# 3D Flow of Hybrid Nanomaterial through a Circular Cylinder: Saddle and Nodal Point Aspects

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{2}/water-based hybrid nanofluid were examined by Asadi et al. [11]. Lund et al. [12] studied the dual solution of the impact of thermal radiation effect over magnetohydrodynamic flow of a Cu–Al

_{2}O

_{3}/H

_{2}O-based hybrid nanofluid. Ramesh et al. [13] investigated the Darcy–Forchheimer model of hybrid nanofluid in a stretchable convergent/divergent channel. They found variations of velocity and solid volume fraction in the channel. Ramesh and Madhukesh [14] investigated the effect of hybrid carbon nanotubes in the presence of activation energy and heat source/sink. They found that hybrid CNTs nanomaterials have a greater rate of heating/cooling than single CNTs nanomaterials.

## 2. Problem Formulation

**Continuity equation**

**Momentum equation**

**Temperature equation**

**Concentration equation**

## 3. Numerical Procedure and Validation of Code

- Convert the BVP into IVP of the first order.
- Apply the shooting procedure to guess the missing boundary values.
- Apply the RKF-45 method to obtain the solution to IVP.
- Find the residuals for all the boundary conditions.
- If the residual error is greater than the error tolerance, adjust the initial guesses.
- If the residual error is less than error tolerance, numerical results are obtained.

## 4. Results and Discussion

## 5. Final Remarks

- A rise in the value of the streamline parameter upsurges the flow velocity and decreases the thermal distribution and concentration.
- Better thermal gradient and concentration are seen in the enhancement of volume fraction.
- Boundary layer thickness and concentration are decreased with increase in the Schmidt number.
- Thermal distribution and concentration are more in the saddle point than in the nodal point of a hybrid nanofluid.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${a}^{*}\&{b}^{*}$ | Free stream dependent constants |

$c$ | Gradient of streamline |

$C$ | Concentration |

${C}_{w}$ | Wall concentration |

${C}_{\infty}$ | Ambient concentration |

${C}_{p}$ | Specific heat |

${C}_{fx}\&{C}_{fy}$ | Skin friction along x and y direction |

$D$ | Diffusivity |

E | Activation energy parameter |

${E}_{a}$ | Activation energy |

$f$ | Fluid |

$f\left(\eta \right)$ | Dimensionless velocity |

$g\left(\eta \right)$ | Dimensionless velocity |

$hnf$ | Hybrid nanofluid |

${j}_{w}$ | Mass transfer |

$k$ | Thermal conductivity |

${\kappa}_{r}^{2}$ | Reaction rate |

$K$ | Boltzmann constant |

$n$ | Fitted rate constant |

$N{u}_{x}$ | Nusselt number |

Pr | Prandtl number |

${q}_{w}$ | Surface heat flux |

${Q}_{0}$ | Uniform heat source/sink coefficient |

$\mathrm{Re}$ | Reynolds number |

${s}_{1}$ | Solid particle of ${\mathrm{Fe}}_{3}{\mathrm{O}}_{4}$ |

${s}_{2}$ | Solid particle of $\mathrm{Go}$ |

$S$ | Heat source/sink parameter |

$Sc$ | Schmidt number |

$S{h}_{x}$ | Sherwood number |

$T$ | Temperature |

${T}_{w}$ | Wall temperature |

${T}_{\infty}$ | Ambient temperature |

${u}_{e}^{*}{v}_{e}^{*}$ | Free stream velocity |

$u,v\&w$ | Velocity components |

$x,y\&z$ | Coordinate axis |

Greek symbols | |

$\alpha $ | Thermal diffusivity |

$\mu $ | Dynamic viscosity |

$\rho $ | Density |

$\nu $ | Kinematic viscosity |

$\rho {C}_{p}$ | Heat capacitance |

$\gamma $ | Thermal slip |

$\lambda $ | Thermal slip parameter |

$\delta $ | Temperature difference parameter |

$\beta $ | Reaction rate |

${\tau}_{wx}$ | Shear stresses surface in the x-direction |

${\tau}_{wy}$ | Shear stresses surface in the y-direction |

${\varphi}_{1}$ | The solid volume fraction of ${\mathrm{Fe}}_{3}{\mathrm{O}}_{4}$ |

