8
Fluid Dynamics Problems in Uncertain Environment

Perumandla Karunakar1, Uddhaba Biswal2, and Snehashish Chakraverty2

1Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Chennai, Tamil Nadu, 601103, India

2Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, 769008, India

8.1 Introduction

In general, fluid flow between two plates is one of the important problems in the field of fluid dynamics. The study of fluid flow in convergent and divergent channels is very important because of their vast application in engineering and industrial field such as heat exchangers for milk flowing, cold drawing operation in polymer industry, etc. Many authors have shown interest to study two‐dimensional incompressible fluid flow between two inclined planes. Jeffery [1] and Hamel [2] were the first persons to discuss about this problem, so it is known as Jeffery–Hamel problem. Jeffery–Hamel problems for heat transfer fluids have been studied by different authors by using different numerical methods. However, the heat transfers fluids, viz., water, oil, and ethylene glycol mixture, are poor heat transfer fluids. With the increasing global competition, industries have a strong need to develop advanced heat transfer fluid with significantly higher thermal conductivity than that of water, oil, etc. It is well known that metals in room temperature have higher thermal conductivity than that of fluids. Moreover, the thermal conductivity of metallic liquids is much greater than that of nonmetallic liquids. Therefore, the fluids that contain suspended metallic particles could be expected to have higher thermal conductivity than that of existing pure fluids. In 1995, Choi [3] have proposed a new class of fluid, termed as nanofluid, i.e. a fluid with suspended nanoparticles in the base fluids and reported that the nanofluids have superior thermal properties as compared to the base fluids. As such, in recent years, Jeffery–Hamel problems for nanofluids have taken attention of many researchers.

In fluid dynamics, most of the problems are related to nonlinear differential equations. Because of this nonlinearity, they do not admit analytical solution. In such situations, researchers relay on numerical methods. Moghimi et al. [4] presented the application of homotopy analysis method (HAM) to Jeffery–Hamel problem for fluid flow between two nonparallel planes. Esmaeilpour and Ganji [5] have used optimal homotopy asymptotic method to analyze the solution of fluid flow between two rigid nonparallel planes. Rostami et al. [6] have used two methods, namely, homotopy perturbation method (HPM) and Akbari–Ganji's method, to solve Jeffery–Hamel problem in the presence of magnetic field and nanoparticle. Differential transformation method (DTM) has been used by Umavathi and Shekar [7] to study Jeffery–Hamel flow of nanofluid with magnetic effect. Hatami and Ganji [8] used weighted residual methods, viz., Galerkin's, least square, and collocation methods to investigate Jeffery–Hamel flow in the presence of magnetic field and nanoparticles. Moradi et al. [9] have used DTM to study the effect of nanoparticle on Jeffery–Hamel flow in the absence of magnetic field.

From the literature review, Jeffery–Hamel problem with nanofluid has been investigated for crisp case only. It is worth mentioning here that every particular application of nanofluid depends on the physical properties of nanoparticle as well as base fluid, viz., effective viscosity, thermal conductivity, density, etc., and value of these parameters depend on the value of nanoparticle volume fraction. There exist few models for these parameters of nanofluid derived by different researchers with the assumption that nanoparticles are uniformly distributed over the considered base fluid. However, in practical case, it may not be possible. Therefore, nanoparticle volume fraction may be taken as an uncertain parameter. As per our knowledge, there is no paper dealing with the Jeffery–Hamel flow for an uncertain case. In this regard, we are motivated to investigate Jeffery–Hamel problem in an uncertain environment, which makes it more challenging. Here, nanoparticle volume fraction is taken as an uncertain parameter in term of fuzzy number. A semianalytical method known as HPM has been used to solve the governing fuzzy differential equation related to the Jeffery–Hamel problem for nanofluid.

