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Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model

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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 8 Issue 1

Abstract

In this paper, a spectral method based on the rational Legendre functions is discussed to approximate the solution of the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a stretching sheet. At first, because of the model is a nonlinear ordinary differential equation, we construct a sequence of linear ordinary differential equations (ODEs) by using the quasilinearization method (QLM). By applying QLM on the ODEs at each iteration, the equations convert to a system of linear algebraic equations. The present results have shown the high attention and convergence of our method, and the residual error of the present results are very small. In addition, the effect of the Eyring-Powell fluid material parameters is explained.

Keywords: Boundary Layer Flow; Eyring-Powell Non-Newtonian Fluid; Rational Legendre Functions; Quasilinearization Method
MSC 2010: 76A05; 74S25; 76D05; 76M55; 34B40

1 Introduction

Considering the engineering and technological applications one of the most increasing interest for the last few decades the flows of non-Newtonian fluids. Continuous stretching of plastic films, artificial fibers, metal extrusion, glass blowing are some important applications in industries [1]. In length manufacture of these sheets, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. The mechanical properties of the final product strongly depend on the stretching and cooling rates in the process [2, 3]. Industrial applications, such as combustion systems, metallurgy, nuclear reactor safety, solar collector and chemical engineering are some of applications of stretched flows in different engineering and industrial applications [4]. Crane [5] at first studied the steady boundary layer flow of the incompressible flow. There are two types of fluid whose flows are investigated: Newtonian and non-Newtonian. The Newtonian law is the only unique relationship existing in the paper [6]. Significant contributions to the study of non-Newtonian fluid models with a diversity of rheological properties have been made by Harris [7] and Bird et al. [8]. Mathematical systems for non-Newtonian fluids are of higher order and intricate in comparison to the Newtonian fluids. The non-Newtonian fluid models differ in their intricacy and capability to catch different physical phenomena. Different models are used to represent different characteristics of the non-Newtonian fluids because no single model can capture all the features of the non-Newtonian fluids complexities [6, 9]. Non-Newtonian fluids gained much attention due to their various applications in physiological and industrial processes [10] such as coal water, jellies, toothpaste, ketchup, and food products [6]. The analytic/numerical studies of non-Newtonian fluids models show exciting challenges to mathematicians, physicists, numerical simulists and modelers [11]. One of the most important models for non-Newtonian fluids is the Eyring-Powell model. Eyring-Powell model deduced from kinetic theory, can be reduced to the Newtonian behavior for low and high shear rates. Takhar et al. [12] studied the MHD flow and mass transfer in a viscous fluid over a stretching surface. In addition, the flow and heat transfer of the Powell-Eyring fluid in the boundary layers on an exponentially stretching continuous permeable surface was studied by Megahed [13]. Newly, researches on the Lie group analysis in the field of fluid science to encountered heat and mass transfer effects simultaneously have been done [14] and also Ellahi et al. [15] worked on the MHD flow of an Eyring-Powell fluid between parallel heated plates by pseudo-spectral collocation method. Also, the studies have been done which aims to investigate the effects of magnetic field and anisotropic slip on the flow of Eyring-Powell fluid and heat transfer over an infinite rotating disk [16], the research on the deals with the fully-developed two-layer Eyring-Powell fluid in a vertical channel divided into two equal regions have been done [17], and the double diffusive Darcian convection flow of Eyring-Powell fluid from a cone embedded in a homogeneous porous medium with the effects of Soret and Dufour has been studied [18]. Nonlinear problems are important in fluid mechanics and heat transfer. These nonlinear equations should be solved using numerical methods as most of them do not have analytical solutions. To approximate linear and nonlinear ordinary differential equations (ODEs) The collocation approaches based on variant functions, such as radial basis functions(RBF), Laguerre functions, etc., have been used in various sciences like astronomy, physics, and biology [19]. The rational Legendre functions are the most useful orthogonal functions, which are very efficient for solving different problems in the finite and semi-infinite intervals as they can approximate the solutions with high accuracy [20, 21, 22, 23].

In this paper, we want to study the boundary layer flow of an Eyring-Powell fluid over a stretching sheet in an unbounded domain by the rational Legendre functions collocation method with the QLM to find an approximate solution for the problem. In the continue, this paper are organized as follows: in section 2, we express the mathematical model and in section 3, preliminaries are explained. Section 4 contains the numerical solution of the problem. Afterward, numerical results are explained in section 5. The conclusion is presented in the last section.

