Wavelet Operational Matrices and Lagrange Interpolation Differential Quadrature-Based Numerical Algorithms for Simulation of Nanofluid in Porous Channel

(is work analyses the features of nanofluid flow and thermal transmission (NFTT) in a rectangular channel which is asymmetric by developing two numerical algorithms based on scale-2 Haar wavelets (S2HWs), Lagrange’s interpolation differential quadrature technique (LIDQT), and quasilinearization process (QP). In the simulation procedure, first of all, using similarity transformation (ST), the governing unsteady 2D flowmodel is changed into two highly non-linear ODEs. After that, QP is applied to linearize the non-linear ODEs, and finally S2HWs and LIDQT are used to simulate the non-linear system of ODEs. In results and discussion section, the parameters Reynolds number (R), expansion ratio and Nusselt (Nu), and nanoparticle volume fraction (φ) are analysed with respect to velocity and temperature profiles. (e proposed techniques are easy to implement for fluid and heat transfer (FHT) problems.


Introduction
In current century, the fluid flow and heat transfer (FFHT) are regarded as the extremely crucial problems of engineering and industries. is is the main reason that NFTT is a hot area of research amongst engineers and research community. It has wide applications in various scientific and industrial areas, such as power plant operations, manufacturing, lubrications, heat exchangers, micro-electromechanical technology, advanced nuclear energy systems, macromolecular science, and so on.
Due to presence of nonlinear thermal radiation and viscous flow in the NFTT models, the analytical studies (in form of solutions) of these models are not an easy task. Secondly, the governing equations (GEs) of NFTT models are in the form of non-linear partial differential equations (PDEs) involving emerging parameters, which may be quite complex, and exact solutions of such type of problems are difficult to calculate. at is why a number of experimental and numerical studies on NFTT models have been conducted in the last few years [1][2][3][4][5][6][7][8][9][10][11].
Carbon nanotubes (CNTs) with a cylindrical nanostructure were first studied by Venkataraman et al. and his associates [12] in 1991. Enough efforts were done to study the NFTT models through exact as well as classical numerical methods till date. Sheikholeslami Kandelousi [1] used the Runge-Kutta 4th order (RK4) scheme to study the characteristics of NFTT between two horizontal parallel plates.
e study reflects that heat transmission (HT) growth increases as we increase Reynolds number when m � 0, but the adverse tendency is inspected for different values of power law index m. Ahmed and his group [2] studied the magneto-hydrodynamic (MHD) fluid flow problem in a domain of rectangular shape. Basically, this technique is an iterative scheme and gives series solution, but this technique is quite complex due to series solution.
Ahmed et al. [3] examined the NFTT in a rectangular channel with the help of the Galerkin method, while Hatami and his associates [4] studied the NFTT between two parallel plates by using the Galerkin and the least square methods. Khan and his group [5,6] simulated the NFTT model with CNT-based nanofluids in non-parallel stretchable walls under the effects of velocity slip in a channel using differential transform (DT) and Runge-Kutta-Fehlberg schemes. Srinivas and his associates [13] analysed the thermal-diffusion and diffusion-thermo effects of a viscous fluid in a 2D channel between slowly expanding or contracting walls with weak permeability.
e main contribution of this work is to cultivate two numerical algorithms based on S2HWs, LIDQT, and QP to simulate the unsteady nanofluid FHT in an asymmetric rectangular porous channel. S2HWs and LIDQT-based algorithms are powerful tools that fulfil all the required conditions of an efficient technique. Herein, the authors analysed the features of the unsteady NFTT model in a rectangular channel using S2HWs and LIDQT-based algorithms. In the simulation procedure, first of all, using similarity transformation (ST), the governing unsteady 2D flow model is transformed into two ODEs in which one is non-linear and other is linear. en, QP is applied to linearize the non-linear ODEs, and finally S2HWs and LIDQT are used to simulate the non-linear system of ODEs. In results and discussion section, the parameters Reynolds number (R), expansion ratio and Nusselt (Nu), and nanoparticle volume fraction (φ) are analysed with respect to velocity and temperature profiles. e proposed techniques are easy to implement than the techniques available in the literature [3,7,16,17] for fluid and heat transfer problems.