Shallow water wave (SWW) equations are widely used for many physical phenomenon such as simulation of tsunami‐wave propagation, shock waves, tidal flows, and coastal waves. To describe SWW, many models have been introduced, such as Boussinesq equations, Korteweg–de Vries (KdV) equations, and Kadomtsev–Petviashvili (KP) equations, etc. In this regard, another set of coupled differential equations that describe SWW are coupled Whitham–Broer–Kaup (CWBK) equations. These are given by Whitham [10], Broer [11], and Kaup [12]. Further, many works are reported for solving CWBK equations. As such, Xie et al. [13] obtained four pair of solutions of CWBK equations using hyperbolic function method. New generalized transformation has been proposed to find the exact solutions of CWBK equations in shallow water by Yan and Zhang [14]. Yıldırım et al. [15] and Ganji et al. [16] applied HPM to find explicit and numerical travelling wave solutions of CWBK equations. Approximate travelling wave solutions of CWBK equations with the help of reconstruction of variational iteration method (RVIM) have been obtained in [17].

In the above‐discussed literature, the constants related to diffusion power are crisp numbers, but they may not be crisp always because these are measured values. To handle these involved uncertainties here, we have considered them as interval numbers, which transform the governing CWBK equations to interval CWBK equations. Solving interval differential equation is a challenging task. As such, in this chapter, homotopy perturbation transform method (HPTM) [1822] has been applied to handle uncertain differential equations with the help of parametric concept.

8.2 Preliminaries

In this section, we will discuss some basic concept of interval/fuzzy theory and some notations that we have used in further discussion.

8.2.1 Fuzzy Set

A fuzzy set images is a set consisting of ordered pairs of the elements s of a universal set say S and their membership value is written as [2325]

equation

where m(s) is a defined membership function for the fuzzy set images.

8.2.2 Fuzzy Number

Fuzzy number is a fuzzy set that is convex, normalized, and defined on real line R. Moreover, its membership function must be piecewise continuous. There are different types of fuzzy numbers based on membership function, viz., triangular, Gaussian, quadratic, exponential fuzzy number, etc. Here, we have used triangular fuzzy number (TFN) and membership function of a TFN images is defined as [2325]

equation

8.2.3 δ‐Cut

δ‐Cut of a fuzzy set is defined as the crisp set given by images.

By using δ‐cut, TFN images may be converted into interval form as [24,25] images.

8.2.4 Parametric Approach

In general, an interval images may be transformed into crisp form by the help of parametric concept as [24,25]

equation

where γ is a parameter that lies in the closed interval [0, 1].

It can also be written as images, where images is the radius of I.

8.3 Problem Formulation

In this section, two problems, namely, Jeffery–Hamel problem and CWBK shallow water equations, have been discussed.

As such, Figure 8.1 represents the schematic diagram of the Jeffery–Hamel problem where two plane rigid walls are inclined at an angle of 2ω. Let us consider a system of cylindrical polar coordinates (r, θ, z) in which steady two‐dimensional flow of an incompressible conducting viscous fluid from a source or sink at channel forms by the nonparallel walls. The plane walls are considered to be convergent if ω < 0 and divergent if ω > 0. We have assumed purely radial motion that has no change in the flow parameter along the z‐axis. Here, the flow depends on r and θ so that v = (u(r, θ), 0), and moreover, it is assumed that there is no magnetic effect on the z‐direction. Now, the continuity equation, Navier–Stokes equation, and Maxwell equations in polar coordinate may be written as

(8.1)equation

where u(r, θ) is the velocity, P is the fluid pressure, B0 is the electromagnetic induction, σ is the conductivity of the fluid, and ρnf and υnf stand for effective density and kinematic viscosity of nanofluid, respectively.

Schematic diagram of two plane rigid walls inclined at an angle of 2ω depicting two-dimensional flow of an incompressible conducting viscous fluid from a source or sink at channel forms by the
nonparallel walls.

Figure 8.1 Geometry of the problem.

The boundary conditions are

  • at the center line of the channel: images, and
  • at the boundary of the channel: u = 0.

The effective density and kinematic viscosity of nanofluid may be given as [9]

where φ stands for nanoparticle volume fraction. μnf is the effective dynamic viscosity of nanofluid, and by Brinkman's model [26], it may be given as

(8.5)equation

Considering only radial flow, from continuity equation, we may have

(8.6)equation

Now introduce dimensionless degree as images and dimensionless form of velocity parameter may be obtained as

By using Eqs. (8.4)(8.7) and eliminating the pressure term from Eqs. (8.2) and (8.3), nondimensional governing ordinary differential equation for the Jeffery–Hamel problem may be obtained as

where Re denotes the Reynolds number, Ha stands for the Hartmann number based on electromagnetic parameter, and A1 is the ratio of effective density of nanofluid to density of base fluid. These parameters are introduced as

equation

and the reduced boundary conditions are

(8.9)equation

Here, these boundary conditions mean that maximum values of velocity are obtained at centerline η = 0, and hence, its derivative will be zero at that point. F(1) = 0 says that no‐slip condition at boundary is considered.

