDM) was applied to solve the three-dimensional Burger’s equation in the determination of the 3D velocity of the flow; so that spatial terms were discretized by the differential quadrature method, and the temporal term was discretized by the finite difference method, and the resulting nonlinear equations were solved using the Newton–Raphson method. All variables were considered as dimensionless in this equation. The solution results were compared with solution results of the two-dimensional equation in the two other numerical methods available in the literature which provided an acceptable accuracy. Also, the results of the mentioned numerical method were compared with those of the fully implicit finite difference method that was solved for larger than or equal viscosities of 0.1. The results showed that by increasing time and viscosity, the longitudinal, depth and transverse velocities were decreased. The occurrence of the upward flow was observed especially in the υ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upsilon$$\end{document} = 0.05 in the close of the bed, end of the length and width that in the presence of very fine particles of the clay and silt shows suspension of these particles in some spaces. The position of the longitudinal, depth and transverse velocities in the plan for the passing plates through the section depth for different viscosities and times showed that by increasing viscosity and time, the position of the maximum velocities became closer to the middle of the section width. Also, stream lines were plotted in all of sections and then analyzed.


Introduction
One of the most important aims in the fluid dynamics is the prediction of physical values including pressure, velocity and temperature of the flow. The physical processes are dependent on different parameters which model in the mathematical language with its equation. Since solving of some nonlinear partial differential equations is so difficult that is impossible to obtain their analytical solution except in specific conditions, such equations can be solved by numerical methods. Numerical methods are one of the lower-cost methods for solving engineering problems which occupy a very important part of the scientific fields, and its advantages and superiority have been proven as compared to the other methods with spending slight time and the solving of various geometries. The Burger's equation is same the Navier-Stokes incompressible equation without the pressure gradient term and continuity equation which have convection and diffusion terms of the Navier-Stokes incompressible equations. This equation was first introduced by Bateman (1915). Then, Burgers (1948) proposed this equation to describe turbulent flow in a channel. For this reason, it was known as "Burger's equation." In recent years, many researchers have solved numerically Burger equations in different ways. The finite element numerical method was described by Arminjon and Beauchamp (1979) for solving the two-dimensional Burger equations. Mittal and Singhal (1993) solved the one-dimensional Burger equation using the technique of finitely reproducing nonlinearities and received good accuracy by comparing their results with those of the finite element method. The one-and two-dimensional Burger's equation were solved using the methods of distributed approximation functionals (DAFs) and the Taylor expansion by Wei et al. (1998). Kutluay et al. (1999) solved the one-dimensional Burger equation using the explicit finite difference method and compared it with the exact solution results. They found that their results is close to the exact solution. The numerical solution of the Burger's equation as a model of turbulence and shock wave using the finite element method was proposed by Öziş et al. (2003). They compared their results with those of Cole's analytical solution and received good accuracy. The quasi-linear parabolic Burger equation was solved using the variational method by Aksan and Özdeş (2004), and the results for different viscosities were compared with the results of the exact solution. Hassanien et al. (2005) proposed a fourth-order finite difference numerical method as the unconditional stable method for solving one-dimensional Burger's equation. Aksan (2006) solved this equation by converting the Burger equation to a set of ordinary differential equations by using the quadratic B-spline finite element method. The collocation method using quartic B-splines to solve the Burger equation was proposed by Saka and Dağ (2007). Khater et al. (2008) used Chebyshev's collocation method to solve some different kinds of the Burger's equation; they presented numerical solution results for each kind of equation and showed high accuracy of their results as compared to other previous results. Solving a kind of the generalized nonlinear Benjamin-Bona-Mahony-Burger's equation (BBMB) using the Exp-function method was proposed by Ganji et al. (2009). Rady et al. (2010) introduced a kind of Burger equation called the Boussinesq-Burgers and proposed the solution of this equation by the generalized tan h method. Tamsir and Srivastava (2011) applied the semi-implicit finite difference method to solve the twodimensional Burger's equation. They found that this method is useful by comparing of the results with those of exact solution and other previous numerical. Another kind of Burger equation called the Burgers-Fisher equation was solved using Chebyshev-Legendre pseudo-spectral method by Zhao et al. (2012); they presented their numerical results for three different boundary conditions sets. Inan and bahadir (2013) presented the solution of the one-dimensional Burger's equation by two methods of implicit and fully implicit exponential finite difference. They found that both the methods are precise and reliable by comparing of their results with exact solution. Numerical solution of the generalized two-and three-dimensional Burger's equation using the Hopf-Cole transformation method was presented by Wazwaz (2014). Zhanlav et al. (2015) used a high-order finite difference scheme to solve the unsteady Burger equation; they found that their results are close to the exact solution results for different Reynolds numbers. Singh and Kumar (2016) solved the nonlinear Burger's equation by using the modified extended cubic B-spline differential quadrature method (MECBDQ). A novel finite difference scheme on the infinite domain with two nonlinear boundary conditions and linear initial condition was applied to solve the one-dimensional nonhomogeneous Burger's equation by Zheng et al. (2017). Lakshmi and Awasthi (2018) used a combination of quintic splines and Crank Nicolson schemes for spatial and temporal discretization, respectively, to solve the nonlinear modified Burger's equation. To solve the BBMB equation, a method based on the lumped Galerkin technique using cubic B-spline finite elements was used for spatial approximation by Karkoc and Bhowmik (2019). Also, Parand and Nikarya (2019) used the generalized Bessel functions (GBF) and the collocation method and the Jacobian free Newton-Krylov sub-space (JFNK) to solve the nonlinear time-fractional Burger equation. They also show the accuracy of the mentioned method by solving some examples of time-fractional Burger equation and compare their results with others. To solve the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation, the quintic Hermite collocation method (QHCM) with weighted finite difference scheme was used