A novel approach on micropolar fluid flow in a porous channel with high mass transfer via wavelet frames

1 Introduction

The examinations on the liquid stream with a high mass transfer in a permeable channel are generally listening cautiously on Newtonian fluids, and an equivalent difficulty for non-Newtonian fluids is in the lot to-be preferred situation. Various kinds of non-Newtonian fluids are couple-stress fluids, viscoelastic fluids, micropolar fluids, power-law fluids, etc. In the current paper, we have considered a micropolar liquid, a non-Newtonian fluid, as the working fluid. The hypothesis of micropolar fluids was first determined by Eringen [1, 2].

In the soul, the pivoting non-Newtonian progression of fluids consists of micro constituents gave the original incentive to the development of the theory; however, the succeeding study has fruitfully useful the model to a wide scope of zones composites, microemulsions, polymer blends, permeable rocks, etc. Broad utilizations of the micropolar liquid hypothesis specified in the books of Eringen [3] and Lukaszewicz [4].

Micropolar flow in a permeable can althrough bulky mass transferis studied using the perturbation analysis [5] and Homotopy analysis method [6]. Idris [7] considered the importance of a non-uniform temperature gradient on micropolar fluids under convective heat transfer. Sajid et al. [8] felt the boundary layer flow of a micropolar fluid in a permeable channel by Homotopy analysis. Joneidi et al. [9] obtained numerical solutions and Homotopy analysis for micropolar fluid flow in a porous channel. They brought the velocities and rotation results for different values of well-known physical parameters. Recently, Sobamowo et al. [10] studied the micropolar fluid flow with high mass transfer in a porous channel driven by suction/injection. Sreenivasulu et al. [11] examined the combined effects of viscous dissipation and Joule heating on MHD three-dimensional laminar flow of a viscous incompressible nonlinear radiating Casson nanofluid past a nonlinear stretching porous sheet. Gireesha and Sindhu [12] studied the natural convection flow of Casson fluid through an annular microchannel formed by two cylinders in the presence of the magnetic field. Khan et al. [13] introduced a new Haar wavelet method to solve Jeffery-Hamel flow and heat transfer in Eyring-Powell fluid in the presence of an external magnetic field. Khan and Razzaq [14] explained that the Legendre wavelet method to reduce the fuzzy differential equations to the fuzzy algebraic equations system made the problem easily computable to attain the solutions more rapidly than the other existing methods in the literature. Khan et al. [15] investigate the three-dimensional axisymmetric steady flow of micropolar fluid over a rotating disk in a slip-flow regime. Zhang et al. [16] studied the blood flow's entropy analysis through an anisotropically tapered artery under the suspension of magnetic Zinc-oxide nanoparticles.

Nonlinear marvel plays an essential part in material science, applied arithmetic, and design issues. A few of the appropriate territories to wave marvels encase liquid mechanics, optical filaments, plasma, versatile media, and so forth. What's more, getting precise answers to these issues is genuinely safe. Regardless, in current years, mathematical strategies radically have been urbanized to be utilized for a nonlinear coupled system of equations. A few of them are solved using numerical methods Murthy and Singh [17], and some are solved via the perturbation method Magyari and Keller [18].

Wavelets are special functions in a restricted domain; that is, a wave function, instead of oscillating everlastingly, drops to zero. Newly, we have been facing various kinds of wavelets with two parameters such as, k is the translation parameter and n is dilation parameter. The hypothesis and uses of wavelets is a reasonably youthful new branch in the signal processing and numerical analysis. It has been functional in engineering applications, such as signal analysis, time-frequency analysis, and engineering mathematics. A wavelet is a powerful tool in the field of numerical methods which helps to find the numerical solutions for the various forms of the differential equations such as, Laguerre wavelet method, Hermite wavelet method, Cardinal B-spline wavelet-based numerical method, CAS wavelets analytic solutions by Shiralashetti and Kumbinarasaiah [19, 20, 21, 22], Chebyshev wavelets approach Biazar and Ebrahimi [23]. Berman [24] studied the Laminar flow in channels with porous walls. Preliminaries of Laguerre wavelets and frames are given in [25]. Recently, very attractive fluid flow problems are solved by using wavelet technique [26, 27, 28].