Description of the Problem and Mathematical Model
Consider a rectangular channel shown as in Figure 1. e main end of the channel is supported by an impenetrable flexible membrane. Unsteady and laminar nanofluid is flowing through the channel. Other details can be found in [3]. For the geometrical explanation, take the middle part of the head end as original displayed in Figure 1. Here, we assume that the temperature T of the lower wall A l is greater than the upper wall A u . In these considerations, the basic mass, momentum, and energy governing equations of the problem are Here, U is the horizontal component of velocity and V is the vertical component of velocity. e viscosity μ nf , density ρ nf , and effective thermal conductivity k nf of nanofluids and effective heat capacity (ρC p ) nf are in the following forms: Equations (1)-(4) are connected with the BCs where A l and A u represent the penetrability of lower and top walls, respectively, and _ a � da/dt is the velocity of moving wall.
e values of parameters of base fluid and nanoparticles are taken from Table 1. Now, to reduce the independent variables, choose the stream function Φ � (]ξ/a)G(y, t), with y � η/a(t) and kinematic viscosity ] � μ nf /ρ nf which satisfies continuity equation (1). Using these transformations, the velocity components become  Journal of Mathematics where G y � zG/zy. Substitute the value of the velocity components U and V from (8) into (2) and (3) and eliminate the pressure term: BCs given in equations (6) and (7) are reduced to where (1 − ϕ) 2.5 are constant parameters, α � a _ a/] is the wall expansion ratio which is very +ve and − ve for expansion and contraction, and R l � aV l /] and R u � aV u /] are the permeation RNs relevant to the bottom and top walls which are +ve due to injection of cooled fluid. Following the work of Uchida and Aoki and Pandit and Sharma [18,19], to normalize equations (9) and (10), substitute g � G/R which leads to where g � g(y) and g (r) , r � 1, 2, 3, 4 denotes the derivatives of g with respect to y and the BCs are Now, to reduce equation (4) in one variable, use the following similarity transformation (ST): Substitute equation (13) into energy equation (4) which leads the following ODE: with BCs where

Quasilinearization Process for Linearization
To simulate equations (1)-(4), we reduce the equations into equations (11), (12), (14), and (15). Now, the next purpose is to solve non-linear ODE (11) and linear ODE (14). In this direction, the first step is to linearize non-linear ODE (11) by using QP proposed by Bellman and Kalaba [20]. e usefulness of the QP is that it linearizes the problem which is easy to solve than the non-linear problem. e non-linear terms of equation (11) are linearized as follows: where n denotes the number of iterations of the quasilinearization technique.

Wavelet Family and Uniform S2HWs
Wavelet-based techniques are powerful tools to solve differential equations (DEs). Generally, scaling function χ(t) and mother wavelet ψ(t) are important to generate a family of wavelets. ese two functions satisfy the following conditions: where χ(ω) and ψ(ω) are Fourier transforms of scaling function χ(t) and mother wavelet ψ(t), respectively. e wavelet family can be generated by the following relation: where j, k are integers and these are called dilation and translation parameters, respectively.
Any function g(t) ∈ L 2 (− ∞, ∞) can be represented into wavelets as where c k , d j,k are scaling and wavelet coefficients, respectively. ese coefficients can be calculated by the orthogonal property of wavelets.
where d T Integrate successively equation (28) from 0 to y to obtain the lower-order derivatives. e lower-order derivatives in terms of S2HW operational matrices are as follows: Similarly, we can obtain the approximation in terms of S2HW operational matrices for equation (14) as follows: where e T (2M) � [e 1 , e 2 , . . . , e 2M ] are unknown S2HW coefficients.

Convergence Analysis of S2HW Approximation.
is section describes the convergence criteria of the S2HWs via the following result.

Proof.
Given g(y) � ∞ l�1 d l ψ l (y) and G 2M (y) � 2M l�1 d l ψ l (y), then the error estimation at Jth level resolution is Using the inner product, orthonormal condition of S2HWs, mean value theorem, and Lemma 1, we have Assume C 1 � Sup l K l , and again using Lemma 1, we conclude that is reflects that as J increases, the error goes to zero. Hence, the solution via S2HWs converges to exact solution.

LIDQT for Simulation of the Problem
In this section, first we will give brief details of LIDQT and then use the technique for the simulation purpose.