by Arora et al. (2020); also the results obtained are found to be in good agreement with the exact solutions. Also, Ömer Oruç (2020) solved the modified Burgers equation (MBE) using two meshless method based on the delta-shaped basis function pseudo-spectral method (DBF-PSM) and the barycentric rational interpolation method (BRIM); also they obtained accurate results using fewer collocation points by comparing with other studies available in the literature such as finite element. A class of the time-fractional generalized Burger's equation using the weak Galerkin (WG) finite element method was studied by Wang et al. (2021). They proved the stability of proposed method by comparing their method with another numerical solutions. Also, Vaghefi et al. (2021) solved the two-dimensional Burger's equation using the incremental differential quadrature method (IDQM) and investigated effect of the time and viscosity parameters on longitudinal and depth velocity for passing axis from the middle of the length and depth. In this paper, the ability of the differential quadrature method (DQM) is used to solve the three-dimensional Burger's equation. Because of the high potential of the mentioned method, this method can be a good substitute for finite difference (FDM) and finite element (FEM) methods. This method approximates differential equations governing the problem as sum of the function values at specified points, which is a high-order method. This method were first introduced by Bellman and Casti (1971) and Bellman et al. (1972) in the early 1970s which was quickly developed after the 1980s. Bert and Malik (1996) presented a comprehensive review of the differential quadrature method and its application. Quan and Chang (1989) developed the approximation the weighted coefficients. Shu (2000) developed application of this method in engineering. In the present study, it should be noted that spatial terms of the three-dimensional Burger's equation are discretized using the differential quadrature method, while the temporal term of this equation is discretized using the finite difference method due to enlargement of the matrix dimensions derived from the Newton-Raphson method. Therefore, the method used in this paper is named "DQ-FDM." The three-dimensional Burger's equation that is considered in this research is as follows: Equation (1) consists of three functions of the longitudinal velocity (u), depth velocity (v) and transverse velocity (w) which there is their first and second derivatives relative to space (x, y, z) and their first derivative relative to time (t) in the three dimensions and is the dimensionless kinematics viscosity. All the parameters in Eq. (1) are considered "dimensionless." Figure 1 shows a schematic of the studied range and the triple directions in velocity calculation. The points shown in this figure are the reagent points which are used as a sample for comparison. (1) Initial conditions: Boundary conditions: In this research, two numerical methods of the differential quadrature and the finite difference have been used in combination to discretize the spatial and temporal terms of Burger's equation in three dimensions' mode. Also, showing the stream lines in different directions is one of the interesting and important points in this study.

Materials and methods
In this paper, a combination of the differential quadrature (DQ) and the finite difference (FD) methods was used to solve the three-dimensional Burger's equation. In this method, spatial terms were discretized by the differential quadrature method, and the temporal term was discretized by the finite difference method. In order to solve nonlinear Eqs. (4), the Newton-Raphson method was used.
In Eq. (4), Nx , Ny and Nz are the number of discrete points along the length, depth and width, respectively. i, j and k are the spatial node coordinates along the x, y and z, respectively, and n is the time node coordinates. Other parameters already are introduced in the introduction. (4)

Verification
Since in the three-dimensional conditions and by other numerical methods like the finite element, data required were not found for verification of the current research; in order to perform verification of the DQ-FD method for the two-dimensional Burger's equation, the results obtained by Arminjon and Beauchamp (1979) and Wei et al. (1998) were used which solved the mentioned equation by the finite element method (FEM) and the method of distributed approximating functional (DAFM), respectively, with consideration of number of the different points. The verification is presented in a series of the available points and for t = 0.01 and = 1 in Table 1 for both the depth and longitudinal velocity values. Figure 2A shows the distribution of the longitudinal velocity along the length and for passing plate through the middle of the width and = 0.05. In y = 0.5, values of the longitudinal velocity are more than y = 0.2, 0.8. For example at t = 0.25, the maximum longitudinal velocity is about 88% more than its value in y = 0.2. As can be seen, by increasing time, the value of the longitudinal velocity was decreased, and the position of the maximum longitudinal velocity was transmitted into downstream. For example at t = 0.1 and in y = 0.2, 0.5 and 0.8, the maximum longitudinal velocity was decreased about 30, 20 and 20 percent, respectively, as compared to its value at t = 0, and their position was transmitted to downstream about 20, 12 and 8 percent, respectively, from the middle of the section length.

Numerical results
In Fig. 2b and y = 0.2, the existence of position of the negative depth velocity (upward flow) was seen from distance of 70% of the beginning of the length up to its end to  For example, at t = 0.25, the maximum depth velocity was decreased about 80% as compared to its initial value at t = 0, and its position was transmitted downward about 8% as compared to its position in the initial time. In y = 0.5 and t = 0.05 and 0.1, the maximum depth velocity was increased about 50 and 45 percent, respectively, as compared to its value in t = 0 and then was decreased. For example, in t = 0.25, the maximum depth velocity was decreased about 45% as compared to t = 0.1. In y = 0.8, the process of the depth velocity changes differs from the other two depths studied; so that there was upward flow up to t = 0.1 and up to about 70% of the beginning of the flow length, and after that, values of the velocity was slight. But from t = 0.25 to the end of the dimensionless time, only downward flow of particles was observed. In Fig. 2c and t = 0, firstly, the transverse velocity is increased to reach to peak and after that began to decrease, and about 35% of beginning of the length reaches to zero and increased again and then reaches to maximum transverse velocity which is about 60% of beginning of the length; then, it was decreased and after reaching zero, becomes negative. It means that its direction was inclined to the left bank. This process continues up to t = 0.1. But from t = 0.25 onward, the process of the transverse velocity changes is the same as the longitudinal velocity. As can be seen, values of the maximum transverse velocity had severe decreasing in all times as compared to the initial time. For example, in y = 0.5 and t = 0.05, the maximum transverse velocity was decreased about 65% as compared to its value at t = 0. In y = 0.8, meanwhile that values of the transverse velocity at t = 0 are about zero, had small values in the other times.