2 Mathematical model of the problem

For the incompressible fluid obeying Eyring-Powell model, the equation of continuity and the x-momentum equation can be simplified using the boundary layer approximation as (see Fig. 1) [2]:

Fig. 1 Velocity boundary layers on a stretching sheet.
Fig. 1

Velocity boundary layers on a stretching sheet.

∂u∂x+∂v∂y=0,(1)
u∂u∂x+v∂u∂y=(α+1ρβH)∂2u∂y2−12ρβH3(∂u∂y)2∂2u∂y2.(2)

By following boundary conditions:

u=Uωx=bx, v=0 at  y=0,u→0 at  y→∞.(3)

We proceed for solutions through stream function satisfying bellow:

u=∂ψ∂y,    v=−∂ψ∂x,(4)

and

ψ=(bα)12xg(η).   (5)

In addition, η is defined as

η=(bα)12y.

The following equation is obtained using the above equations:

(1+ε)d3gdη3−εδ(d2gdη2)2d3gdη3−(dgdη)2+gd2gdη2=0,(6)

where ε and δ, the material fluid parameters, are defined as:

ε=1μβH,   δ=b3x22H2α.

The boundary conditions for Eq. (6) are defined as follows:

g(0)=0,g′(0)=1,g′(η)=0η→∞.(7)

In the above model, μ is the dynamic viscosity of the fluid, ρ is the fluid density, α = μρ is the kinematic viscosity of the fluid, H and β are the material parameters, b is the linear stretching velocity, Uω is the stretching velocity, and ε and δ are the non-dimensional fluid parameters.

In the next section, the solution to Eq. (6) will be approximated by the rational Legendre functions collocation method.

3 Preliminaries

In this section, the Legendre polynomials, the rational Legendre functions, and the QLM method are explained.

3.1 Rational Legendre functions

Legendre polynomials are orthogonal in the interval [–1, 1] with respect to the weight function w(x) = 1 and can be specified by the help of the following recurrence formula [24]:

       P0(x)=1,P1(x)=x,Pn+1(x)=(2n+1n+1)xPn(x)−(nn+1)Pn−1(x),n≥1.(8)

Legendre polynomials are orthogonal on the interval [–1,1]:

∫−11Pn(x)Pm(x)dx=22n+1δmn.

The new basis functions, denoted by RPn(x), are explained as:

RPn(x)=Pn(x−Lx+L),

where the constant parameter L sets the length scale of the mapping. Boyd in page 377 of his famous book [25] offered guidelines for optimizing the map parameter L for rational Chebyshev functions: “The experimental trial-and-error method”, that is also useful for rational Legendre functions.

The nth eigenfunction of the singular Sturm-Liouville problem is RPn(x):

(x+L)2L(x(RPn′(x))′+n(n+1)RPn(x)=0,x∈[0,∞],n=0,1,⋯

and by Eq. (8), they satisfy in the following recurrence relation:

RP0(x)=1,RP1(x)=x−Lx+L,RPn+1(x)=(2n+1n+1)(x−Lx+L)RPn(x)−(nn+1)RPn−1(x),n≥1.

3.2 The quasilinearization method

Quasilinearization method (QLM) is a technique to find a solution for nonlinear n-th order differential equation, ordinary or partial, in N dimensions as a limit of a sequence of linear differential equations [26]. Bellman and Kalaba expanded this method at first, which is based on the Newton-Raphson method [27]. Also, the best property of this approach is its quadratic convergence [28, 29, 30].

We suppose to have a nonlinear differential equation on domain [0, ∞) as follows:

d3ydη3=G(y″(η),y′(η),y(η),η),(9)

and the boundary conditions:

y(0)=A,    y′(0)=B,    y′(∞)=C,

where A, B, C are real constants.

The solution of Eq. (9) determines by using the QLM as a solution of the linear ODE at (i + 1)-th iterations as follows:

d3yi+1dη3=G(yi″,yi′,yi,η)+(yi+1−yi)Gy(yi″,yi′,yi,η)+(yi+1′−yi′)Gy′(yi″,yi′,yi,η)+(yi+1″−yi″)Gy″(yi″,yi′,yi,η),(10)

and the boundary conditions

yi+1(0)=A,    yi+1′(0)=B,    yi+1′(∞)=C,

where the functions

Gy=∂G∂y,    Gy′=∂G∂y′,    Gy″=∂G∂y″,

are functional derivatives of the function G(yn″,yn′,yn,η), and the prime explains the ordinary derivative with respect to η, and n = 0, 1, ⋯.