Next, CWBK equations that describe SWW equations which read as [1017]

Table 8.1 Some properties of the considered fluid and nanoparticles [27].

Material Symbol Density (kg/m3) Thermal conductivity (W/m K)
Copper Cu 8933 401
Sodium alginate SA 989 0.6376

It may be worth mentioning here that a small change in the value of nanoparticle volume fraction may affect the numerical solution of the considered problem. Similarly, in the case of CWBK equations, the constants α and β representing diffusion power may also not be crisp always. As such, we are motivated to handle such challenging problems in uncertain environment by considering volume fraction φ as a fuzzy number and α and β as interval numbers.

Accordingly, fuzzy form of the governing differential equation (8.8) may be written as

with boundary conditions as

(8.13)equation

here, “∼” represents the fuzzy form.

And the interval form of CWBK equations is

Some physical properties of base fluid and nanoparticle have been presented in Table 8.1 [27].

8.4 Methodology

In this section, two efficient semianalytical methods, viz., HPM and HPTM, which are useful in solving the above‐discussed fluid‐related problems, have been briefly illustrated.

8.4.1 Homotopy Perturbation Method

To delineate briefly the idea of HPM, let us consider the differential equation [2730]

with given boundary condition

where A is a differential operator that can be divided into two parts, viz., linear (L) and nonlinear (N), B stands for boundary operator, f(x) is a known analytical function, and Γ is the boundary of the domain Ω.

By splitting A into linear and nonlinear part, Eq. (8.16) may be written as

(8.18)equation

Now, we construct a homotopy v(r, q) :  Ω × [0, 1] → R satisfying

where q is an embedding parameter lies between 0 and 1 and u0 is an initial approximation satisfying boundary condition Eq. (8.17).

From Eq. (8.19), one may observe that

when q = 0, L(v) = L(u0) and for q = 1, A(v) − f(x) = 0 that is, when q converges to 1, we may get approximate solution of Eq. (8.16).

As q is a small parameter, the solution of Eq. (8.19) can be expressed as a power series in q

equation

By setting q = 1 results the best approximation of Eq. (8.16) that is

equation

8.4.2 Homotopy Perturbation Transform Method

Here, Laplace transform method and HPM are combined to have a method called HPTM for solving nonlinear differential equations. It is also called as Laplace homotopy perturbation method (LHPM).

Let us consider a general nonlinear partial differential equation with source term g(x, t) to illustrate the basic idea of HPTM as follows [1822]

subject to initial conditions

(8.21)equation

where D is the linear differential operator images, R is the linear differential operator whose order is less than that of D, and N is the nonlinear differential operator.

The HPTM methodology consists of mainly two steps. The first step is applying Laplace transform on both sides of Eq. (8.20) and the second step is applying HPM where decomposition of nonlinear term is done using He's polynomials.

First by operating Laplace transform on both sides of (8.20), we obtain

equation

Assuming that D is a second‐order differential operator and using differentiation property of Laplace transform, we get

Applying inverse Laplace transform on both sides of (8.22), we have

where G(x, t) is the term arising from first three terms of right‐hand side of (8.22).

Next, to apply HPM, first, we need to assume the solution as a series that contains embedding parameter p ∈ [0, 1] as

and the nonlinear term may be decomposed using He's polynomials as

where Hn(u) represents the He's polynomials [1822] which are defined as follows:

(8.26)equation

Substituting Eqs. (8.24) and (8.25) in Eq. (8.23) and combining Laplace transform with HPM, one may obtain the following expression:

Comparing the coefficients of like powers of “p” on both sides of (8.27), we may obtain the following successive approximations:

equation

Finally, the solution of the differential equation (8.20) may be obtained as follows:

(8.28)equation

In the next session, we apply HPM and HPTM to governing equations of Jeffery–Hamel problem and CWBK equation in both crisp and uncertain environments.