Distribution of the longitudinal velocity along the depth is shown in Fig. 3a for passing plate through the middle of the length and = 0.05. By increasing time, the value of the longitudinal velocity was decreased, and the position of the maximum longitudinal velocity was moved into more depths which means that it became closer to the bottom of the bed. For example at t = 0.25 and z = 0.2, 0.5 and 0.8, the maximum longitudinal velocity was decreased about 70, 55 and 20 percent, respectively, as compared to its value at t = 0 and its position became closer about 20, 25 and 10 percent, respectively, to bottom of the bed from the middle of the depth at the initial time.
In Fig. 3b and z = 0.2, by increasing time, the value of the depth velocity was decreased and the position of the maximum depth velocity was moved into more depths, and at t = 1 was reached to 30% of the bottom of the bed. For example in t = 0.25, the maximum depth velocity was decreased about 60%, and its position was reached from about 30% of the beginning of the depth to its 65%. In z = 0.5 and t = 0.05 and 0.1, the maximum depth velocity was increased about 10 and 15 percent, respectively, and after that decreased. Also, by increasing time, the position of the maximum depth velocity was inclined toward the bottom of the bed. For example at t = 0.5, the maximum depth velocity was decreased about 50% as compared to its value in t = 0.1 and its position became closer to the bottom of the bed about 25%. In Fig. 3c and in z = 0.2 and 0.5 by increasing time, the value of the transverse velocity was decreased which the value of decreasing is more severe in z = 0.5. For example at t = 0.05 and widths of 0.2 and 0.5, the maximum transverse velocity was decreased about 35 and 70 percent, respectively, as compared to its value at t = 0. In z = 0.8, since the transverse velocity is zero in the initial time, the maximum transverse velocity was decreased by progressing time. For example at t = 0.5, the maximum transverse velocity was decreased about 30% as compared to its value in t = 0.05. Figure 4a expresses distribution of the longitudinal velocity along the width and for passing plate through the middle of the depth and = 0.05. As can be seen, in x = 0.2 and 0.5, the maximum longitudinal velocity was decreased by progressing time after initial time, and its position was transmitted to more widths, that is, it was inclined into the right bank. For example at t = 0.25 and x = 0.2 and 0.5, the maximum longitudinal velocity was decreased about 60 and 50 percent, respectively, as compared to its value in t = 0 and its position became closer to the right bank about 25 and 20 percent, respectively, from the middle of the width in the initial time. In x = 0.8 and t = 0.05 and 0.1, the maximum longitudinal velocity was increased about 5 and 10 percent, respectively, as compared to t = 0 and after that decreased. For example at t = 1, the maximum longitudinal velocity was decreased about 85% as compared to its value at t = 0.1.
In Fig. 4b and x = 0.2, 0.5 and t = 0.05, the maximum depth velocity was increased about 40 and 75 percent, respectively, as compared to its value in t = 0 and after that decreased. For example, in x = 0.2 and 0.5 and t = 0.25, the value of the maximum depth velocity was decreased about 70 and 60 percent, respectively, as compared to its value in t = 0.05. In x = 0.8 up to t = 0.1 and up to about 65% of the beginning of the length, there was upward flow which after that up to end of the length, there was downward flow with very small values.
In Fig. 4c and x = 0.2 and 0.5 in effect of progressing time, the value of the transverse velocity was decreased and the position of the maximum transverse velocity was transmitted into the right bank. For example, in x = 0.2 and 0.5 and t = 0.1, the value of the maximum transverse velocity was decreased about 60 and 70 percent, respectively, as compared to its value in initial time, and its position was moved into the right bank about 15 and 10 percent, respectively, as compared to t = 0. Figure 5a shows the comparison of DQ-FD and FD methods for distribution of the longitudinal velocity along the length for passing plate through the middle of section width at t = 0.05 and = 0.1. In y = 0.2, longitudinal velocity obtained by DQ-FD and FD methods has a slight difference; so that the maximum longitudinal velocity obtained from the finite difference method is about 3% lesser than the differential quadrature method. In the passing plate through y = 0.5, the longitudinal velocities obtained by the two methods have a slight difference, which is not easily visible in the corresponding diagram. For example, the difference of the maximum longitudinal velocity in the two methods is about 0.4% which is so slight. In the passing plate through y = 0.8, the longitudinal velocity obtained by the finite difference method presented lower values than the differential quadrature method. The maximum longitudinal velocity obtained from the two methods has a difference about 3%. Comparison of Figs. 5a and 2a expresses that with an increase in viscosity, the value of the longitudinal velocity is decreased. For example, in y = 0.5 by twofold increasing in the viscosity (from = 0.05 to = 0.1), the maximum longitudinal velocity was decreased about 8%. As shown in Fig. 5b, values of the longitudinal velocity by FDM are lesser than DQ-FDM for the passing plate through y = 0.2 and 0.5. For example, the maximum depth velocities obtained from the two methods for y = 0.2 and 0.5 have a difference about 3 and 1 percent, respectively. In the passing plate through y = 0.8, the negative depth velocity which its position is up to about 70% of the beginning of the length, in the FDM, has more values as compared to DQ-FDM. But from 70% of the beginning of the length to end of the section length, the positive depth velocity in the two methods has a slight difference. For example, maximum negative and positive depth velocities have a difference about 9 and 3 percent, respectively. By comparing Figs. 5b and 2b, one can find that with increasing viscosity, less upward flow occurred, and the value of the depth velocity is decreased along the length. For example, in y = 0.2, the maximum longitudinal velocity is decreased about 13% in the viscosity of 0.1 compared with its value in the viscosity of 0.05. In Fig. 5c and y = 0.2 and 0.5, values of the transverse velocity obtained from the FDM are greater than DQ-FDM; but for y = 0.8, values obtained from the FDM are smaller than values obtained from DQ-FDM with a nearly high difference. For example, the maximum longitudinal velocity obtained from two methods for passing plates through y = 0.2, 0.5 and 0.8 has a difference about 15, 8 and 10 percent, respectively. Comparison of Figs. 5c and 2c expresses that by increasing viscosity, the value of the transverse velocity is decreased. For example, in y = 0.8, by twofold increasing viscosity (from = 0.05 to = 0.1), the maximum transverse velocity was decreased about 55%. Figure 6a shows the comparison of the DQ-FD and FD methods for distribution of the longitudinal velocity along the depth for passing plate through the middle of the length at t = 0.05 and = 0.1. As can be seen, for passing plate through z = 0.5, the difference of longitudinal velocities derived from DQ-FDM and FDM is slight and about 0.02%. The maximum longitudinal velocities derived from two methods for passing plates through z = 0.2 and 0.8 have a difference about 3 and 2 percent, respectively. Comparison of Figs. 6a and 3a shows a decrease in the longitudinal velocity due to the increase in viscosity. For example, in z = 0.2, 0.5 and 0.8 by twofold increasing viscosity, the maximum longitudinal velocity was decreased about 10, 8 and 6 percent, respectively.
