Abstract

The authors regret in the published paper referenced above and agree with the discussion by Pantokratoras (2019, “Discussion: “Computational Analysis for Mixed Convective Flows of Viscous Fluids With Nanoparticles” (Farooq, U., Lu, D. C., Ahmed, S., and Ramzan, M., 2019, ASME J. Therm. Sci. Eng. Appl., 11(2), p. 021013),” ASME J. Therm. Sci. Eng. Appl., 11(5), p. 055503). In this Closure, the non-similar mathematical model is developed to describe the mixed convective nanofluid flow over vertical sheet which is stretching at an exponential rate. In the published article referenced above, similarity transformations are utilized to convert the governing nonlinear partial differential equations (PDEs) into ordinary differential equations (ODEs). The important physical numbers such as magnetic field (M2), Brownian motion parameter (Nb), thermophoresis (Nt), Eckert number (Ec), ratio of mass transfer Grashof to heat transfer Grashof (N), buoyancy parameter (λ), and Reynolds number (Re) appearing in the dimensionless ODEs are still functions of coordinate “x”; therefore, the problem is non-similar. In this corrigendum, the non-similar model is developed by using ξ(x) as non-similarity variable and η(x, y) as pseudo-similarity variable. The dimensionless non-similar model is numerically simulated by employing local non-similarity via bvp4c. The graphical results show no change in behavior. The important thermal and mass transport quantities such as Nusselt number and Sherwood number have been computed for the non-similar model, and results are compared with the published article.

1 Introduction

In the literature, most of the boundary layer problems have been tackled using similarity transformations which change the governing partial differential equations (PDEs) to ordinary differential equations (ODEs) [1,2]. Numerical and analytical methods are generally used to evaluate these ODEs. Analytical methods are mostly used to solve similar flows due to calculation and ideal simplicity. Sometimes it may happen that after applying similarity variables, the physical parameters in the governing equations are still functions of coordinate “x” (see Refs. [3,4]), and these problems are known as non-similar. Despite of this fact that non-similar flows are extensive in nature and they have vast applications in real life, there are still lesser publications in this direction compared with similar flows. Many numerical techniques have been developed which can tackle non-similar flows. The most prominent technique is known as local non-similarity established by Sparrow et al. [5]. In this corrigendum, local non-similarity via bvp4c is employed. The main drawback of the local similarity approach is that it neglects the non-similar part. To overcome this drawback, Sparrow and Yu [6] presented a new technique termed as “method of local non-similarity” to obtain results for non-similar flows. Later non-similar flows were studied by numerous authors (see for instance, Refs. [5,711]). In this corrigendum, non-similar transformations have been proposed for mixed convective boundary layer flow over vertical surface which is stretching at an exponential rate. The local non-similarity technique via bvp4c is employed to obtain non-similar solutions.

2 Mathematical Formulation

Consider two-dimensional, laminar, incompressible magnetized nanofluid flow which is produced by the exponential expansion of the sheet with velocity “Uw”. The buoyancy in terms of temperature and concentration is incorporated in the momentum equation. The impacts of magnetic and viscous dissipations are incorporated in the energy equation. The physical interpretation of the model is described as follows:
ux+vy=0
(1)
uux+vuy=v(2uy2)σBo2vfu+gβc(CC)+gβT(TT)
(2)
uTx+vTy=α(2Ty2)+μρfCp(uy)2+τDB(CyTy)+τDTT(Ty)2+σBo2Cpρfu2(3)
(3)
uCx+vCy=DB(2Cy2)+DTT(2Ty2)
(4)

In preceding expressions, u and v represent velocities in x- and y-directions, and α, v, T, σ, Cp, T, DB, C, DT, and ρf represent thermal diffusivity, kinematic viscosity, temperature, electrical conductivity, specific heat, freestream temperature, Brownian diffusion coefficient, nanoparticle volume fraction, thermophoretic diffusion coefficient, and fluid density, respectively.