To the best of our konwledge, no one solved this type of problem using the Laguerre wavelet exact Parseval frame in the view of literature. This impetus us to solve such equations via the Laguerre wavelet exact Parseval frame. The present work's interest is to solve the coupled nonlinear ordinary differential equations governing micropolar fluid flow in a porous channel with high mass transfer by using the Laguerre wavelet exact Parseval frame method, and the results have a comparison with solutions of Joneidi et al. [9]. We wish to establish how the extra material constants of the micropolar fluid influence the flow for large mass transfer through the channel walls.

This article's motivation is to illustrate the present algorithm in solving some following systems of two highly nonlinear ODEs. The proposed algorithm is useful for obtaining numerical solutions of the system of nonlinear differential equations. Significantly, this algorithm yields an exact solution for a system of ordinary differential equations which are having solutions as a polynomial of finite degree [25]. The obtained results are compared with the Optimal Homotopy Asymptotic method (OHAM), Runge-Kutta solutions Joneidi et al. [9].

2 Problem formulation

The highly nonlinear coupledordinary differential equations which explain the Micropolar boundary layer fluid flow in a porous channel with high mass transfer can be summarized as introduced by Joneidi et al. [9]:

The boundary conditions are

or the flow is symmetric about the channel

where the prime (′) denotes differentiation concerning η. The non-dimension Reynolds number Re=ρqhμ and micropolar parameters N1=κμ , N2=νsμh2 , N3=jh2 . Analytical treatment is carried out by taking s = 0. For known cross-flow Reynolds number Re, the physical problem contains four extra micropolar dimensionless parameters N₁, N₂, N₃ and s. Re < 0 corresponds to injection and Re > 0 to suction. The parameters may depend on the concentration and shape of the microelements.

3 Laguerre wavelet exact Parseval frame method

The Laguerre wavelet exact Parseval frame method is used to solve Eq. (1) subjected to the boundary conditions given in Eq. (2). First, assume the solution in the following form:

F (η) and G (η) are truncated as

where ψn, m(η)¯ is the Laguerre wavelet exact Parseval frame defined in [25], n = 1, 2, 3, ..., 2^k−1, k is any positive integer and M represents Laguerre polynomial degree. Substituting Eq. (5) in Eq. (1) we get,

Discrete each of the above equations using the following collection points

which leads to (2^k−1 M − 4) + (2^k−1 M − 2) number of equations. But, by data 6 equations are furnished by boundary conditions. On solving this system containing 2 (2^k−1 M) number of nonlinear algebraic equations by the suitable solver, we get 2 (2^k−1 M) Laguerre wavelet exact Parseval frame coefficients then substitute these coefficients in Eq. (5) leads the Laguerre wavelet exact Parseval frame numerical solution for Eq. (1).

4 Results and discussion

In organize to authenticate the computer code urbanized in this study, the outcomes for various values of η for unchanging values of N₁ = N₂ = 1, N₃ = 0.1 and Re = −1 are compared with the consequences of Joneidi et al. [9] in Tables 1–2 and Figures 1–3. This accuracy gives us high confidence in the validity of this problem and reveals an excellent engineering accuracy agreement. This investigation is completed by depicting the effects of some important parameters to evaluate how these parameters influence this fluid. The velocities F (η), F^′ (η) and rotation G (η) profiles for various parameters Re, N₁, N₂ and N₃ by Laguerre wavelet exact Parseval frame method are shown in Figures 4–9. In these figures, we removed comparing our results with the analytical/numerical solution because of having an excellent agreement.

Figure 1

Comparison between the solutions of OHAM, numerical solution, and Laguerre wavelet exact Parseval frame method for F (η) when N₁ = N₂ = 1, N₃ = 0.1 and Re = −1.

Figure 2

Comparison between the solutions of OHAM, numerical solution, and Laguerre wavelet exact Parseval frame method for G (η) when N₁ = N₂ = 1, N₃ = 0.1 and Re = −1.