${\varphi}_{2}$ | The solid volume fraction of $\mathrm{Go}$ |

$\theta \left(\eta \right)$ | Dimensionless temperature |

$\chi \left(\eta \right)$ | Dimensionless concentration |

## References

- Choi, S.U.S.; Eastman, J. Enhancing thermal conductivity of fluids with nanoparticles. In Proceedings of the ASME International Mechanical Engineering Congress and Exposition, San Francisco, CA, USA, 12–17 November 1995; Volume 66. [Google Scholar]
- Choi, S.U.S. Nanofluids: From Vision to Reality Through Research. J. Heat Transf.
**2009**, 131, 033106. [Google Scholar] [CrossRef] - Koriko, O.K.; Shah, N.A.; Saleem, S.; Chung, J.D.; Omowaye, A.J.; Oreyeni, T. Exploration of bioconvection flow of MHD thixotropic nanofluid past a vertical surface coexisting with both nanoparticles and gyrotactic microorganisms. Sci. Rep.
**2021**, 11, 16627. [Google Scholar] [CrossRef] [PubMed] - Rehman, F.U.; Nadeem, S.; Rehman, H.U.; Haq, R.U. Thermophysical analysis for three-dimensional MHD stagnation-point flow of nano-material influenced by an exponential stretching surface. Results Phys.
**2018**, 8, 316–323. [Google Scholar] [CrossRef] - Fetecau, C.; Shah, N.A.; Vieru, D. General Solutions for Hydromagnetic Free Convection Flow over an Infinite Plate with Newtonian Heating, Mass Diffusion and Chemical Reaction. Commun. Theor. Phys.
**2017**, 68, 768. [Google Scholar] [CrossRef] - Sheikholeslami, M. Magnetic field influence on nanofluid thermal radiation in a cavity with tilted elliptic inner cylinder. J. Mol. Liq.
**2017**, 229, 137–147. [Google Scholar] [CrossRef] - Chamkha, A.J.; Abbasbandy, S.; Rashad, A.M.; Vajravelu, K. Radiation Effects on Mixed Convection over a Wedge Embedded in a Porous Medium Filled with a Nanofluid. Transp. Porous Media
**2011**, 91, 261–279. [Google Scholar] [CrossRef] - Animasaun, I.L.; Shah, N.A.; Wakif, A.; Mahanthesh, B.; Sivaraj, R.; Koriko, O.K. Ratio of Momentum Diffusivity to Thermal Diffusivity: Introduction, Meta-Analysis, and Scrutinization, 1st ed.; Chapman and Hall/CRC: London, UK, 2022. [Google Scholar] [CrossRef]
- Saeed, M.; Kim, M.-H. Heat transfer enhancement using nanofluids (Al
_{2}O_{3}-H_{2}O) in mini-channel heatsinks. Int. J. Heat Mass Transf.**2018**, 120, 671–682. [Google Scholar] [CrossRef] - Sarkar, J.; Ghosh, P.; Adil, A. A review on hybrid nanofluids: Recent research, development and applications. Renew. Sustain. Energy Rev.
**2015**, 43, 164–177. [Google Scholar] [CrossRef] - Asadi, A.; Alarifi, I.M.; Foong, L.K. An experimental study on characterization, stability and dynamic viscosity of CuO-TiO
_{2}/water hybrid nanofluid. J. Mol. Liq.**2020**, 307, 112987. [Google Scholar] [CrossRef] - Lund, L.A.; Omar, Z.; Khan, I.; Sherif, E.-S.M. Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow. Symmetry