The relevant conditions take the following forms:
Aty=0,u=Uw(x)=U0exp(xl),v=0,C=Cw(x),kTy=hf(TwT)
(5)
Asy,v0,u0,TT,CC
(6)
where
Tw=T+T0exp(2xl),Cw=C+C0exp(2xl)
In earlier mentioned conditions, Cw and Tw are the concentration and temperature on the wall and hf is the convective heat transfer coefficient. Introducing two dimensionless numbers ξ and η the non-similarity variable and pseudo-similarity variable, respectively, non-similarity transformations are then proposed as
{ξ=exp(xl),η=U02νlyexp(x2l)u=U0fη(ξ,η)exp(xl)v=νU02lexp(x2l)(f(ξ,η)+ηfη+2ξfξ)T=θ(ξ,η)(TwT)+TC=ϕ(ξ,η)(CwC)+C
(7)
Using Eq. (7) in Eqs. (1)(6), they are transformed as follows:
3fη3+f2fη22(fη)22M2ξ1fη+2λ(θ+Nϕ)=2ξ(fη2fξη2fη2fξ)
(8)
2θη2+Pr(fθη4θfη+2M2Ecξ1(fη)2+ξ2(Nbϕηθη+Nt(θη)2)+Ec(2fη2)2)=2Prξ(fηθξθηfξ)
(9)
2ϕη2+NtNb2θη2Sc(4ϕfηfϕη)=2Scξ(fηϕξϕηfξ)
(10)
The boundary conditions are as follows:
fη(ξ,0)=1,f(ξ,0)+2ξfξ(ξ,0)=0,ϕ(ξ,)=0,ϕ(ξ,0)=1fη(ξ,)=0,θ(ξ,)=0,θη(ξ,0)=γξ1/2(1θ(ξ,0))
(11)
The parameters Re, λ, Nb, Pr, Nt, M, Le, N, and Ec represent Reynolds number, buoyancy parameter, Brownian motion, Prandtl number, thermophoresis, magnetic field, Lewis number, ratio between mass transfer Grashof number and heat transfer Grashof number, and Eckert number, respectively. The dimensionless parameters are represented as
λ=GrRe2,Pr=να,Nb=τDBC0ν,Nt=τToDTνT,M2=σBo2lρfUo,Ec=Uo2CpTo,LePr=ScSc=νDB,Re=lUoν,N=Gr*Gr,γ=hfk2νlUo,Gr=gβTT0l3ν2,Gr*=gβcC0l3ν2
(12)
The local Nusselt number, skin friction, and local Sherwood number coefficients are
Nu=xqIk(TIT),Cf=2τIρUI2,Sh=xjIDB(CIC)
(13)
where jI is the mass flux, τI is the wall skin friction, and qI is heat flux from surface, defined below:
τI=μ(uy)y=0,jI=DB(Cy)y=0,qI=k(Ty)y=0
(14)
(2Re)(1/2)Cf=2f(ξ,0),Re(1/2)Sh=12ln(ξ)ϕ(ξ,0),Re(1/2)Nu=12θ(ξ,0)ln(ξ)
(15)

3 Non-Similarity Method

To interpret several non-similarity boundary layers, many scientists utilize this method (see for instance, Refs. [1214]). The principle of local similarity is usually considered for outcomes of non-similarity boundary layer. With the help of the local non-similar technique, we have approximated non-similar PDEs through ODEs. In this method, the right-edge of Eqs. (8)(10) are supposed to be sufficiently small; therefore, it can be estimated to zero, and hence leading PDEs become ODEs. The final resulting non-dimensional set of coupled nonlinear PDEs is then resolved with the utilization of bvp4c function in matlab. A technique for finding locally non-similar boundary layer is discussed and demonstrated by Farooq et al. [15].

4 Results and discussion

Local non-similarity via bvp4c is utilized for the influences of numerous parameters on temperature profile θ, velocity profile f′, and concentration profile ϕ. Figures 110 are generated for distinct values of γ, N, Nb, Nt, λ, and Le. The graphs depict similar behavior to the original problem [16].