In Fig. 6b, the difference of the depth velocity derived from two methods of DQ-FDM and FDM was slight for passing plates through z = 0.2 and 0.5, but for passing plate through z = 0.8, it has more difference compared with two other widths. For example, the maximum depth velocity for passing plates through z = 0.2, 0.5 and 0.8 has a difference about 1, 0.02 and 2 percent, respectively. Decreasing the value of the depth velocity is seen by comparison of Figs. 6b and 3b. For example, in z = 0.2, 0.5 and 0.8, the maximum depth velocity at = 0.1 was decreased about 20, 10 and 50 percent, respectively, compared with its values at = 0.05.
As shown in Fig. 6c, in z = 0.2 and 0.5, values of the transverse velocity derived from FDM are greater than DQ-FDM; but in z = 0.8, the FDM provides lower values compared with DQ-FDM. The maximum transverse velocity obtained from two methods has a difference about 2, 5 and 10 percent, respectively, for passing plates through z = 0.2, 0.5 and 0.8. Comparison of Figs. 6c and 3c expresses decrease in the transverse velocity by increasing viscosity. For example, in z = 0.2, by twofold increasing viscosity, the maximum transverse velocity was decreased about 40%. Figure 7a shows the comparison of the DQ-FD and FD methods for distribution of the longitudinal velocity along the width for passing plates through the middle of the depth in t = 0.05 and = 0.1. As can be seen, for passing plates through x = 0.2 and 0.8, the longitudinal velocity derived from FDM provides lesser values than DQ-FDM. For example, the maximum longitudinal velocity obtained from two methods for passing plates from x = 0.2 and 0.8 has a difference about 2%. For passing plates through the middle of the length, a slight difference is seen between two numerical methods; so that the values obtained from finite difference method are greater than the differential quadrature method. Comparison of Figs. 7a and 4a shows that by increasing viscosity, the longitudinal velocity is decreased. For example, in x = 0.2, 0.5 and 0.8, the maximum longitudinal velocity at = 0.1 is decreased about 7, 7 and 10 percent, respectively, as compared to its values at = 0.05. As shown in Fig. 7b, for passing plates through x = 0.2 and 0.5, values of the depth velocity obtained from FDM are smaller than DQ-FDM, and difference value is very low. For example, the maximum depth velocity derived from two methods for x = 0.2 and 0.5 has a difference about 2 and 1 percent, respectively. In the passing plates through x = 0.8, the negative depth velocity which its position is up to about 70% beginning of the depth, in the FDM, more values were seen compared with DQ-FDM. But from 70% of flow surface to the bottom of the bed, the positive depth velocity has a slight difference in two methods. For example, the maximum negative and positive depth velocities obtained from two methods have a difference about 5 and 2 percent, respectively. By comparison of Figs. 7b and 4b, one can find that by increasing viscosity, the value of the depth velocity is decreased. For example, in x = 0.2, 0.5 and 0.8 by twofold increasing viscosity, the maximum depth velocity is decreased about 30, 25 and 50 percent, respectively.
In Fig. 7c and x = 0.2 and 0.5, values of the transverse velocity derived from FDM are greater than DQ-FDM. But in x = 0.8, FDM provides lower values than DQ-FDM. The maximum transverse velocity derived from two methods for passing plates in x = 0.2, 0.5 and 0.8 has a difference about 4, 5 and 15 percent, respectively. Decreasing the transverse velocity with increasing viscosity is seen by comparison of Figs. 7c and 4c. For example, in x = 0.8 by twofold increasing viscosity, the maximum transverse velocity is decreased about 15%. Figure 8a shows comparison of the DQ-FD and FD methods for distribution of the longitudinal velocity along the length for passing plates through the middle of the width in t = 0.05 and = 0.25. As can be seen, the difference of values of the longitudinal velocity obtained from FDM and DQ-FDM is low for all three depths that this difference is the lowest in the middle of the depth (y = 0.5) and not easily visible. For example, the maximum longitudinal velocity derived from two methods has a difference about 3, 1 and 3 percent, respectively, for passing plates through y = 0.2, 0.5 and 0.8. Comparison of Figs. 8a and 5a expresses that by increasing viscosity, the longitudinal velocity is decreased along the length. For example, in the passing plate through y = 0.2, the maximum longitudinal velocity at = 0.25 was decreased about 15% as compared to its value at = 0.1.
In Fig. 8b, there is a slight difference between FD and DQ-FD methods for passing plates through all three depths. For example, the maximum depth velocity obtained from two methods has a difference about 2, 0.2 and 0.5 percent, respectively, for passing plates through y = 0.2, 0.5 and 0.8. By comparison of Figs. 8b and 5b, one can find that by increasing viscosity, the value of the depth velocity is decreased. For example, in y = 0.5, by 2.5-fold increasing Figs. 6a and 3a shows decreasing the longitudinal velocity along the depth by increasing viscosity. For example, in z = 0.2, by 5-and 2.5-fold increasing viscosity, the value of the maximum longitudinal velocity was decreased about 30 and 20 percent, respectively. In Fig. 9b, the difference of the depth velocity derived from DQ-FD and FD methods is more than two other study lengths in passing plate through z = 0.8. For example, the maximum depth velocity obtained from two methods has a difference about 3, 0.2 and 20 percent, respectively, for passing plates through z = 0.2, 0.5 and 0.8. Comparison of Fig. 9b with Figs. 6b and 3b expresses decreasing depth velocity along the depth by increasing viscosity. For example, in z = 0.5, by 5-and 2.5-fold increasing viscosity, the maximum depth velocity was decreased about 55 and 40 percent, respectively.