**2020**, 12, 276. [Google Scholar] [CrossRef] [Green Version] - Ramesh, G.K.; Shehzad, S.A.; Tlili, I. Hybrid nanomaterial flow and heat transport in a stretchable convergent/divergent channel: A Darcy-Forchheimer model. Appl. Math. Mech.
**2020**, 41, 699–710. [Google Scholar] [CrossRef] - Ramesh, G.; Madhukesh, J. Activation energy process in hybrid CNTs and induced magnetic slip flow with heat source/sink. Chin. J. Phys.
**2021**, 73, 375–390. [Google Scholar] [CrossRef] - Alghamdi, M. Significance of Arrhenius Activation Energy and Binary Chemical Reaction in Mixed Convection Flow of Nanofluid Due to a Rotating Disk. Coatings
**2020**, 10, 86. [Google Scholar] [CrossRef] [Green Version] - Rekha, M.B.; Sarris, I.E.; Madhukesh, J.K.; Raghunatha, K.R.; Prasannakumara, B.C. Activation Energy Impact on Flow of AA7072-AA7075/Water-Based Hybrid Nanofluid through a Cone, Wedge and Plate. Micromachines
**2022**, 13, 302. [Google Scholar] [CrossRef] [PubMed] - Ramesh, G. Analysis of active and passive control of nanoparticles in viscoelastic nanomaterial inspired by activation energy and chemical reaction. Phys. A Stat. Mech. Its Appl.
**2020**, 550, 123964. [Google Scholar] [CrossRef] - Alsaadi, F.E.; Ullah, I.; Hayat, T.; Alsaadi, F.E. Entropy generation in nonlinear mixed convective flow of nanofluid in porous space influenced by Arrhenius activation energy and thermal radiation. J. Therm. Anal.
**2019**, 140, 799–809. [Google Scholar] [CrossRef] - Asma, M.; Othman, W.; Muhammad, T. Numerical study for Darcy–Forchheimer flow of nanofluid due to a rotating disk with binary chemical reaction and Arrhenius activation energy. Mathematics
**2019**, 7, 921. [Google Scholar] [CrossRef] [Green Version] - Ramzan, M.; Gul, H.; Zahri, M. Darcy-Forchheimer 3D Williamson nanofluid flow with generalized Fourier and Fick’s laws in a stratified medium. Bull. Pol. Acad. Sci. Tech. Sci.
**2020**, 68, 327–335. [Google Scholar] - Jagan, K.; Sivasankaran, S.; Bhuvaneswari, M.; Rajan, S. Effect of Non-linear Radiation on 3D Unsteady MHD Nanoliquid Flow over a Stretching Surface with Double Stratification. Trends Math.
**2019**, 109–116. [Google Scholar] [CrossRef] - Nayak, M.K.; Shaw, S.; Chamkha, A.J. 3D MHD Free Convective Stretched Flow of a Radiative Nanofluid Inspired by Variable Magnetic Field. Arab. J. Sci. Eng.
**2019**, 44, 1269–1282. [Google Scholar] [CrossRef] - Irfan, M.; Khan, W.A.; Khan, M.; Gulzar, M.M. Influence of Arrhenius activation energy in chemically reactive radiative flow of 3D Carreau nanofluid with nonlinear mixed convection. J. Phys. Chem. Solids
**2019**, 125, 141–152. [Google Scholar] [CrossRef] - A Alwawi, F.; Alkasasbeh, H.T.; Rashad, A.; Idris, R. Heat transfer analysis of ethylene glycol-based Casson nanofluid around a horizontal circular cylinder with MHD effect. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2020**, 234, 2569–2580. [Google Scholar] [CrossRef] - Tiwari, R.K.; Das, M.K. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf.
**2007**, 50, 2002–2018. [Google Scholar] [CrossRef] - Nadeem, S.; Abbas, N. On both MHD and slip effect in micropolar hybrid nanofluid past a circular cylinder under stagnation point region. Can. J. Phys.
**2019**, 97, 392–399. [Google Scholar] [CrossRef] - Tlili, I.; Waqas, H.; Almaneea, A.; Khan, S.U.; Imran, M. Activation Energy and Second Order Slip in Bioconvection of Oldroyd-B Nanofluid over a Stretching Cylinder: A Proposed Mathematical Model. Processes
**2019**, 7, 914. [Google Scholar] [CrossRef] [Green Version] - Shah, N.A.; Wakif, A.; El-Zahar, E.R.; Thumma, T.; Yook, S.-J. Heat transfers thermodynamic activity of a second-grade ternary nanofluid flow over a vertical plate with Atangana-Baleanu time-fractional integral. Alex. Eng. J.
**2022**, 61, 10045–10053. [Google Scholar] [CrossRef] - Rashid, U.; Baleanu, D.; Iqbal, A.; Abbas, M. Shape Effect of Nanosize Particles on Magnetohydrodynamic Nanofluid Flow and Heat Transfer over a Stretching Sheet with Entropy Generation. Entropy
**2020**, 22, 1171. [Google Scholar] [CrossRef] - Manohar, G.R.; Venkatesh, P.; Gireesha, B.J.; Madhukesh, J.K.; Ramesh, G.K. Performance of water, ethylene glycol, engine oil conveying SWCNT-MWCNT nanoparticles over a cylindrical fin subject to magnetic field and heat generation. Int. J. Model. Simul.
**2021**, 1–10. [Google Scholar] [CrossRef] - Mathews, J.H.; Fink, K.D. Numerical Methods Using MATLAB, 3rd ed.; Prentice Hall: Hoboken, NJ, USA, 1998. [Google Scholar]
- Ramesh, G.; Madhukesh, J.; Das, R.; Shah, N.A.; Yook, S.-J. Thermodynamic activity of a ternary nanofluid flow passing through a permeable slipped surface with heat source and sink. Waves Random Complex Media
**2022**, 1–21. [Google Scholar] [CrossRef] - Gangadhar, K.; Nayak, R.E.; Rao, M.V.S.; Kannan, T. Nodal/Saddle Stagnation Point Slip Flow of an Aqueous Convectional Magnesium Oxide–Gold Hybrid Nanofluid with Viscous Dissipation. Arab. J. Sci. Eng.
**2021**, 46, 2701–2710. [Google Scholar] [CrossRef] - Bhattacharyya, S.; Gupta, A. MHD flow and heat transfer at a general three-dimensional stagnation point. Int. J. Non-Linear Mech.
**1998**, 33, 125–134. [Google Scholar] [CrossRef] - Dinarvand, S. Nodal/saddle stagnation-point boundary layer flow of CuO–Ag/water hybrid nanofluid: A novel hybridity model. Microsyst. Technol.
**2019**, 25, 2609–2623. [Google Scholar] [CrossRef] - Bachok, N.; Ishak, A.; Nazar, R.; Pop, I. Flow and heat transfer at a general three-dimensional stagnation point in a nanofluid. Phys. B Condens. Matter
**2010**, 405, 4914–4918. [Google Scholar] [CrossRef]