8.5 Application of HPM and HPTM

8.5.1 Application of HPM to Jeffery–Hamel Problem

To handle fuzziness involved in Eq. (8.12), first we have used δ‐cut to convert the fuzzy differential equation given in (8.12) into interval form and then parametric approach may be used to convert the interval form into crisp form.

For simplicity, by putting images and images in Eq. (8.12), we may have the fuzzy differential equation as

Now, by using δ‐cut for fuzzy form, Eq. (8.29) may be converted into interval form as

Further, by introducing parametric concept for involved intervals in differential equation (8.30), we may have its crisp form as

where δ and γ are parameters that lie between 0 and 1.

Let us denote

equation

and

equation

By using these notations, Eq. (8.31) may be written as

with boundary conditions F(0) = 1, F(0) = 0, F(1) = 0.

Now, we apply HPM to solve Eq. (8.32). Homotopy for Eq. (8.32) may be constructed as

According to this method, assumed series solution of Eq. (8.33) may be written as

Afterward, our goal is to find the unknown functions used in the assumed series solution (8.34), viz., F0(γ, δ, η), F1(γ, δ, η), F2(γ, δ, η), F3(γ, δ, η)….

Substituting Eq. (8.34) in Eq. (8.33) and collecting the coefficients of various powers of q, we may get

Further, by equating the coefficient of various powers of q with zero and using proper boundary conditions, we may get the functions F0(γ, δ, η), F1(γ, δ, η), F2(γ, δ, η), F3(γ, δ, η)… explicitly.

Equating the coefficient of q0 in Eq. (8.35) to zero, we may have

with boundary conditions as images.

By solving Eq. (8.36) with boundary conditions, we may obtain

(8.37)equation

Further, by equating the coefficient of q1 with zero, we may have

equation

with boundary conditions as images.

By solving Eq. (8.38) with boundary conditions, we may obtain

(8.39)equation

Again from coefficient of q2 with boundary conditions as images, we may obtain

(8.40)equation

By proceeding like this for coefficients of various powers of q with appropriate boundary conditions, we may have the functions F3(γ, δ, η), F4(γ, δ, η), F5(γ, δ, η)….

Hence, the approximate solution of Eq. (8.12) may be given as

Here, three‐term approximate solution of Eq. (8.12) may be found as

equation

where F0(γ, δ, η), F1(γ, δ, η), and F2(γ, δ, η) are given in Eqs., respectively.

It may be noted that by increasing the number of terms, one may get more appropriate approximate results. Moreover, here, the fuzzy solutions are control by the parameters δ and γ.

8.5.2 Application of HPTM to Coupled Whitham–Broer–Kaup Equations

Now, we use HPTM to find the solution of CWBK equations given in 8.10,8.11 [1017] subject to initial conditions as [15,16]

(8.42)equation
(8.43)equation

Applying the Laplace transform to Eqs. (8.10) and (8.11), we get

Simplifying (8.44) and (8.45), we obtain

Taking inverse Laplace transform on both sides of (8.46) and (8.47), we may have

Now applying HPM to (8.48) and (8.49) gives

where H1n(u) and H2n(uv) are He's polynomials for the nonlinear terms uux and (uv)x, respectively. First, few terms of H1n(u) and H2n(uv) are given by

(8.52)equation
(8.53)equation

Comparing the coefficients of like powers of p on both sides of Eqs. (8.50) and (8.51), we obtain

(8.54)equation
(8.55)equation
(8.56)equation
(8.57)equation
equation

where

equation

Solutions of CWBK equations (8.10) and (8.11) may be obtained as

Exact solutions of CWBK equations (8.10) and (8.11) may be given as [15,16]

(8.60)equation
(8.61)equation

Next, we solve interval CWBK equations (8.14) and (8.15) using HPTM with the help of parametric approach defined above. In this regard, first, we transform the interval CWBK equations to crisp form using parametric approach, and then, these transformed equations have been solved using HPTM.

The interval numbers images and images may be written in crisp form using parametric approach as

(8.62)equation

Further, for simplicity denoting

Similarly, interval solutions images and images of (8.14) and (8.15) may be denoted as

Using Eqs. (8.63) and (8.64), the interval CWBK equations (8.14) and (8.15) may be transformed to crisp form as

subject to initial conditions

Next, we apply HPTM to (8.65) and (8.66) subject to initial conditions (8.67) and (8.68).