In Fig. 9c, the difference of the transverse velocity derived from DQ-FD and FD methods is about 1, 0.5 and 10 percent, respectively, in passing plates through z = 0.2, 0.5 and 0.8. By comparison of Figs. 9c and 6c, one can find that by increasing viscosity, the transverse velocity is decreased along the depth. For example, in z = 0.8, by 2.5fold increasing viscosity, the maximum transverse velocity was decreased about 25%. Figure 10a shows comparison of the DQ-FD and FD methods for distribution of the longitudinal velocity along the width for passing plates through the middle of the depth in t = 0.05 and = 0.25. As can be seen, there is a slight difference in all passing plates through three study lengths between the longitudinal velocities obtained from FDM and DQ-FDM. For example, the maximum longitudinal velocity derived from two methods has a difference about 3, 1 and 2 percent, respectively, for passing plates through x = 0.2, 0.5 and 0.8. Comparison of Fig. 10a with Figs. 7a and 4a shows decrease in the longitudinal velocity with an increase in viscosity. For example, in x = 0.5, by 5-and 2.5-fold increasing viscosity, the maximum longitudinal velocity was decreased about 30 and 20 percent, respectively.
As can be seen in Fig. 10b, the depth velocity obtained from FDM shows lower values as compared to DQ-FDM. For example, the maximum depth velocity derived from two methods has a difference about 2, 0.5 and 5 percent, respectively, for passing plates through x = 0.2, 0.5 and 0.8. By comparing Fig. 10b with Figs. 7b and 4b, one can find that depth velocity is decreased by increasing viscosity. For example, in x = 0.2, by 5-and 2.5-fold increasing viscosity, the value of the maximum depth velocity was decreased about 55 and 80 percent, respectively.
As shown in Fig. 10c, the FDM shows lower values as compared to the DQ-FDM for the transverse velocity along the width. For example, the maximum transverse velocity derived from two methods has a difference about 0.1, 0.5 and 10 percent, respectively, for passing plates through x = 0.2, 0.5 and 0.8. Comparison of Fig. 10c with Fig. 7c expresses decrease in the transverse velocity by increasing the viscosity. For example, in x = 0.2, by 2.5-fold increasing viscosity, the maximum transverse velocity was decreased about 40%. Table 2 shows the longitudinal, depth and transverse velocities for nine reagent points and different times for = 0.05. As can be seen, the values of the velocity are decreased by increasing the time. Of course, in the central point (x = y = z = 0.5) at first, the depth velocity is increased in t = 0.05 and t = 0.1 as compared to t = 0 (about 80 and 110 percent, respectively) and then begins to decrease. After As shown in Table 3, in both of two methods, decreasing the longitudinal, depth and transverse velocities by increasing time occurred in = 0.1. For example, in central point (x = y = z = 0.5), by tenfold increasing time (from t = 0.05 to t = 0.5), longitudinal, depth and transverse velocities were decreased about 80, 74 and 50 percent, respectively, in the  DQ-FD method and about 80, 75 and 55 percent, respectively, in the FD method. This table shows that there is a low difference between the two mentioned methods at = 0.1. For example, the longitudinal, depth and transverse velocities obtained from two methods have a difference about 10, 12 and 15 percent, respectively, in the point 2 (x = y = z = 0.1) and t = 0.25. Table 4 shows values of the longitudinal, depth and transverse velocities in = 0.25 by using two numerical methods. Comparison of this table and Table 3 is seen that by increasing viscosity, the value of the longitudinal, depth and transverse velocity is decreased. For example, at t = 0.05 by 2.5-fold increasing viscosity, (from = 0.1 to = 0.25) in the point 9 (x = y = z = 0.9), longitudinal, depth and transverse velocities are decreased about 20, 85 and 15 percent, respectively, in the DQ-FD method and about 20, 85 and 30 percent, respectively, in the FD method. Also, from this table, one can find that the longitudinal, depth and transverse velocities are decreased by increasing time. For example, in the point 1 (x = y = 0.1, z = 0.9), by fivefold increasing  Figure 11a shows the position of the maximum longitudinal velocity in the plan for t = 0.05 and = 0.05. As can be seen, the range of the maximum longitudinal velocity for three considered depths is between 45 to 75 percent of the left bank. In this state, the position of the maximum longitudinal velocity for passing plate through y = 0.2 was transmitted to the right bank as compared to y = 0.5, 0.8. For example, in x = y = 0.5, the position of the maximum longitudinal velocity from about 70% of the left bank in y = 0.2 was reached to about its 55%. Generally, by approaching to the end of the longitudinal section, the position of the maximum longitudinal velocity became closer to the left bank. For example, in x = 0.9 and y = 0.2, the position of the maximum longitudinal velocity from about 60% of the left bank in x = 0.65 was reached to about its 45%. In Fig. 11b and in y = 0.2 and 0.5, the position of the maximum depth velocity in 20% of the end of the longitudinal section is related to the upward flow. But in y = 0.8, the position of the maximum depth velocity up to 80% of the beginning of the length is related to negative depth velocity and after that is related to positive depth velocity. In y = 0.2 and 0.5, the position of the maximum depth velocity in x = 0.65 is closer to the right bank than its upstream and downstream. For example, for passing plates through y = 0.2 and 0.5 and x = 0.65, the position of the maximum depth velocity from about 30% of the left bank in x = 0.5 was reached to about its 40%.