The QLM iteration needs an initialization or initial guess y0(η) that it is chosen from physical and mathematical considerations or the boundary conditions in the equation. Furthermore, Mandelzweig and Tabakin in their paper [26] have fully studied the convergence of the QLM and they showed that QLM is convergent from rate O(h2), and also Canuto et al. [31] have proved the stability and convergence analysis of spectral methods.

4 Application of the method

In this section, we solved the equation (6) by QLM based on rational Legendre collocation method.

First, by using QLM, the solution of Eq. (6) specifies the (i+1)-th iterative approximation gi+1(η) as a solution for the following linear ODE (i = 0, 1, ⋯):

gi+1‴(η)=G+(gi+1(η)−gi(η))dGdgi(η)+(gi+1′(η)−gi′(η))dGdgi′(η)+(gi+1″(η)−gi″(η))dGdgi″(η),(11)

in which G = gi′2(η)−gi(η)gi″(η)ε+1−εδgi″2(η).

Furthermore, the boundary conditions are defined as follows:

gi+1(0)=0,gi+1′(0)=1,gi+1′(∞)=0.(12)

At the beginning, we need an initial guess for the QLM, so we considered as g0(η) = η which satisfies some of the boundary conditions of the problem. Mandelzweig and Tabakin [26] have shown that if the initial guess is true in one of the boundary conditions then the QLM is convergent.

We approximate the function y(η) defined over a semi-infinite interval as follows:

y(η)=∑j=0∞ajRPj(η),(13)

where aj is the unknown coefficient. By truncating Eq. (13) in the N-th period, we have:

y(η)=∑j=0N−1ajRPj(η)=A(RP(η))T,(14)

where A = [a0, a1, ⋯, aN–1] and RP(η) = [RP0 (η),RP1(η), ⋯, RPN–1(η)]. Due to the Eq. (14), gi+1(η) can be defined as follows:

gi+1(η)=η+η2∑j=0N−1ajRPj(η).(15)

Eq. (15) satisfies the boundary conditions (Ritz method), other than g′i+1(∞) = 0. We consider a sufficiently large number M and suppose gi+1′(M)=0 to satisfy the last boundary condition. In this paper, we assumed M is equal to the largest root of RPN(η) on the domain (0, ∞). Eq. (15) is replaced in Eq. (11); so, the residual function is obtained.

Resi+1(η)=−gi+1‴(η)+G+(gi+1(η)−gi(η))dGdgi(η)+(gi+1′(η)−gi′(η))dGdgi′(η)+(gi+1″(η)−gi″(η))dGdgi″(η),(16)

where G = (gi′(η))2−gi(η)gi″(η)ε+1−εδ(gi″(η))2.

In this paper, the roots of RPN(η) are considered as the collocation points. By collocating the points in residual function, a system of N linear algebraic equations is obtained. We approximated the function g by solving this system.

5 Numerical results and discussion

In this part, the numerical simulation and the results are presented. Moreover, the effects of the non-dimensional fluid parameters on the derivatives of the dimensionless stream function are shown.

Tables 2 and 3 contain the values of g′(η) and g″(η) for different values of η by the different number of collocation points (20, 50, and 60) in the 20-th iteration.

Table 1

Obtained values of g′(η) for different values of N for iteration = 20 by RP functions.

ηN = 20N = 50N = 60
0111
0.20.838598989364891020.838598994446472510.83859899444647171
0.40.703401022012074200.703401032131495330.70340103213149372
0.60.590091710348996640.590091725600553940.59009172560055149
0.80.495090693711572980.495090714751516640.49509071475151330
1.00.415417644091646440.415417671123738920.41541767112373467
1.20.348586125216100880.348586158074340990.34858615807433578
1.40.292518333744076720.292518372427302950.29251837242729676
1.60.245475862256928080.245475907081098750.24547590708109156
1.80.206002996661902310.206003048029315730.20600304802930752
2.00.172879979335048990.172880037452816020.17288003745280679
2.20.145084299854233720.145084364620276360.14508436462026612
2.40.121758519033515450.121758590105315140.12175859010530392
2.60.102183452570169560.102183529520741030.10218352952072884
2.80.857557780950001030.085755860554018510.08575586055400539
3.00.071969309513718330.071969397237662210.07196939723764819
3.20.060399321999074350.060399414864763100.06039941486474821
3.40.050689420953985580.050689518901362600.05068951890134690
3.60.042540536247823720.042540639206965440.04254063920694898
3.80.035701694261036990.035701802091993990.03570180209197681
4.00.029962278481391030.029962390942480020.02996239094246217
4.20.025145537298384960.025145654045739130.02514565404572066
4.40.021103137276666250.021103257888806240.02110325788878719
4.60.017710593112708720.017710717129531300.01771071712951171
4.80.014863432907973760.014863559873149630.01486355987312956
5.00.012473980291749020.012474109789681980.01247410978966146

Table 2

Obtained values of g″(η) for different values of N for iteration = 20 by RP functions.

