The Analysis of Fractional-Order Navier-Stokes Model Arising in the Unsteady Flow of a Viscous Fluid via Shehu Transform

Sunthrayuth, Pongsakorn; Shah, Rasool; Zidan, A. M.; Khan, Shahbaz; Kafle, Jeevan

doi:https://doi.org/10.1155/2021/1029196

Journal of Function Spaces

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 1029196 | https://doi.org/10.1155/2021/1029196

The Analysis of Fractional-Order Navier-Stokes Model Arising in the Unsteady Flow of a Viscous Fluid via Shehu Transform

Pongsakorn Sunthrayuth,¹Rasool Shah,²A. M. Zidan,^3,4Shahbaz Khan,²and Jeevan Kafle⁵

Academic Editor: Nehad Ali Shah

Received30 Apr 2021

Accepted22 Jul 2021

Published10 Aug 2021

Abstract

This paper presents a new method that is constructed by combining the Shehu transform and the residual power series method. Precisely, we provide the application of the proposed technique to investigate fractional-order linear and nonlinear problems. Then, we implemented this new technique to obtain the result of fractional-order Navier-Stokes equations. Finally, we provide three-dimensional figures to help the effect of fractional derivatives on the actions of the achieved profile results on the proposed models.

1. Introduction

Fractional calculus is an engineering and sciences area that deals with the integral and derivative of arbitrary orders. Fractional differential equations (FDEs) have gained popularity and significance, primarily due to their proven implementations in applied mathematics. For instance, these problems are more and more utilized to an equation in fluid flow, diffusion, rheology, oscillation, relaxation, reaction-diffusion, anomalous diffusion, diffusive transport akin to diffusion, turbulence, polymer physics, electric networks, electrochemistry of corrosion, chemical physics, propagation of seismic waves, relaxation processes in complex systems, porous structures, and dynamical processes in self-similar and various other physical procedure. The most significant benefit of FDEs in these and other uses is their nonlocal property. It is well-known that the differential operator of integer-order is a local operator, but the fractional-order differential operator is nonlocal [1–5]. For example, the nonlinear oscillation of earthquakes can be modeled with fractional derivatives [6], and the fluid dynamic traffic model with fractional derivatives [7] can calculate the insufficiency arising from the assumption of continuum traffic flow. Experimental data fractional partial differential equations for seepage flow in porous media are shown in [8]. Fractional differential equations have created attention among the researcher due to the exact description of nonlinear phenomena, especially in nanohydrodynamics where continuum assumption does not well, and fractional model can be considered the best candidate. These findings invoked the growing interest in studies of fractal calculus in many branches of engineering and science. This is more realistic and it is one reason why fractional calculus has become more and more popular [9–17].

A famous governing equation of motion of viscous fluid flow called the Navier-Stokes (NS) equation has been derived in 1822 [18]. The equation can be regarded as the Momentum equation and is a combination of Newton’s second law of motion for fluid substances, the energy equation, and continuity equation. This equation describes many physical things such as ocean currents, liquid flow in pipes, blood flow, and airflow around the wings of an aircraft. The NSE is the computational fluid dynamics initial equation, relating external forces and pressure acting on a fluid to the fluid flow response. The Navier-Stokes and continuity models are described by where the Caputo fractional derivative is , time is , the velocity vector is , kinematics viscosity is , pressure is , and density is . In this article, we consider two special cases. First, a viscous fluid in a tube moves in an unsteady, one-dimensional motion. The fluid flow in the tube of fractional NSEs in cylindrical coordinates that govern is defined by with initial condition where and is a function that depends only on .