In Fig. 11c and in passing plate through each three depths, the position of the maximum transverse velocity on 10% of the end of the longitudinal section is relative to negative transverse velocity. This position occurred in the range of 50-70 percent of the left bank. As can be seen, in x = 0.65 and 0.8 for passing plates through y = 0.2, 0.5 As shown in Fig. 11d, the position of the maximum longitudinal velocity is located in the range of 45-70 percent of the left bank. In this figure, the position of the maximum longitudinal velocity is progressed into the left bank by increasing depth. For example, the position of the maximum longitudinal velocity in y = 0.5 and x = 0.2, from about 70% of the left bank in y = 0.2, was reached about its 60%. Generally, in each depth by moving into downstream, the position of the maximum longitudinal velocity was transmitted into the left bank. For example, in y = 0.8 and x = 0.6, the position of the maximum longitudinal velocity from the middle of the width was reached to about 55% of the left bank in x = 0.9.
In Fig. 11e, the position of the maximum depth velocity in the plan occurs in the range of 25 to 40 percent of the left bank. As can be seen, the position of the maximum depth velocity in x = 0.65, as compared to its upstream and downstream, is closer to the right bank. For example, in y = 0.2 and x = 0.65, the position of the maximum depth velocity was moved into the right bank about 5% as compared to its place in x = 0.5. In 20% of the end of the length, the position of the maximum depth velocity is related to negative depth velocity which occurred in about 30% of the left bank.
In Fig. 11f, the position of the maximum transverse velocity in the plan in the range of 50 to 75 percent of the left bank. In passing plates through y = 0.2 and 0.5, the position of the positive maximum transverse velocities is same and is located in about 70% of the left bank.
As shown in Fig. 11g, in the passing plates through y = 0.2, the position of the maximum longitudinal velocity up to the end of the length is fixed (about 60% of the left bank) and after that, moved toward lower widths and from 80% of beginning to end of the length is fixed and it is in the middle of the width. By comparing of Fig. 11g and d, one can find that by increasing viscosity, the position of the maximum longitudinal velocity became closer to z = 0.5. For example, in y = 0.5 and x = 0.2 for = 0.25, the position of the maximum longitudinal velocity from about 70% of the left bank in = 0.1 was reached to about its 60%.
In Fig. 11h for passing plates through y = 0.2 and 0.5, the position of the maximum depth velocity in 20% of end of the length and for y = 0.8 in 70% of the beginning of the length is related to negative depth velocity.
According to Fig. 11i, the position of the maximum transverse velocity in the plan is in range of 55 to 65 percent of the left bank. Comparison of these figures and Fig. 11e expresses that by increasing viscosity, the position of the maximum transverse velocity was transmitted into the left bank. For example, in x = y = 0.5, the position of the maximum transverse velocity from about 70% of the left bank at = 0.1 was reached to about its 60%. In y = 0.5 and 0.8, the position of the maximum transverse velocity occurred in about 60% of the left bank for all lengths. Figure 12a expresses the position of the maximum longitudinal velocity in the plan for t = 0.05 and = 0.05 which is in range of 60 to 80 percent of the left bank. In y = 0.2, 0.5 by moving into downstream, the position of the maximum longitudinal velocity transmitted into the left bank. For example, in y = 0.5 and x = 0.8, the position of the maximum longitudinal velocity from about 70% of the left bank in x = 0.65 was reached to about its 65%. Also, by increasing the depth, the position of the maximum longitudinal velocity was moved into the left bank. For example, in x = y = 0.5, the position of the maximum longitudinal velocity from about 75% of the left bank in y = 0.5 was reached to about its 70%.
According to Fig. 12b, the maximum depth velocity was transmitted into the right bank by moving into downstream. For example, in y = 0.5 and x = 0.8, the position of the maximum depth velocity from about 55% of the left bank in x = 0.65 was reached to about 60% of the left bank.
In Fig. 12c, the position of the maximum transverse velocity in the plan is in the range of 55 to 75 percent of the left bank. In this state, by increasing the depth, the maximum transverse velocity was moved into the left bank. For example, in x = 0.65 and y = 0.8, the position of the maximum transverse velocity from about 70% of the left bank in y = 0.5 was reached to about its 60%.
As can be seen in Fig. 12d, the maximum longitudinal velocity is in range of the 50 to 65 percent of the left bank. In this state, for passing plate through y = 0.5, the maximum longitudinal velocities occurred in the one place which is about 60% of the left bank. Comparison of Fig. 12d and a shows that by increasing viscosity, the maximum longitudinal velocity was transmitted into the left bank. For example, in y = 0.2 and x = 0.5, the position of the maximum longitudinal velocity from about 75% of the left bank at = 0.05 was reached to about its 65%.
In Fig. 12e, the position of the maximum depth velocity is in range of 45-50 percent of the left bank. As can be seen, the path of the maximum depth velocities is same for passing plates through three depths. From the beginning of the length to x = 0.35, the maximum depth velocity occurred about 45% of the left bank and from x = 0.5 to the end of the length occurred in the middle of the width.
A comparison between Fig. 12f and c shows that range of the position of the maximum transverse velocity is in less range than = 0.05. As can be seen, the maximum transverse velocities from x = 0.65 up to end of the length for passing plates through three depths occurred in one place which is about 60% of the left bank.
The role of increasing viscosity and the effect of high viscosities in the position of the maximum longitudinal velocities is shown in Fig. 12. In this state, the position of the maximum longitudinal velocity is the same in all lengths for passing plates through three depths and is exactly in the middle of the width.
As shown in Figs. 12h and i, depth and transverse velocities occurred exactly in the middle of the width in all lengths and passing plates from each three depths. Figure 13a shows the position of the maximum longitudinal velocity in the plan for t = 1 and = 0.05 which is in range of 63 to 70 percent of the left bank. For passing plates through y = 0.2, the position of the maximum longitudinal velocity is same for all lengths and is about 65% of the left bank except x = 0.35 and 0.5 which is about 70% of the left bank. In passing plates through y = 0.5, in all lengths except x = 0.02 and 0.1, maximum longitudinal velocity is in one place and about 70% of the left bank. In x = 0.5 for passing plates through each three depths, the maximum longitudinal velocity is in one place and about 70% of the left bank.