ηN = 20N = 50N = 60
0-0.88078499747125922-0.88078497244601874-0.88078497244602272
0.2-0.73755704202254789-0.73755701571027815-0.73755701571028217
0.4-0.618003787858089621-0.61800376335769558-0.61800376335769972
0.6-0.518061915396319429-0.51806188800884449-0.51806188800884880
0.8-0.434423093230810111-0.43442306325249482-0.43442306325249933
1.0-0.364372384768658295-0.36437235516508404-0.36437235516508873
1.2-0.305668577673621639-0.30566854883410831-0.30566854883411316
1.4-0.256453314708564489-0.25645328499869334-0.25645328499869830
1.6-0.215180623218004124-0.21518059144444689-0.21518059144445195
1.8-0.180561289199228789-0.18056125571972253-0.18056125571972762
2.0-0.151518308569813979-0.15151827481996751-0.15151827481997256
2.2-0.127150795231806599-0.12715076270648465-0.12715076270648965
2.4-0.106704476071187646-0.10670444560744341-0.10670444560744831
2.6-0.089547404015357987-0.08954737563760644-0.08954737563761120
2.8-0.075149862220013972-0.07514983540086583-0.07514983540087041
3.0-0.063067671647562489-0.06306764571285020-0.06306764571285460
3.2-0.052928285960463070-0.05292826042390378-0.05292826042390796
3.4-0.044419184026325635-0.04441915875546242-0.04441915875546637
3.6-0.037278165624955666-0.03727814083905698-0.03727814083906069
3.8-0.031285229374198879-0.03128520552943696-0.03128520552944044
4.0-0.026255769535789220-0.02625574716651940-0.02625574716652264
4.2-0.022034874327473658-0.02203485389997206-0.02203485389997506
4.4-0.018492545485008100-0.01849252729709350-0.01849252729709627
4.6-0.015519689089803195-0.01551967322725212-0.01551967322725466
4.8-0.013024752557373998-0.01302473890149214-0.01302473890149447
5.0-0.010930903247790180-0.01093089151937285-0.01093089151937498

In Fig. 2, the graphs of the profile of the g(η), g′(η) and g″(η) functions for various values of η and ε = 0.3, L = 10 and δ = 0.1 are shown. An approximation of the optimal value of L is calculated by the experimental trial-and-error method which is provided in page 377 from Ref. [25].

Fig. 2 The g(η), g′(η), g″(η) functions for N = 60 and iteration = 20.
Fig. 2

The g(η), g′(η), g″(η) functions for N = 60 and iteration = 20.

Fig. 3 demonstrates the logarithm residual errors for N = 20, 50 and 60 in the 20-th iteration. We can see that by increasing the number of collocation points, the values of residual errors are decreasing. Since the residual errors are very close to zero, it is obtained that the presented method has a high accuracy, and also the method is convergent and that means by increasing collocation points the values of residual errors tends to zero.

Fig. 3 The logarithm residual errors for RP for N = 20, 50, and 60 and iteration = 20.
Fig. 3

The logarithm residual errors for RP for N = 20, 50, and 60 and iteration = 20.

In Fig. 4, the effects of the value of ε on g′(η), g″(η) with δ = 0.1 are shown.

Fig. 4 The effect of the value ofε on g′(η),g″ (η) (δ = 0.1]
Fig. 4

The effect of the value ofε on g′(η),g″ (η) (δ = 0.1]

Fig. 5 The effects of the value of δ on g′(η), g″(η) (ε = 0.3).
Fig. 5

The effects of the value of δ on g′(η), g″(η) (ε = 0.3).