Consider the two-dimensional fractional NSE is given as with initial conditions where , , where time, pressure and constant density define by , , and , respectively. and are the spatial components and and are two functions dependent only on and . El-Shahed and Salem [19] generalized the classical NSEs by substituting a Caputo fractional derivative of order for the first time derivative, where . Momani and Odibat [20] use the Adomian decomposition approach to solve fractional-order NSEs. Ganji et al. [21] used the Homotopy perturbation procedure, an analytical technique, to solve fractional NSEs in polar coordinates, and the solutions were given in the closed-form. Singh and Kumar [22] implemented the reduced differential transform method to obtain an analytical solution of fractional multidimensional NSEs. Ahmad and Jaber used the residual power series approach to solve the nonlinear fractional two-dimension NSEs [23]. For a class of NSEs with fractional-order derivatives, Wang and Zhang [24] proposed a numerical method. Some investigators combined two strong methods to produce a new method for solving fractional-order NSE equations. Below, we expressed some of these combinations: a mixture of the Adomian decomposition method and Laplace transform; Kumar et al. [25] introduce homotopy perturbation transformation technique, combine the Homotopy perturbation technique with Laplace transform, to solve fractional NSEs. Jena and Chakraverty [26] applied the homotopy perturbation transform technique and this method consists in the combination of homotopy perturbation technique and Elzaki transform; Prakash et al. [27] proposed q-homotopy analysis transform method to obtain a solution of fractional system NSEs. This method combined the homotopy analysis technique and Laplace transform.

The residual power series method (RPSM) is an efficient and straightforward method for constructed a Power Series solution for extremely linear and nonlinear equations without linearization, perturbation, and discretization. The RPS method, unlike the classical power series method, does not involve comparing the coefficients of the corresponding terms or a recursion relation. The power series coefficients are calculated using a series of algebraic equations in one or more variables. The methodology’s key benefit is its dependence on derivation, which is much simpler and more accurate than integration, which is the basis of most other solutions techniques. This approach is an alternative technique for solving fractional-order partial differential equations theoretically [28].

The RPSM was used as an essential tool for determining power series method values for fuzzy differential equations [29]. It has been successes used in the analytical solution of the generalize Lane-Emden equations [30], which is significantly nonlinear singular differential equations, in the exact work of higher-order regular differential equations [31], in the result of composite and noncomposite time-fractional differential equations [32], in estimating and demonstrating the variety of solutions for fractional boundary value equation [33], in the approximate investigation of the nonlinear time-fractional Burgers and KdV equations [34], in further to few other implementations [35–37], and recently, it has been implemented to investigate the analytical solution of a fractional two-component evolutionary system [38]. This paper presents an analytical investigation of one- and two-dimensional, fractional-order model of Navier-stokes equations by using RPSTM.

The fractional calculus fundamental definitions are defined in Section 2, write the general methodologies in Section 3, and many test models to show the effectiveness of suggested techniques are given in Section 4, and finally, the conclusion is given in Section 5.

2. Basic Definitions

2.1. Definition

The fractional-order Riemann-Liouville integral is defined as [39, 40].

2.2. Definition

The fractional-order Caputo’s derivative of is given as [39, 40].

2.3. Definition

Shehu transformation is new and identical to other integral transformations defined for exponential order functions. In Set A, the function is defined by [41–43]

The Shehu transform which is described as for a function is defined as

The Shehu transform of a function is ; then, is called the inverse of which is given as

2.4. Definition

Shehu transform for th derivatives is given as [41–43]

2.5. Definition

The fractional-order derivatives of Shehu transformation are defined as [41–43]

2.6. Definition

A power series is given as where and is called fractional-order power series about , where are the constant called the coefficient of the series. If then the fractional-order power series will be reduced to the fractional-order Maclaurin series.

2.7. Theorem

Suppose that has a fractional power series expression of the form at

If , for are continuous on , then the coefficient can be defined as

2.8. Definition

The power series expansion is given as is called to be multi fractional power series at , where are the fractional power series coefficients.

2.9. Theorem

Let us suppose that has the fractional-order power define at is given as

If , are the continuous on , then the coefficient can be written as where .

So we can write the fractional power expansion of of the form which is the generalized form of the Taylor expansion.

2.10. Corollary

Let assume that has a power series multi fractional represent about as

If , are continuous on , then where .

3. The General Methodologies of RPSTM

In this section, we will go through the steps that RPSTM takes to solve the FPDEs. with initial condition we use the Shehu transformation to (22)

By the fact that and apply the initial condition (23), we can write (10) as where

Second, we expressed the transformation function as the following equation

The th-truncated series of (26), we write as

As stated in [24], the definition of Shehu residual function to (26) is and the th-Shehu residual function of (28) is

Third, we explain some properties of the residual power series method (i) and for each (ii), ,

Furthermore, to calculate the coefficient function the following system can be solved recursively.

