# Inviscid Modes within the Boundary-Layer Flow of a Rotating Disk with Wall Suction and in an External Free-Stream

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Problem Definition

#### 1.2. The Importance of the Problem

## 2. Formulation of the Problem and Boundary Layer Approximation

#### 2.1. Positive and Negative Strength of Axial Flow (${T}_{s}$)

#### 2.2. Various Suction/Injection (${W}^{*}$) within a Positive Fixed Axial Flow

#### 2.3. Linear Disturbance Equations

## 3. Inviscid Model

#### 3.1. Leading Order Eigenmodes

#### 3.2. First Order Eigenmodes

#### 3.2.1. Rotating Disk in Axial Flow When ${T}_{s}>0.25$ and ${T}_{s}<0$

#### 3.2.2. Rotating Disk with a Varying Suction/Injection Parameter ${W}^{*}$

## 4. Conclusions

#### 4.1. Positive and Negative Strength of Axial Flow

- For the rotating disk within axial flow ${T}_{s}$ between (0.30–0.65), the stability parameter values for a rotating disk are gradually increased by increasing axial flow.
- For negative axial flow, the efficiency is degraded substantially and using ${T}_{s}$ parameters less than ${T}_{s}=-0.04$ could not achieve convergence compared to ${T}_{s}=0.3-0.65$ which is consistent with the physical interpretation of positive axial flow stabilising according to Hussain [15], Hussain et al. [31], and Al Malki [42].
- Positive and negative ${T}_{s}$ shifts the curves vertically upwards. In general, our results are consistent with Hussain et al. [31] in that axial flow causes a stabilisation of the type I cross flow instability mode.

#### 4.2. Rotating Disk with a Suction/Injection Parameter ${W}^{*}$

- Injection at the boundary is easier to obtain convergence of stability when increasing ${W}^{*}$, for example when compared with a negative axial flow in the bulk fluid.
- The required number of iterations for the maintenance of convergence for the injection parameter is comparatively better than for suction parameters.
- For injection, the efficiency is degraded substantially. Convergence was also not achieved for injection parameters less than $-0.55$.
- Increasing ${W}^{*}$ shifts the curves vertically downwards. In general, our results are consistent with Al-Malki [43] for Blasius flow, in that suction causes a stabilisation of the type I cross flow instability mode.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