6 Conclusion

In this paper, the rational Legendre functions (RLFs) collocation method has been presented for solving a nonlinear ordinary differential equation in a semi-infinite interval. The explained problem is the boundary layer flow of a non-Newtonian fluid over a linear stretching sheet. The quasilinearization method is discussed to convert the equation to a sequence of linear differential equations. This sequence is solved by the rational Legendre collocation method. The residual errors and the results indicate the highly admitted accuracy and convergence of the proposed method. The present method has some advantages, for example:

  • The effects of non-Newtonian fluid material parameters are investigated on the velocity and stress profiles of a non-Newtonian Eyring-Powell fluid over a linearly stretching sheet.

  • The effects of Powell-Eyring fluid parameters on velocity and stress profiles are quite opposite.

  • The roots of the RLFs were used in the interval [0, ∞) as collocation points to solve the equation and do not reform the problem to a finite domain.

  • An approximation of the optimal value of L is calculated in the rational Legendre functions RPnα(x,L) to solve the equation.

  • The convergence of the obtained results is shown with the reduction of the residual errors.

  • An accurate solution for g(η), g′(η) and g″(η) is obtained.

  • This method is a good experience and method for the other sciences.

A The quasilinearization method

Most algorithms for solving nonlinear equations or systems of equations are variations of Newton’s method. When we apply the collocation method for solving a nonlinear differential equation, the result is a nonlinear system of algebraic equations. The aim of the quasilinearization method (QLM) pioneered by Bellman and Kalaba [27] for solving the n-order nonlinear differential equations in N dimensions as the limit of a sequence of linear differential equations. The general properties of the method, particularly its uniform and quadratic convergence, which often also is monotonic [26]. The QLM is easily understandable since there is no difficult technique for obtaining the general solution of a nonlinear equation in terms of a finite set of particular solutions.

To know how the QLM works, imagine that we have one nonlinear ordinary differential equation in one variable on the interval [0, b] as [26, 27]:

L(n)y(x)=f(y(x),y(1)(x),⋯,y(n−1)(x),x),(17)

with n boundary conditions

gk(y(0),y(1)(0),⋯,y(n−1)(0))=0,    k=1,⋯,lgk(y(b),y(1)(b),⋯,y(n−1)(b))=0,    k=l+1,⋯,n,

where L(n) is n-order linear ordinary differential operator and f and g1, g2, ⋯, gn are nonlinear functions of y(x) and its (n+1) derivatives y(s)(x), s = 1, ⋯,n. The QLM determines the (r + 1)th iterative approximation yr+1(x) to the solution of Eq. (17) as a solution of the linear differential equation:

L(n)yr+1(x)=f(yr(x),yr(1)(x),⋯,yr(n−1)(x),x)+∑s=0n−1(yr+1(s)(x)−yr(s)(x))fy(s)(yr(x),yr(1)(x),⋯,yr(n−1)(x),x),

where yr(0) (x) = yr(x), with linearized two-point boundary conditions:

∑s=0n−1(yr+1(s)(0)−yr(s)(0))gky(s)(yr(0),yr(1)(0),⋯,yr(n−1)(0),0),     k=1,⋯,l

and

∑s=0n−1(yr+1(s)(b)−yr(s)(b))gky(s)(yr(b),yr(1)(b),⋯,yr(n−1)(b),b),     k=l+1,⋯,n,

where the functions fy(s) = ∂f∂y(s) and gky(s) = ∂gk∂y(s)s = 1,2, ⋯,n–1 are functional derivatives of the functionals f(yr(x), yr(1) (x), ⋯, yr(n−1) (x),x) and gk(yr(x), yr(1) (x), ⋯, yr(n−1) (x),x), respectively. The QLM iteration needs an initialization or initial guess y0(η) that it is chosen from physical and mathematical considerations or the boundary conditions in the equation.

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Received: 2018-01-06
Revised: 2018-06-01
Accepted: 2018-06-07
Published Online: 2018-07-12
Published in Print: 2019-01-28

© 2019 K. Parand et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

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  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
  5. Effect of Heterogeneity on Imbibition Phenomena in Fluid Flow through Porous Media with Different Porous Materials
  6. Electro-osmotic flow of a third-grade fluid past a channel having stretching walls
  7. Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation
  8. Local convergence for an eighth order method for solving equations and systems of equations
  9. Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions
  10. Influence of Non-linear Boussinesq Approximation on Natural Convective Flow of a Power-Law Fluid along an Inclined Plate under Convective Thermal Boundary Condition
  11. A reliable analytical approach for a fractional model of advection-dispersion equation
  12. Mass transfer around a slender drop in a nonlinear extensional flow
  13. Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study
  14. A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation
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  24. Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load
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  27. Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness
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  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
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  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
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https://doi.org/10.1515/nleng-2018-0005
Keywords for this article
Boundary Layer Flow; Eyring-Powell Non-Newtonian Fluid; Rational Legendre Functions; Quasilinearization Method
Creative Commons
BY 4.0