Overview of LIDQT.
e differential quadrature method (DQM) was introduced by Bellman et al. [28] in 1972, and it is based on Lagrange's interpolation. at is why it is called LIDQT. LIDQT became famous for solving differential equations (DEs) after 1972. e main use of LIDQT is to approximate the derivatives of unknown functions in the problems identical to integral quadrature rules. To understand the idea of the LIDQT, consider a sufficiently smooth function f(y) defined on the domain [α, β] and let the interval be divided into the grid nodes as α � y 1, < y 2 , . . . , < y p � β. By using the DQ technique, the qth order derivative f q (y) is approximated at y � y i in the following way: Journal of Mathematics where p is the number of nodes in the interval [α, β], y i is an arbitrary node, and ω (q) ij are the weighting coefficients (WCs) of qth order derivative. In DQ technique methodology, the first step is to calculate the WCs ω (q) ij . In the literature, there are many ways [16,17,20,[28][29][30][31][32][33] to compute the WCs. Here, we used the following Lagrange's test functions (LTFs) to compute the WCs: where N(y) � (y − y 1 )(y − y 2 ) . . . (y − y p ), (38)

Stability Criteria of LIDQT.
Stability analysis is a major issue for solving the fluid and heat problems. e stability of the LIDQT depends on the eigenvalues (EVs) of WC matrices, and these EVs totally depend on the distribution of node points used for domain discretization. is concept had been studied and shown by Chang [34]. In this book, Shu et al. showed that the uniform node distribution does not produce stable solution, but the stable solution can be acquired by using CGL nodes which we have used for simulation purpose.

Simulation and Remarks
e main goal of this segment is to present the simulated results of model via the developed numerical algorithms based on S2HWs, LIDQT, and QP. e computational work is done with 31 and 64 grid points for LIDQT and S2HWs, respectively. e maximum absolute error L ∞ and relative error (RE) are computed by the formulas 6 Journal of Mathematics where u analytic and u appx. denote analytical and approximated solutions, respectively. e simulation part has been done by writing MATLAB routines. First, to accredit the developed algorithms for the model (11), let us assume g(y) � y 4 as exact solution (ES) of equation (11).
is assumption coverts homogeneous equation (11) into a non-homogeneous equation as follows: where with BCs e main reason of this assumption is to validate the developed algorithms for the problem since now we can compute absolute (L ∞ ) and relative errors (REs) of the problem. After that, systems (44)-(46) are solved by the developed algorithms. Obtained outcomes are given in Table 2 and Figures 2 and 3. Table 2 Figures 6 and 7 conclude that the maximum velocity lies at the centre of the channel, while near the wall, GV profile decreases as EC ratio α increases. Figure 8 discusses the case when contraction is combined with injection and suction, and the figure demonstrates that the velocity decreases with increasing value of R. Figure 9 reveals that as we decrease the value of R, the profile of GV goes down till R � 1 and again goes up for R � 0 in both the cases of SWCNTs and MWCNTs. Figures 2-9 show the impact of parameters on velocity g(y) and GV g ′ (y) profiles. Figure 10 shows the effect of NVF φ on temperature profile h(y) of problem (14).  It is concluded that enhancement in NVF φ causes the rapid increase of temperature for MWCNTs as compared to SWCNTs. Figure 11 demonstrates that both SWCNTs and MWCNTs exhibit the same phenomenon when altering the value of Reynolds number for lower and upper walls in problems (11)-(14).

Concluding Remarks
is work analysed the nanofluid FHT model by developing two numerical algorithms based on QP, S2HWs, and LIDQT. e developed algorithms are influenced by study done in [21,22,29] for fluid problems. Further, the impact of NVF, Reynolds number, and EC ratio has been investigated on the nanofluid FHT. e precision of developed algorithms has been tested by choosing the ES of equation (11) and by finding the L ∞ and REs. It is concluded that SWCNTs have an exalted rate of heat transmission as compared to MWCNTs. Enhancement in the NVF does not influence the velocity outline, but it has plausible impact on temperature profile. Further, both single as well as multi-walled carbon nanotubes present the same demeanour for different values of Reynolds number for lower and upper walls. e proposed algorithms are easy to implement than the techniques used in [3,7,16,17] for fluid and heat transfer problems. e proposed algorithms with some modifications can be useful for different types of fluid flow problems and higher-order non-linear mathematical models.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

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