Fig. 1
Physical configuration
Fig. 1
Physical configuration
Close modal
Fig. 2
Graph of f′(η) for distinct values of N
Fig. 2
Graph of f′(η) for distinct values of N
Close modal
Fig. 3
Graph of f′(η) for distinct values of λ
Fig. 3
Graph of f′(η) for distinct values of λ
Close modal
Fig. 4
Graph of θ(η) for distinct values of γ
Fig. 4
Graph of θ(η) for distinct values of γ
Close modal
Fig. 5
Graph of θ(η) for distinct values of Nb
Fig. 5
Graph of θ(η) for distinct values of Nb
Close modal
Fig. 6
Graph of θ(η) for distinct values of N
Fig. 6
Graph of θ(η) for distinct values of N
Close modal
Fig. 7
Graph of ϕ(η) for distinct values of Nt
Fig. 7
Graph of ϕ(η) for distinct values of Nt
Close modal
Fig. 8
Graph of ϕ(η) for distinct values of N
Fig. 8
Graph of ϕ(η) for distinct values of N
Close modal
Fig. 9
Graph of ϕ(η) for distinct values of Nb
Fig. 9
Graph of ϕ(η) for distinct values of Nb
Close modal
Fig. 10
Graph of ϕ(η) for distinct values of Le
Fig. 10
Graph of ϕ(η) for distinct values of Le
Close modal

Tables 13 present the comparison between local similar skin friction, Nusselt and Sherwood numbers of Farooq et al. [16] and present non-similar results.

Table 1

Comparison of local similar and non-similar Cfx (local skin friction) for different values of λ versus N when M = Ec = Nb = γ = Nt = 0.1, ξ = Sc = 1, Pr = 3

ParametersFarooq et al. [16]Present
λNCfx(Rex(1/2))Difference (%)
11−0.4909596405−0.664721590926.14
20.1349248240−0.1404157755196.08
30.72073455290.3502839440105.75
210.1691179725−0.1207441687240.06
21.32061512040.836295773957.91
32.41779118861.730114439439.74
ParametersFarooq et al. [16]Present
λNCfx(Rex(1/2))Difference (%)
11−0.4909596405−0.664721590926.14
20.1349248240−0.1404157755196.08
30.72073455290.3502839440105.75
210.1691179725−0.1207441687240.06
21.32061512040.836295773957.91
32.41779118861.730114439439.74
Table 2

Comparison of local similar and non-similar Nu (local Nusselt number) for different values of Nb versus Ec when Pr = 3, Nt = M = 0.1, N = λ = Sc = ξ = γ = 1

ParametersFarooq et al. [16]Present
EcNbNu(Rex(1/2))Difference (%)
10.1−0.5659884824−0.358765726857.75
0.2−0.5433373428−0.341311333259.19
0.3−0.5260558981−0.333527297557.71
20.1−0.5594207277−0.1566273066257.16
0.2−0.5080076972−0.1506193302237.27
0.3−0.4827801575−0.1455588629231.67
ParametersFarooq et al. [16]Present
EcNbNu(Rex(1/2))Difference (%)
10.1−0.5659884824−0.358765726857.75
0.2−0.5433373428−0.341311333259.19
0.3−0.5260558981−0.333527297557.71
20.1−0.5594207277−0.1566273066257.16
0.2−0.5080076972−0.1506193302237.27
0.3−0.4827801575−0.1455588629231.67
Table 3

Comparison of local similar and non-similar Sh (local Sherwood number) for different values of Sc versus Nt when Nb = Ec = M = γ = 0.1, Pr = N = λ = ξ = 1

ParametersFarooq et al. [16]Present
ScNtSh(Rex(1/2))Difference (%)
30.1−1.8928151531−3.047064622237.88
0.3−1.5128377645−2.967541343449.02
0.5−1.1221686851−2.832690922960.38
50.1−2.5964060620−4.119237462336.96
0.3−2.3333398044−4.068008544342.64
0.5−2.0125506640−3.954414561249.10
ParametersFarooq et al. [16]Present
ScNtSh(Rex(1/2))Difference (%)
30.1−1.8928151531−3.047064622237.88
0.3−1.5128377645−2.967541343449.02
0.5−1.1221686851−2.832690922960.38
50.1−2.5964060620−4.119237462336.96
0.3−2.3333398044−4.068008544342.64
0.5−2.0125506640−3.954414561249.10

5 Conclusion

The authors agree with the concerns of Pantokratoras [17] on Farooq et al. [16]. Therefore, in this corrigendum, the non-similar model is developed in which the physical parameters are independent of “x”. Non-similar solutions depict similar graphical behavior by comparing it with the published article. The important thermal and mass transport quantities such as Nusselt number and Sherwood number have been computed for the non-similar model.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (Approval No.12062018) and the Natural Science Foundation of Inner Mongolia (Approval No. 2018LH01016).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. Data provided by a third party are listed in Acknowledgment.

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