In Fig. 13b, the position of the maximum depth velocity is in range of 50 to 60 percent of the left bank. By moving into According to Fig. 13c, the position of the maximum transverse velocity in the plan is in range of 59 to 65 percent of the left bank. In passing plate through y = 0.2, the maximum transverse velocity occurred in one place and about 60% of the left bank. For passing plate through y = 0.5, after x = 0.2, the position of the maximum transverse velocity was moved into the right bank and was fixed up to end of the length. For example, in y = 0.5 and x = 0.35, the position of the maximum transverse velocity from about 60% of the left bank in x = 0.2 was reached to about its 65% and stayed fixed up to end of the length. In y = 0.8, the same procedure that mentioned for y = 0.5 occurs after x = 0.35.
As can be seen in Fig. 13d, the path of the maximum longitudinal velocity is the horizontal line for all lengths According to Figs. 13e and f, the position of the maximum depth and transverse velocities is the same for all lengths and passing plates through three depths and is in the middle of the width and 55% of the left bank, respectively.
As can be seen in Fig. 13g-i, the maximum longitudinal, depth and transverse velocities occurred exactly in the middle of the width for all lengths and passing plates through three depths.
By the increase in the viscosity, the velocities decrease drastically and their difference decreases. Therefore, due to insignificant of this difference, the velocity values get close to each other. This is obvious in sections g, h and i of Fig. 12. Also, the end of the time interval even at the viscosity of 0.1 is obvious by the investigated time approaches (sections d, e and f of Fig. 13). Figure 14a shows stream lines in the longitudinal section and for passing plates through the width in = 0.05 and t = 0.05. In this figure, it was recognized that in addition to the downward flow, there is the upward flow in some places. In z = 0.2 and 0.5, the flow consists of two parts. In the first part, up to y = 0.7 and up to x = 0.7, the downward flow is seen at first and then the upward flow. In the second part, from distance of about y = 0.7 up to the bottom of the bed and up to x = 0.7, there was the upward flow at first and then there was the downward flow. Stream lines in z = 0.5 have a lower slope relative to the horizon as compared to z = 0.2 which is due to the decreasing difference between the depth velocity and the longitudinal velocity. For example, in the center of longitudinal section (x = y = 0.5) and z = 0.5, the depth velocity was decreased about 20%, and longitudinal velocity was increased about 90% as compared to their value in z = 0.2. In z = 0.8, the path of the stream lines is different from two other widths. In this state, up to about y = 0.7 and to x = 0.7, there is the upward flow. Generally, one can say that in this width, values of the longitudinal velocity is greater than depth velocity.
In Fig. 14b and z = 0.2, 0.5 same as Fig. 14a for t = 0.05, flow consists of two parts, but with the difference that occurred less upward flow. In this state, upward flow is seen up to y = 0.6 and from distance of x = 0.8 to the end of the length and as well seen from distance of y = 0.8 up to the bottom of the bed and up to x = 0.6. Stream lines in z = 0.5 have a lower slope as compared to z = 0.2 which is due to the decreasing difference between the depth velocity and the longitudinal velocity. For example, in the center of the longitudinal section (x = y = 0.5) and z = 0.5, the depth velocity was decreased about 1% as compared to its value in z = 0.2, and longitudinal velocity was increased about 100%. In z = 0.8, in case of existence suspended particles, those are less suspended as compared to t = 0.05 in Fig. 14a and moved from upstream to downstream in all depths.
In Fig. 14c and z = 0.2 and 0.5, there is upward flow only up to y = 0.4 and from the longitudinal distance of about x = 0.8 up to end of the length. By comparing this figure with Figs. 14a and b, one can find that by increasing time, the effect of particles suspension was decreased. In z = 0.8, the upward flow was not seen. Figure 15a expresses stream lines in the longitudinal section and for passing plates through the width in = 0.25 and t = 0.05. Up to z = 0.5, the flow consists of two parts. Up to y = 0.8 and up to x = 0.8, there is downward flow and from distance of about x = 0.8 to end of the length there is upward flow. From distance of about y = 0.8 up to the bottom of the bed and up to x = 0.8, there is upward flow and then downward flow. In fact, in 80% of the beginning of the depth, values of the depth velocity were so small as compared to longitudinal velocity and almost horizontal flow occurred. Due to the decrease in depth velocity, by progressing width, the slope of stream lines was decreased relative to the horizon. For example, in x = y = 0.5, z = 0.5 the depth velocity is decreased about 5% as compared to its value in z = 0.2. In z = 0.8, particles moved from upstream toward downstream. Actually, in this state, the longitudinal velocity dominates the depth velocity. Generally, by comparing Figs. 15a and 14a, one can find that by increasing viscosity, less upward flow occurred. It means, in case of existence fine particles, those are less suspended.
By comparing Figs. 15b and 14b, one can find that in high viscosities, in the longitudinal section even in the initial times, there is no upward flow, and particles are not suspended. In z = 0.2 and 0.5, the path of stream lines is almost the similar, but in 20% of the right bank, the slope of stream lines relative to horizon is decreased significantly as compared to two other widths.
By comparing Figs. 15c and 14c, it was seen that in t = 0.25, stream lines at = 0.25 have less curvature as compared to = 0.05. In all three widths studied, the stream lines are almost similar. In this state, values of the depth velocity are about twice the longitudinal velocity. The slope of the stream lines relative to horizon in z = 0.5 and 0.8 was decreased to a low amount as compared to z = 0.2 which shows decreasing depth velocity. For example, in x = 0.2 and y = 0.4 for z = 0.8, the depth velocity was decreased about 40% as compared to its value in z = 0.5.