Figure 3

Comparison between the solutions of OHAM, numerical solution, and Laguerre wavelet exact Parseval frame method for F′ (η) when N₁ = N₂ = 1, N₃ = 0.1 and Re = −1.

Figure 4 shows the rotation G (η) profile of the fluid for different values of the Reynolds number. The rotation profile curves in the (η, G (η))-plane show an upward concave shape and G (η) increases with a decrease in the Reynolds number's values. For an unchanging value of the Reynolds number, the rotation profile G (η) decreases with increasing η, up to around 0.6, and subsequently increases with increasing η. Also, from these figures, we observe that with an increase in the Reynolds number, the point at which minimum rotation occurs does not move away from the origin of the channel.

Figure 4

Rotation profile, G (η) via Laguerre wavelet exact Parseval frame method for different Reynolds number values when N₁ = N₂ = 1, N₃ = 0.1.

Figure 5 shows the velocity F (η) curve of the fluid for a variety of values of the Reynolds number. The velocity F (η) curve decreases with decreasing the values of Reynolds number and η. But when suction occurs, as is shown in this figure, inverse results are obtained.

Figure 5

Velocity profile, F (η) via Laguerre wavelet exact Parseval frame method for different Reynolds number values when N₁ = N₂ = 1, N₃ = 0.1.

Figure 6 shows the velocity distribution F^′ (η) profile of the fluid for various values of the Reynolds number Re. The velocity distribution F^′ (η) profile decreases with increasing the value of η. The velocity distribution F^′ (η) increases with increasing the value of the Reynolds number. All the above results are obtained when N₁ = N₂ = 1, N₃ = 0.1.

Figure 6

Velocity profile, F′ (η) via Laguerre wavelet exact Parseval frame method for different values of Reynolds number when N₁ = N₂ = 1, N₃ = 0.1.

Figures 7 and 8 show the rotation G (η) curve for various values of N₁ and N₂. The behavior of the rotation profile is similar to Figure 4. For a fixed value of N₁ (Figure 7) or N₂ (Figure 8), the rotation G (η)attains a minimum value concerningη. The rotation G (η) increases with increasing the value of N₂, but the opposite trend is noticed with increasing N₁. Figure 9 shows the rotation G (η) profile for different values of N₃. The rotation G (η) profile increases with decreasing the value of N₃. Moreover, micropolar parameters have no vital role in the velocity profiles and are not given in this paper.

Figure 7

Effects of different values of (a) N₁ and (b) N₂ on G (η) when Re = 1, N₁ = 1, N₃ = 0.1.

Figure 8

Effects of different values of (a) N₁ and (b) N₂ on G (η) when Re = 1, N₂ = 1, N₃ = 0.1.

Figure 9

Effects of different values of N₃ on G (η) when N₁ = N₂ = 1, and Re = 1.

5 Conclusions

In this paper, we functional the Laguerre wavelet exact Parseval frame method to solve coupled nonlinear ODEs from the Berman's similarity solution of micropolar fluid flow in a permeable canalbulky mass transfer. This method is easy to apply and yields the preferred accuracy. It has seemed that increasing the value of Re leads to an increase in the velocities F (η) and F^′ (η). However, Micropolar parameters have no critical role in the velocities profiles. For each value of the Reynolds number, rotation G (η) attains the smallest value does not go away from the center of the channel. The rotation G (η) decreases with decreasing the value of N₂, but the opposite trend is noticed with increasing N₁ and N₃. Further the comparisons between the OHAM, Runge-Kutta solutions Joneidi et al. [9] and the present work, it was shown that the current work provided a promising approach for this type of equations. While the Laguerre wavelet exact Parseval frame leads to more accurate results, it seems that the accuracy and rapidity of the Laguerre wavelet exact Parseval frame method (LWPM) are higher than that of the OHAM method in this problem. This method is an extension work of [25]. The process in [25] applied only to initial valued problems. But, in the present approach, we extended for boundary value problems. As per the literature survey, we have not found any article of frames applied to micropolar fluid flow problems. So, this is the first paper on micropolar fluid flow problems via wavelet frames.