Particles | $\mathit{\rho}(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3})$ | ${\mathit{c}}_{\mathit{p}}(\mathbf{J}/\mathbf{k}\mathbf{g}\mathbf{K})$ | $\mathit{k}(\mathbf{W}/\mathbf{m}\mathbf{K})$ |
---|---|---|---|

${\mathrm{Fe}}_{3}{\mathrm{O}}_{4}$ | 5180 | 670 | 9.7 |

$\mathrm{Go}$ | 1800 | 717 | 5000 |

Water | 997.1 | 4179 | 0.613 |

**Table 2.**Numerical validation for various values of $c$ in the absence of $S$, $\lambda $, ${\varphi}_{1}$, and ${\varphi}_{2}$.

Gangadar et al. [33] | Bhattacharyya and Gupta [34] | Dinarvand [35] | Bachok et al. [36] | Present Study | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Parameter | $\mathit{c}$ | $\mathit{c}$ | $\mathit{c}$ | $\mathit{c}$ | $\mathit{c}$ | |||||

−0.5 | 0.5 | −0.5 | 0.5 | −0.5 | 0.5 | −0.5 | 0.5 | −0.5 | 0.5 | |

$\sqrt{{\mathrm{Re}}_{x}}{C}_{fx}$ | 1.2302 | 1.2669 | 1.2312 | 1.2679 | 1.2325 | 1.2681 | - | 1.2681 | 1.2308 | 1.2675 |

$\sqrt{{\mathrm{Re}}_{x}}{C}_{fy}$ | 0.0558 | 0.4991 | 0.0557 | 0.4993 | 0.0557 | 0.4993 | - | 0.4994 | 0.0555 | 0.4990 |

$\frac{N{u}_{x}}{\sqrt{{\mathrm{Re}}_{x}}}$ | 1.1227 | 1.2938 | 1.1235 | 1.3302 | 1.1237 | 1.3301 | - | 1.3302 | 1.1231 | 1.2979 |

**Table 3.**Computational values of ${f}^{\u2033}(0),{g}^{\u2033}(0),-{\theta}^{\prime}(0)$ and $-{\chi}^{\prime}(0)$ for different values of $c$.

$\mathit{c}$ | ${\mathit{f}}^{\u2033}(0)$ | ${\mathit{g}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\chi}}^{\prime}(0)$ |
---|---|---|---|---|

−0.5 | 1.127855 | −0.102224 | 0.523666 | 0.551725 |

−0.2 | 1.123800 | 0.307432 | 0.521301 | 0.550452 |

0.0 | 1.130049 | 0.523008 | 0.524605 | 0.565968 |

0.2 | 1.140688 | 0.698789 | 0.530197 | 0.587699 |

0.5 | 1.161475 | 0.915078 | 0.540786 | 0.624976 |

**Table 4.**For nodal point, computational values of ${f}^{\u2033}(0),{g}^{\u2033}(0),-{\theta}^{\prime}(0)$ and $-{\chi}^{\prime}(0)$ with $\mathrm{Pr}=6.2,n=0.2,\alpha =0.5$ and ${\varphi}_{1}=0.1$.

${\mathit{\varphi}}_{2}$ | $\mathit{S}$ | $\mathit{\delta}$ | $\mathit{S}\mathit{c}$ | $\mathit{\beta}$ | $\mathit{E}$ | ${\mathit{f}}^{\u2033}(0)$ | ${\mathit{g}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\chi}}^{\prime}(0)$ |
---|---|---|---|---|---|---|---|---|---|

0.01 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 1.161475 | 0.915078 | 0.540786 | 0.624976 |

0.02 | 1.173017 | 0.924171 | 0.527588 | 0.631374 | |||||

0.03 | 1.181275 | 0.930677 | 0.514390 | 0.637650 | |||||

0.01 | −0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 1.161475 | 0.915078 | 0.634573 | 0.627710 |

0.0 | 1.161475 | 0.915078 | 0.591876 | 0.626489 | |||||

0.5 | 1.161475 | 0.915078 | 0.540786 | 0.624976 | |||||

0.01 | 0.5 | 0.0 | 0.5 | 0.5 | 0.5 | 1.161475 | 0.915078 | 0.857072 | 0.613593 |

0.5 | 1.161475 | 0.915078 | 0.540786 | 0.624976 | |||||

1.0 | 1.161475 | 0.915078 | 0.395014 | 0.629594 | |||||

0.01 | 0.5 | 0.5 | 0.1 | 0.5 | 0.5 | 1.161475 | 0.915078 | 0.540786 | 0.315530 |

0.3 | 1.161475 | 0.915078 | 0.540786 | 0.505093 | |||||

0.5 | 1.161475 | 0.915078 | 0.540786 | 0.624976 | |||||

0.01 | 0.5 | 0.5 | 0.5 | 0.0 | 0.5 | 1.161475 | 0.915078 | 0.540786 | 0.521808 |

0.5 | 1.161475 | 0.915078 | 0.540786 | 0.624976 | |||||

1.0 | 1.161475 | 0.915078 | 0.540786 | 0.716755 | |||||

0.01 | 0.5 | 0.5 | 0.5 | 0.5 | 0.0 | 1.161475 | 0.915078 | 0.540786 | 0.703487 |

0.5 | 1.161475 | 0.915078 | 0.540786 | 0.624976 | |||||

1.0 | 1.161475 | 0.915078 | 0.540786 | 0.579943 |

**Table 5.**For saddle point, computational values of ${f}^{\u2033}(0),{g}^{\u2033}(0),-{\theta}^{\prime}(0)$ and $-{\chi}^{\prime}(0)$ with $\mathrm{Pr}=6.2,n=0.2,\alpha =0.5$ and ${\varphi}_{1}=0.1$.

