## 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 (M^{2}), Brownian motion parameter (N_{b}), thermophoresis (N_{t}), 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. [5,7–11]). 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

*U*”. 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:

_{w}In preceding expressions, *u* and *v* represent velocities in *x*- and *y*-directions, and *α*, *v*, *T*, *σ*, *C*_{p}, *T*_{∞}, *D*_{B}, *C*, *D*_{T}, and *ρ*_{f} 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.

*C*

_{w}and

*T*

_{w}are the concentration and temperature on the wall and

*h*

_{f}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

*λ*,

*N*

_{b}, Pr,

*N*

_{t},

*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

*j*

_{I}is the mass flux,

*τ*

_{I}is the wall skin friction, and

*q*

_{I}is heat flux from surface, defined below:

## 3 Non-Similarity Method

To interpret several non-similarity boundary layers, many scientists utilize this method (see for instance, Refs. [12–14]). 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 1–10 are generated for distinct values of *γ*, *N*, *N*_{b}, *N*_{t}, *λ*, and Le. The graphs depict similar behavior to the original problem [16].

Tables 1–3 present the comparison between local similar skin friction, Nusselt and Sherwood numbers of Farooq et al. [16] and present non-similar results.

Parameters | Farooq et al. [16] | Present | ||
---|---|---|---|---|

λ | N | $Cfx(Rex(1/2))$ | Difference (%) | |

1 | 1 | −0.4909596405 | −0.6647215909 | 26.14 |

2 | 0.1349248240 | −0.1404157755 | 196.08 | |

3 | 0.7207345529 | 0.3502839440 | 105.75 | |

2 | 1 | 0.1691179725 | −0.1207441687 | 240.06 |

2 | 1.3206151204 | 0.8362957739 | 57.91 | |

3 | 2.4177911886 | 1.7301144394 | 39.74 |

Parameters | Farooq et al. [16] | Present | ||
---|---|---|---|---|

λ | N | $Cfx(Rex(1/2))$ | Difference (%) | |

1 | 1 | −0.4909596405 | −0.6647215909 | 26.14 |

2 | 0.1349248240 | −0.1404157755 | 196.08 | |

3 | 0.7207345529 | 0.3502839440 | 105.75 | |

2 | 1 | 0.1691179725 | −0.1207441687 | 240.06 |

2 | 1.3206151204 | 0.8362957739 | 57.91 | |

3 | 2.4177911886 | 1.7301144394 | 39.74 |

Parameters | Farooq et al. [16] | Present | ||
---|---|---|---|---|

Ec | N_{b} | $Nu(Rex\u2212(1/2))$ | Difference (%) | |

1 | 0.1 | −0.5659884824 | −0.3587657268 | 57.75 |

0.2 | −0.5433373428 | −0.3413113332 | 59.19 | |

0.3 | −0.5260558981 | −0.3335272975 | 57.71 | |

2 | 0.1 | −0.5594207277 | −0.1566273066 | 257.16 |

0.2 | −0.5080076972 | −0.1506193302 | 237.27 | |

0.3 | −0.4827801575 | −0.1455588629 | 231.67 |

Parameters | Farooq et al. [16] | Present | ||
---|---|---|---|---|

Ec | N_{b} | $Nu(Rex\u2212(1/2))$ | Difference (%) | |

1 | 0.1 | −0.5659884824 | −0.3587657268 | 57.75 |

0.2 | −0.5433373428 | −0.3413113332 | 59.19 | |

0.3 | −0.5260558981 | −0.3335272975 | 57.71 | |

2 | 0.1 | −0.5594207277 | −0.1566273066 | 257.16 |

0.2 | −0.5080076972 | −0.1506193302 | 237.27 | |

0.3 | −0.4827801575 | −0.1455588629 | 231.67 |

Parameters | Farooq et al. [16] | Present | ||
---|---|---|---|---|

Sc | N_{t} | $Sh(Rex\u2212(1/2))$ | Difference (%) | |

3 | 0.1 | −1.8928151531 | −3.0470646222 | 37.88 |

0.3 | −1.5128377645 | −2.9675413434 | 49.02 | |

0.5 | −1.1221686851 | −2.8326909229 | 60.38 | |

5 | 0.1 | −2.5964060620 | −4.1192374623 | 36.96 |

0.3 | −2.3333398044 | −4.0680085443 | 42.64 | |

0.5 | −2.0125506640 | −3.9544145612 | 49.10 |

Parameters | Farooq et al. [16] | Present | ||
---|---|---|---|---|

Sc | N_{t} | $Sh(Rex\u2212(1/2))$ | Difference (%) | |

3 | 0.1 | −1.8928151531 | −3.0470646222 | 37.88 |

0.3 | −1.5128377645 | −2.9675413434 | 49.02 | |

0.5 | −1.1221686851 | −2.8326909229 | 60.38 | |

5 | 0.1 | −2.5964060620 | −4.1192374623 | 36.96 |

0.3 | −2.3333398044 | −4.0680085443 | 42.64 | |

0.5 | −2.0125506640 | −3.9544145612 | 49.10 |

## 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.