Applying Laplace transform to Eqs. (8.65) and (8.66), we obtain

Simplifying (8.69) and (8.70), we get

Taking inverse Laplace transform on both sides of (8.71) and (8.72) will give

(8.73)equation
(8.74)equation

Next, applying HPM, we get

(8.75)equation
(8.76)equation

where H3n and H4n are He's polynomials like H1n and H2n.

Next, proceeding like crisp case, we may obtain the solutions of (8.14) and (8.15) as

8.6 Results and Discussion

Results obtained for Jeffery–Hamel problem and CWBK equations have been discussed in this section. Also, the solutions obtained by the present methods, viz., HPM and HPTM, are validated by comparing with existing/exact solutions in special cases.

Table 8.2 presents the solution of Eq. (8.12) for different numbers of terms of the series solution (8.41) when images, and δ = γ = 0. It is well known that residual error is defined as the error obtained by substitute approximate solution in the governing differential equation. Figure 8.2 depicts the residual error plots for different numbers of terms in the approximate series solution (8.41). In Table 8.3, we have compared the present result with the existing result by DTM when Ha = 0,  Re  = 50, ω = 50, φ = 0.

Table 8.2 Velocity profile when Ha = 0,  Re  = 110, ω = 30, φ = 0 by taking different numbers of terms in series solution.

F(γ, δ, η) in different numbers of terms in series solution (8.41)
η 2 terms 3 terms 4 terms 5 terms 6 terms 7 terms
0   1.000 000 00 1.000 000 00 1.000 000 00 1.000 000 00 1.000 000 00 1.000 000 00
0.1   0.987 396 04 0.986 785 04 0.986 819 50 0.986 809 78 0.986 810 40 0.986 810 53
0.2   0.950 007 65 0.947 670 97 0.947 833 59 0.947 793 01 0.947 795 51 0.947 795 99
0.3   0.889 071 54 0.884 182 46 0.884 631 18 0.884 531 89 0.884 537 57 0.884 538 59
0.4   0.806 536 55 0.798 643 03 0.799 613 72 0.799 412 91 0.799 423 44 0.799 425 16
0.5   0.704 895 19 0.693 876 84 0.695 667 03 0.695 297 16 0.695 315 62 0.695 318 30
0.6   0.586 947 69 0.572 843 40 0.575 774 80 0.575 137 28 0.575 169 95 0.575 174 12
0.7   0.455 498 74 0.438 248 76 0.442 623 66 0.441 592 19 0.441 650 02 0.441 656 68
0.8   0.312 986 75 0.292 176 24 0.298 237 97 0.296 676 10 0.296 773 43 0.296 784 57
0.9   0.161 045 77 0.135 769 96 0.143 660 04 0.141 455 87 0.141 603 05 0.141 623 09
0.0 −0.000 000 0 −0.031 017 12 −0.021 335 46 −0.024 215 76 −0.024 028 34 −0.023 989 46
Chart depicting the residual error plots for different numbers of terms in the approximate series solution.

Figure 8.2 Residual error plot for different numbers of terms in series solution (8.41) when Ha = 2000,  Re  = 50, ω = 30, φ = 0.01.

Table 8.3 Comparison of velocity profile of present results with existing results by DTM when Ha = 0, Re  = 50, ω = 50, φ = 0.