${\mathit{\varphi}}_{2}$ | $\mathit{S}$ | $\mathit{\delta}$ | $\mathit{S}\mathit{c}$ | $\mathit{\beta}$ | $\mathit{E}$ | ${\mathit{f}}^{\u2033}(0)$ | ${\mathit{g}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\chi}}^{\prime}(0)$ |
---|---|---|---|---|---|---|---|---|---|

0.01 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 1.127855 | −0.102224 | 0.523666 | 0.551725 |

0.02 | 1.139063 | −0.103240 | 0.511081 | 0.557363 | |||||

0.03 | 1.147081 | −0.103967 | 0.498476 | 0.563019 | |||||

0.01 | −0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 1.127855 | −0.102224 | 0.624321 | 0.554988 |

0.0 | 1.127855 | −0.102224 | Not Converging | ||||||

0.5 | 1.127855 | −0.102224 | 0.523666 | 0.551725 | |||||

0.01 | 0.5 | 0.0 | 0.5 | 0.5 | 0.5 | 1.127855 | −0.102224 | 0.814850 | 0.539718 |

0.5 | 1.127855 | −0.102224 | 0.523666 | 0.551725 | |||||

1.0 | 1.127855 | −0.102224 | 0.380486 | 0.511333 | |||||

0.01 | 0.5 | 0.5 | 0.1 | 0.5 | 0.5 | 1.127855 | −0.102224 | 0.523666 | 0.261420 |

0.3 | 1.127855 | −0.102224 | 0.523666 | 0.437491 | |||||

0.5 | 1.127855 | −0.102224 | 0.523666 | 0.551725 | |||||

0.01 | 0.5 | 0.5 | 0.5 | 0.0 | 0.5 | 1.127855 | −0.102224 | 0.523666 | 0.422603 |

0.5 | 1.127855 | −0.102224 | 0.523666 | 0.551725 | |||||

1.0 | 1.127855 | −0.102224 | 0.523666 | 0.659084 | |||||

0.01 | 0.5 | 0.5 | 0.5 | 0.5 | 0.0 | 1.127855 | −0.102224 | 0.523666 | 0.642921 |

0.5 | 1.127855 | −0.102224 | 0.523666 | 0.551725 | |||||

1.0 | 1.127855 | −0.102224 | 0.523666 | 0.497428 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Madhukesh, J.K.; Ramesh, G.K.; Roopa, G.S.; Prasannakumara, B.C.; Shah, N.A.; Yook, S.-J.
3D Flow of Hybrid Nanomaterial through a Circular Cylinder: Saddle and Nodal Point Aspects. *Mathematics* **2022**, *10*, 1185.
https://doi.org/10.3390/math10071185

**AMA Style**

Madhukesh JK, Ramesh GK, Roopa GS, Prasannakumara BC, Shah NA, Yook S-J.
3D Flow of Hybrid Nanomaterial through a Circular Cylinder: Saddle and Nodal Point Aspects. *Mathematics*. 2022; 10(7):1185.
https://doi.org/10.3390/math10071185

**Chicago/Turabian Style**

Madhukesh, Javali K., Gosikere K. Ramesh, Govinakovi S. Roopa, Ballajja C. Prasannakumara, Nehad Ali Shah, and Se-Jin Yook.
2022. "3D Flow of Hybrid Nanomaterial through a Circular Cylinder: Saddle and Nodal Point Aspects" *Mathematics* 10, no. 7: 1185.
https://doi.org/10.3390/math10071185