Finally, we used the Shehu inverse to , to obtain the th-approximated supportive result .

4. Numerical Result

4.1. Example

Consider the one dimensional fractional NS equation

Subject to the initial condition

Applying Shehu transform to Eq. (32) and using the initial condition given in Eq. (33), we get

The th truncated term series of Eq. (34) is and the th Shehu residual function is

Now, to determine , , we put the th-truncated series (35) into the th-Shehu residual function (36), multiply the solution equation by and, then, solve recursively the relation , for . The first few component of the series

Putting the values of in Eq. (35), we get

Applying inverse Shehu transform to Eq. (39), we get

The above equation put , we get which is the actual result of classical Navier-Stokes equations for the velocity field. The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 1, and the fractional-order behavior of Navier-Stokes equations with and is shown in Figures 2 and 3, respectively.

4.2. Example

Consider the one-dimensional fractional NS equation

Subject to the initial condition

Applying Shehu transformation to Eq. (42) and applying the initial condition given in Eq. (43), we get

The th truncated term series of Eq. (44) is and the th Shehu residual function is

Now, to determine , , we substitute the th-truncated series (45) into the th-Shehu residual function (46), multiply the resulting equation by and, then, solve recursively the relation , for . The first several component of the series

Substituting the value of in Eq. (45), we obtain as

Using inverse Shehu transformation to Eq. (49), we get

The above equation put , we get

Figure 4 shows that the actual and analytical solutions graphs at . The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 4, and the fractional-order behavior of Navier-Stokes equations with and is shown in Figure 5, respectively.

4.3. Example

Consider the two-dimensional fractional NS equation with initial condition

Applying Shehu transform to Eq. (52) and using the Eq. (53), we get

The th truncated term series of Eq. (54) is and the th Shehu residual function is

Now, to determine and , , we substitute the th-truncated series (55) into the th-Shehu residual function (56), multiply the resulting equation by , and then solve recursively the relation , for and . The first few component of the series and

Putting the values of and , in Eq. (55), we get

Using inverse Shehu transformation, we get

The exact solution is given as

Figure 6 shows that the actual and analytical solution graphs at with respect to . The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 6, and the fractional-order behavior of Navier-Stokes equations with and is shown in Figure 7, respectively. Similarly, Figure 8 shows that the actual and analytical solution graphs at with respect to . The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 8, and the fractional-order behavior of Navier-Stokes equations with and is shown in Figure 9, respectively.

4.4. Example

Consider the two-dimensional fractional NS equation with initial condition

Applying Shehu transform to Eq. (61) and using the Eq. (62), we obtain as

The th truncated term series of Eq. (63) is and the th Shehu residual function is

Now, to determine and , , we substitute the th-truncated series (64) into the th-Shehu residual function (65), multiply the resulting equation by , and then solve recursively the relation , for and . The first few component of the series and

Putting the values of and , in Eq. (64), we get

Using inverse Shehu transformation, we get

The exact result is given as

Figure 10 shows that the actual and analytical solution graphs at with respect to . The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 10, and the fractional-order behavior of Navier-Stokes equations with and is shown in Figures 11 and 12, respectively. Similarly, Figure 13 shows that the actual and analytical solution graphs at with respect to . The field of velocity behavior of the classical Navier-Stokes equations is show in Figure 13, and the fractional-order behavior of Navier-Stokes equations with and is shown in Figures 14 and 15, respectively.

5. Conclusion

In this paper, the residual power series transform method is implemented to approximate analytical solutions of the fractional-order Navier-Stokes equations, which are widely applied as problems for spatial effects in applied sciences. The method gives a series form of solutions that converge very quickly in physical problems. It is predicted that this paper obtains solutions to analyze the complicated nonlinear physical problems. The calculations of this method are very simple and straightforward. Thus, we assume that this method can be applied to solve many systems of nonlinear fractional-order partial differential equations.

Data Availability

The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

One of the coauthors (A. M. Zidan) extends his appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through research groups program under grant number R.G.P-2/142/42.

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