In Fig. 16a, the upward flow was seen in parts of the transverse section. In x = 0.2 and 0.5, the flow consists of two parts. There is downward flow up to y = 0.7 and up to z = 0.7. Actually, in 30% of the bottom of the bed, the depth velocity was almost equal zero, and the flow was seen only along the width. The path of the stream lines in widths of 0.2 and 0.5 is almost similar. In z = 0.8, the process of particles displacement differs from the other two widths studied. In this state, the particles move more along the width. Meanwhile that up to about 70% of the water surface and up to about 50% of the beginning of the section, upward flow was seen.
In Fig. 16b and x = 0.2 and 0.5, there is downward flow up to about 75% of the depth and up to about 75% of the left bank. Actually, in 25% of the bottom of the bed, the depth velocity was almost zero and flow only seen along the width. By comparing this figure with Fig. 16a, one can find that by increasing time, less upward flow occurs. Generally, in x = 0.8, the value of the transverse velocity is more than depth velocity, and there is flow along the width.
In Fig. 16c, the path of particles displacement is similar in the lengths of 0.2 and 0.5. In this state, the upward flow is not seen, and the depth velocity dominates transverse velocity in most places. In x = 0.8, the slope of the stream lines was decreased significantly as compared to other two lengths studied which is due to severe reduction of the difference of the depth and transverse velocities. For example, in the center of transverse section (z = y = 0.5), difference of the depth and transverse velocities from 0.88 in x = 0.5 reached to 0.04 in x = 0.8.
In Fig. 17a, up to the middle of the length and for about 75% of the beginning of the depth and up to about 80% of the beginning of the width, downward flow is seen. In x = 0.8, the particles move from the left bank toward the right bank, and it means that in this area of the length, the value of the transverse velocity dominates depth velocity.
By comparing of Figs. 17b and 16b, one can find that in the transverse section and the high viscosities even in the initial times such as t = 0.1, there is not upward flow and particles is not suspended. The path of the stream lines in lengths of 0.2 and 0.5 is almost similar. In x = 0.8, the slope of the stream lines was decreased significantly as compared to lengths of 0.2 and 0.5 which is due to severe reduction of depth velocity. For example, in the center of the transverse section (z = y = 0.5), depth velocity was decreased about 80% as compared to its value in x = 0.5.
By comparing of Figs. 17c and 16c, it was seen that for = 0.05, there was upward flow in the parts of the transverse section of the flow which was not seen at = 0.25. In x = 0.8, the slope of the stream lines was decreased significantly as compared to other two widths studied which is due to decreasing depth velocity. For example, in the center of the transverse section (z = y = 0.5), the depth velocity was decreased to about half as compared to its value in x = 0.5. In Fig. 18a and y = 0.5, the slope of the stream lines was decreased as compared to y = 0.2 and in y = 0.8, the path of the particles displacement is similar to y = 0.2. The reason for the decreasing slope in y = 0.5 is the increasing longitudinal velocity. For example, in the center of the plan (x = z = 0.5), the longitudinal velocity in y = 0.5 was increased about 115% as compared to its value in y = 0.2.
Also, in Fig. 18b and y = 0.5, the slope of the stream lines was decreased significantly as compared to y = 0.2 which its reason is increasing longitudinal velocity. For example, in the center of the plan (x = z = 0.5), the longitudinal velocity in y = 0.5 was increased about 140% as compared to its value in y = 0.2.
In Fig. 18c, also by increasing longitudinal velocity in y = 0.5, decreasing slope of the stream lines was seen. For example, in the center of the plan (x = z = 0.5), the longitudinal velocity in y = 0.5 was increased about 125% as compared to its value in y = 0.2.
In Fig. 19a, the slope of the stream lines in y = 0.5 was decreased as compared to y = 0.2 which is due to increasing longitudinal velocity and decreasing transverse velocity. For example, in the center of the plan (x = z = 0.5) in y = 0.5, the longitudinal velocity was increased about 100%, and the transverse velocity was decreased about 17% as compared to its value in y = 0.2.
In Fig. 19b and y = 0.5, the slope of the stream lines was decreased significantly as compared to y = 0.2 which its reason is increasing longitudinal velocity. For example, in the center of the plan (x = z = 0.5), the longitudinal velocity in y = 0.5 was increased about 100% as compared to its value in y = 0.2.
In Fig. 19c, increasing longitudinal velocity in y = 0.5 was seen with decreasing slope of the stream lines. For example, in the center of the plan (x = z = 0.5), the longitudinal velocity in y = 0.5 was increased about 83% as compared to its value in y = 0.2.

Conclusion
In this article, the differential quadrature (DQ) and the finite difference (FD) combined method or "DQ-FDM" was presented for solving of the nonlinear threedimensional Burger's equation with specified initial and boundary conditions. In this method, spatial terms were discretized by using DQ method which is a highorder method, and the temporal term was discretized by FD method, and the resulting nonlinear equations were solved by Newton-Raphson iterative method. The results obtained from the mentioned method were compared with fully implicit finite difference method, and their differences were expressed which for ≥ 0.1, two methods had a low difference. In the present method, generally by increasing time and viscosity, longitudinal, depth and transverse velocities were decreased. For example, in = 0.05, the maximum longitudinal, depth and transverse velocities by fivefold increasing time (from t = 0.05 to t = 0.25), for passing plates through the middle of the transverse section (z = y = 0.5) along the length, were decreased about 80, 95 and 70 percent, respectively. At t = 0.25, the maximum longitudinal, depth and transverse velocities by fivefold increasing viscosity (from = 0.05 to = 0.25), for passing plates through the middle of the plan (x = z = 0.5) along the depth, were decreased about 68, 86 and 60 percent, respectively. The position of the maximum longitudinal, depth and transverse velocities in the plan for passing plates through the depth for different viscosities and times showed that by increasing viscosity and time, the position of the velocities became closer to the middle of the width. By plotting stream lines in longitudinal and transverse sections and plan, with increasing viscosity and time, a less upward flow in the longitudinal and transverse sections was seen; stream lines in the plan for passing plates through the depth showed that transverse velocity dominates longitudinal velocity in the more of places.