Present results for velocity in different numbers of terms in (8.41) Existing result
η 3 terms 4 terms 5 terms 6 terms 7 terms DTM [9]
0   1.0   1.0   1.0   1.0   1.0   1.0
0.1   0.982 614 12   0.982 398 66   0.982 416 1   0.982 445 74   0.982 452 04   0.982 431
0.2   0.931 812 53   0.931 082 99   0.931 168 73   0.931 283 62   0.931 306 24   0.931 226
0.3   0.851 479 9   0.850 263 83   0.850 496 56   0.850 737 31   0.850 779 2   0.850 611
0.4   0.747 505 94   0.746 172 58   0.746 627 58   0.747 003 2   0.747 058 09   0.746 791
0.5   0.626 998 88   0.626 077 02   0.626 767 86   0.627 244 49   0.627 299 38   0.626 948
0.6   0.497 356 84   0.497 240 89   0.498 079 88   0.498 587 82   0.498 629 56   0.498 234
0.7   0.365 347 54   0.366 050 3   0.366 867 81   0.367 326 52   0.367 348 59   0.366 966
0.8   0.236 359 72   0.237 463 79   0.238 083 44   0.238 427 36   0.238 432 12   0.238 124
0.9   0.113 978 73   0.114 830 55   0.115 147 47   0.115 334 15   0.115 330 53   0.115 152
0.0 −0.000 000 0 −0.000 000 0 −0.000 000 0 −0.000 000 0 −0.000 000 0 −0.000 000 0

Velocity profiles of SA–Cu nanofluid with different values of nanoparticle volume fraction and fixed values Ha = 500,  Re  = 50, ω = 50 have shown in Figure 8.3. Figure 8.4 present the effect of Hartmann number on nondimensional velocity of SA–Cu nanofluid flow between the inclined planes. Effect of Reynolds number on dimensionless velocity has been displayed in Figure 8.5. Further, Figure 8.6 represents fuzzy plots of velocity profile for different values of η when nanoparticle volume fraction is a TFN considered as images and Ha = 100,  Re  = 50, ω = 30. Moreover, by putting δ = 0, γ = 0 in F(γ, δ, η), we may get lower bound of the nondimensional velocity, whereas by putting δ = 0, γ = 1, we may get an upper bound for different values of η. Solution bounds for velocity profile when Ha = 100,  Re  = 50, ω = 30, and images are given in Table 8.4.

Chart depicting the velocity profiles of SA–Cu nanofluid with different values of nanoparticle volume fraction and fixed values.

Figure 8.3 Effect of nanoparticle volume fraction on velocity profile for SA–Cu nanofluid when Ha = 500,  Re  = 50, ω = 50.

Chart depicting the effect of Hartmann number
on nondimensional velocity of SA–Cu nanofluid flow between inclined planes.

Figure 8.4 Effect of Hartmann number on the velocity profile for SA–Cu nanofluid when Re = 50, ω = 50, φ = 0.1.

Chart depicting the effect of Reynolds number on the dimensionless velocity profile for SA–Cu nanofluid.

Figure 8.5 Effect of Reynolds number on the velocity profile for SA–Cu nanofluid when Ha = 50, ω = 50, φ = 0.1.

Charts representing fuzzy plots of velocity profile for SA–Cu nanofluid for different values of η when nanoparticle volume fraction is a TFN.

Figure 8.6 Fuzzy plot of the velocity profile for SA–Cu nanofluid when images and Ha = 100,  Re  = 50, ω = 30. (a) η = 0.2, (b) η = 0.4, (c) η = 0.6, and (d) η = 0.8.

Table 8.4 Solution bounds for SA–Cu nanofluid when Ha = 100, Re  = 50, ω = 30, and images.

η Velocity (lower bound) Velocity (upper bound)
0.0 1.0 1.0
0.1 0.985 773 307 662 548 0.987 740 276 941 951
0.2 0.943 859 937 040 970 0.951 341 473 381 224
0.3 0.876 452 283 883 681 0.891 904 052 869 956
0.4 0.786 868 265 777 731 0.811 132 250 151 742
0.5 0.679 120 068 532 976 0.711 157 455 666 914
0.6 0.557 432 029 106 776 0.594 323 008 426 450
0.7 0.425 783 780 430 919 0.462 953 665 930 697
0.8 0.287 532 866 251 400 0.319 129 944 261 829
0.9 0.145 141 078 684 186 0.164 480 872 391 153
1.0 0.000 000 0.000 000

HPTM solutions (8.58) and (8.59) of CWBK equations given in (8.10) and (8.11) are compared with exact solutions (8.60) and (8.61) for α = 0, β = 0.5, k = 0.2, L = 0.005, and t = 1 in Figure 8.7, whereas Figure 8.8 compares the same for α = 0.5, β = 0, k = 0.2, L = 0.005, and t = 1. Solutions (8.77) and (8.78) of the interval CWBK equations (8.14) and (8.15) are presented in Figure 8.9 for α = [0.1, 0.9], β = 0, k = 0.2, L = 0.005, and t = 0.5. Finally, Figure 8.10 shows the interval solutions (8.77) and (8.78) for α = 0, β = [0.1, 0.9], k = 0.2, L = 0.005, and t = 0.5.