Articles in the same Issue

  1. Chebyshev Operational Matrix Method for Lane-Emden Problem
  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
  5. Effect of Heterogeneity on Imbibition Phenomena in Fluid Flow through Porous Media with Different Porous Materials
  6. Electro-osmotic flow of a third-grade fluid past a channel having stretching walls
  7. Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation
  8. Local convergence for an eighth order method for solving equations and systems of equations
  9. Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions
  10. Influence of Non-linear Boussinesq Approximation on Natural Convective Flow of a Power-Law Fluid along an Inclined Plate under Convective Thermal Boundary Condition
  11. A reliable analytical approach for a fractional model of advection-dispersion equation
  12. Mass transfer around a slender drop in a nonlinear extensional flow
  13. Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study
  14. A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation
  15. An effective technique for the conformable space-time fractional EW and modified EW equations
  16. Fractional variational iteration method for solving time-fractional Newell-Whitehead-Segel equation
  17. New exact and numerical solutions for the effect of suction or injection on flow of nanofluids past a stretching sheet
  18. Numerical investigation of MHD stagnation-point flow and heat transfer of sodium alginate non-Newtonian nanofluid
  19. A New Finance Chaotic System, its Electronic Circuit Realization, Passivity based Synchronization and an Application to Voice Encryption
  20. Analysis of Heat Transfer and Lifting Force in a Ferro-Nanofluid Based Porous Inclined Slider Bearing with Slip Conditions
  21. Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model
  22. Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations
  23. MHD nonaligned stagnation point flow of second grade fluid towards a porous rotating disk
  24. Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load
  25. Swirling flow of couple stress fluid due to a rotating disk
  26. MHD stagnation point slip flow due to a non-linearly moving surface with effect of non-uniform heat source
  27. Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness
  28. Nonhomogeneous porosity and thermal diffusivity effects on stability and instability of double-diffusive convection in a porous medium layer: Brinkman Model
  29. Magnetohydrodynamic(MHD) Boundary Layer Flow of Eyring-Powell Nanofluid Past Stretching Cylinder With Cattaneo-Christov Heat Flux Model
  30. On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain
  31. An adaptive mesh method for time dependent singularly perturbed differential-difference equations
  32. On stretched magnetic flow of Carreau nanofluid with slip effects and nonlinear thermal radiation
  33. Rational exponential solutions of conformable space-time fractional equal-width equations
  34. Simultaneous impacts of Joule heating and variable heat source/sink on MHD 3D flow of Carreau-nanoliquids with temperature dependent viscosity
  35. Effect of magnetic field on imbibition phenomenon in fluid flow through fractured porous media with different porous material
  36. Impact of ohmic heating on MHD mixed convection flow of Casson fluid by considering Cross diffusion effect
  37. Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine
  38. Surface roughness effect on thermohydrodynamic analysis of journal bearings lubricated with couple stress fluids
  39. Convective conditions and dissipation on Tangent Hyperbolic fluid over a chemically heating exponentially porous sheet
  40. Unsteady Carreau-Casson fluids over a radiated shrinking sheet in a suspension of dust and graphene nanoparticles with non-Fourier heat flux
  41. An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering
  42. New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind
  43. Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating
  44. Analysis of Bifurcation and Chaos of the Size-dependent Micro–plate Considering Damage
  45. Non-Similar Comutational Solutions for Double-Diffusive MHD Transport Phenomena for Non-Newtnian Nanofluid From a Horizontal Circular Cylinder
  46. Mathematical model on distributed denial of service attack through Internet of things in a network
  47. Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect
  48. Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation
  49. MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink
  50. A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating
  51. Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
  52. Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
  53. Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability
  54. Thermomechanical nonlinear stability of pressure-loaded CNT-reinforced composite doubly curved panels resting on elastic foundations
  55. Combination synchronization of fractional order n-chaotic systems using active backstepping design
  56. Vision-Based CubeSat Closed-Loop Formation Control in Close Proximities
  57. Effect of endoscope on the peristaltic transport of a couple stress fluid with heat transfer: Application to biomedicine
  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
  64. A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations
  65. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses
  66. The expa function method and the conformable time-fractional KdV equations
  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
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