Charts depicting the comparison of solution of CWBK equations (8.10) and (8.11) with exact solution for α = 0, β = 0.5, k = 0.2, and L = 0.005.

Figure 8.7 Comparison of solution of CWBK equations (8.10) and (8.11) with exact solution for α = 0, β = 0.5, k = 0.2, and L = 0.005.

Charts depicting the comparison of solution of CWBK equations (8.10) and (8.11) with exact solution for α = 0.5, β = 0, k = 0.2, L = 0.005, and t = 1.

Figure 8.8 Comparison of solution of CWBK equations (8.10) and (8.11) with exact solution for α = 0.5, β = 0, k = 0.2, L = 0.005, and t = 1.

Charts depicting the solution of CWBK equations (8.14) and (8.15) along with exact solution for α = [0.1, 0.9], β = 0, and matching with crisp solution for α = 0.5 and β = 0.

Figure 8.9 Solution of interval CWBK equations (8.14) and (8.15) along with exact solution for α = [0.1, 0.9], β = 0, k = 0.2, and L = 0.005.

Charts depicting the solution of CWBK equations (8.14) and (8.15) along with exact solution for α = 0, β = [0.1, 0.9], and matching with crisp solution for α = 0 and β = 0.5.

Figure 8.10 Solution of interval CWBK equations (8.14) and (8.15) along with exact solution for α = 0, β = [0.1, 0.9], k = 0.2, and L = 0.005.

From Table 8.2, one may see the convergence of the result with increase in number of terms in the series solution. It may be observed from Figure 8.2 that by increasing number of terms in series solution (8.41), residual error decreases and it tends to zero, which confirms the convergence of the series solution. The present results by HPM are in good agreement with the existing DTM solution [9], which may be seen in Table 8.3. From Figure 8.3, one may conclude that by increase in value of nanoparticle volume fraction, there is a decrease in the velocity profile of SA–Cu nanofluid in between the considered channel. It may also be seen from Figure 8.4 that increase in Hartmann number causes an increase in the velocity profile. Figure 8.5 confirms that the velocity profile for SA–Cu nanofluid has decreasing nature with increase in the Reynolds number. It is worth mentioning here that the fuzzy plots for velocity profile for different values of η are also TFN, which may be confirmed from Figure 8.6. From Figures 8.7 and 8.8, it may be seen that solutions of CWBK equations by the present method are in good agreement with exact solutions. It may be observed from Figure 8.9 that the center solution of (8.14) and (8.15) for α = [0.1, 0.9] and β = 0 is matching with crisp solution for α = 0.5 and β = 0. Similarly, for α = 0 and β = [0.1, 0.9] also, the center solution is matching with crisp solution for α = 0 and β = 0.5, which can be seen in Figure 8.10. A worth mentioning point from Figures 8.9 and 8.10 may be that lower bounds for the interval solutions images and images are at α = 0.9 and β = 0.9, respectively, whereas upper bounds images and images are at α = 0.1 and β = 0.1, respectively. This may be due to decreasing nature of u and v with the increase in values of α and β.

8.7 Conclusion

HPM has been applied successfully using the concept of δ ‐ cut and parametric approach to solve Jeffery–Hamel problem in uncertain environment by taking φ as a TFN. The obtained fuzzy velocity for this problem is also found to be TFN for different values of η. The results in special case of the fuzzy solution (crisp results) are compared with existing results, and they are found to be in good agreement. For SA–Cu nanofluid, it may be observed that the nondimensional velocity profile increases with the increase in the value of Hartmann number. However, the velocity profile decreases with the increase in the value of φ or Reynolds number. Next, CWBK equations have been solved using HPTM. The results obtained by HPTM are compared with exact solution and are found to be in agreement. Further, constants α and β representing diffusion power in CWBK equations have been considered as uncertain in terms of interval to form interval CWBK equations. Again, HPTM has been applied to interval CWBK equations to find lower and upper bound solutions.

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