- Thompson, W.S. Hydrokinetic solutions and observations. Phil. Mag.
**1871**, 4, 374. [Google Scholar] - Helmholtz, H.L.F. On Discontinuous Movements of Fluids; Academy of Sciences: Berlin, Germany, 1868; Volume 36, pp. 215–228. [Google Scholar]
- Taylor, G.I., VIII. Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. A
**1923**, 223, 289–343. [Google Scholar] - Getling, A.V. Rayleigh—Bénard Convection: Structures and Dynamics, 1st ed.; Scientific: London, UK, 1998; Volume 11. [Google Scholar]
- Lin, C.C. The Theory of Hydrodynamic Stability, 1st ed.; Cambridge University Press: Cambridge, UK, 1955. [Google Scholar]
- Holm, D.D.; Marsden, J.E.; Ratiu, T.; Weinstein, A. Nonlinear stability of fluid and plasma equilibria. Phys. Rep.
**1985**, 123, 1–116. [Google Scholar] [CrossRef][Green Version] - Criminale, W.O.; Jackson, T.L.; Joslin, R.D. Theory and Computation in Hydrodynamic Stability, 1st ed.; Cambridge University Press: Cambridge, UK, 2018. [Google Scholar]
- Swaters, G.E. Introduction to Hamiltonian Fluid Dynamics and Stability Theory; CRC Press: London, UK, 2010. [Google Scholar]
- Drazin, P.G.; Reid, W.H. Hydrodynamic Stability, 1st ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Dryden, H.L. Review of published data on the effect of roughness on transition from laminar to turbulent flow. J. Aero. Sci.
**1953**, 20, 477–482. [Google Scholar] [CrossRef] - Reed, H.L.; Saric, W.S. Stability of three-dimensional boundary layers. Ann. Rev. Fluid Mech.
**1989**, 21, 235–284. [Google Scholar] [CrossRef] - Hall, P.; Malik, M.R. On the stability of a three-dimensionalattachment-line boundary layer: Weakly non-linear theory and a numerical approach. J. Fluid Mech.
**1986**, 163, 257–282. [Google Scholar] [CrossRef] - Lakin, W.; Hussaini, Y. Stability of the laminar boundary layer in a streamwise corner. Proc. R. Soc. Lond.
**1984**, A 393, 101–116. [Google Scholar] - Duraisamy, K. Perspectives on machine learning-augmented Reynolds-averaged and large eddy simulation models of turbulence. Phys. Rev. Fluids.
**2021**, 6, 050504. [Google Scholar] [CrossRef] - Hussain, Z. Stability and Transition of Three-Dimensional Rotating Boundary Layers. Ph.D. Thesis, University of Birmingham, Birmingham, UK, 2010. [Google Scholar]
- Reynolds, O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Philos. Trans. R. Soc.
**1883**, 174, 935–982. [Google Scholar] - Prandtl, L. Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg. Teubner Leipz. Ger.
**1905**, 8, 485–491. [Google Scholar] - Reshotko, E. Boundary-layer instability, transition and control. AIAA J.
**1994**, 94, 0001. [Google Scholar] - Saric, W.S.; Reed, H.L.; White, E.B. Stability and transition of three-dimensional boundary layers. Ann. Rev. Fluid Mech.
**2003**, 35, 413–440. [Google Scholar] [CrossRef][Green Version] - Kobayashi, R.; Izumi, H. Boundary-layer transition on a rotating cone in still fluid. J. Fluid Mech.
**1983**, 127, 353–364. [Google Scholar] [CrossRef] - Hussain, Z.; Garrett, S.J.; Stephen, S.O.; Griffiths, P.T. The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free stream. J. Fluid Mech.
**2016**, 788, 70–94. [Google Scholar] [CrossRef][Green Version] - Hussain, Z.; Garrett, S.J.; Stephen, S.O. The centrifugal instability of the boundary-layer flow over slender rotating cones. J. Fluid Mech.
**2014**, 755, 274–293. [Google Scholar] [CrossRef][Green Version] - Berker, R. Integration des Equations du Movement d’un Fluid Visquent Incompressible. In Hand Book of Fluid Dynamics; Springer: Berlin/Heidelberg, Germany, 1963; Volume 3. [Google Scholar]
- Coirier, J. Rotations non coaxials d’un disque et d’un fluide a I’infini. J. Mec.
**1972**, 11, 317–340. [Google Scholar] - Mohanty, H.K. Hydromagnetic flow between two rotating disks with noncoincident parallel axes of rotation. Phys. Fluids
**1972**, 15, 1456–1458. [Google Scholar] [CrossRef] - Mabood, F.; Rauf, A.; Prasannakumara, B.C.; Izadi, M.; Shehzad, S.A. Impacts of Stefan blowing and mass convention on flow of Maxwell nanofluid of variable thermal conductivity about a rotating disk. Chin. J. Phys.
**2021**, 71, 260–272. [Google Scholar] [CrossRef] - Mabood, F.; Imtiaz, M.; Hayat, T.; Alsaedi, A. Unsteady convective boundary layer flow of Maxwell fluid with nonlinear thermal radiation: A numerical study. Int. J. Nonlinear Sci. Numer. Simul.
**2016**, 17, 221–229. [Google Scholar] [CrossRef] - Ijaz, M.; Ayub, M. Nonlinear convective stratified flow of Maxwell nanofluid with activation energy. Heliyon
**2019**, 5, e01121. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ahmed, A.; Khan, M.; Ahmed, J. Mixed convective flow of Maxwell nanofluid induced by vertically rotating cylinder. Appl. Nanosci.
**2020**, 2, 01320. [Google Scholar] [CrossRef] - Khan, M.; Ahmed, J. Ali, Thermal analysis for radiative flow of magnetized Maxwell fluid over a vertically moving rotating disk. J. Therm. Anal. Calorim.
**2021**, 143, 4081–4094. [Google Scholar] [CrossRef] - Hussain, Z.; Garrett, S.J.; Stephen, S.O. The instability of the boundary layer over a disk rotating in an enforced axial flow. Phys. Fluids
**2011**, 23, 114108. [Google Scholar] [CrossRef][Green Version] - Turkyilmazoglu, M. Analytic approximate solutions of rotating disk boundary layer flow subject to a uniform suction or injection. Int. J. Mech. Sci.
**2010**, 52, 1735–1744. [Google Scholar] [CrossRef] - Garrett, S.J.; Peake, N. The absolute instability of the boundary layer on a rotating cone. Eur. J. Mech. B
**2007**, 26, 344–353. [Google Scholar] [CrossRef] - Garrett, S.J.; Hussain, Z.; Stephen, S.O. Boundary-layer transition on broad cones rotating in an imposed axial flow. IAA J.
**2010**, 48, 1184–1194. [Google Scholar] [CrossRef][Green Version] - Appelquist, E. Direct Numerical Simulations of the Rotating-Disk Boundary-Layer Flow; Royal Institute of Technology KTH Mechanics: Stockholm, Sweden, 2014. [Google Scholar]
- Rosenhead, L. Laminar Boundary Layers, 1st ed.; Oxford University Press: Oxford, UK, 1963. [Google Scholar]
- Evans, H. Laminer Boundary Layer Theory; Addison Wesley: Reading, MA, USA, 1968. [Google Scholar]
- Garrett, S.J.; Hussain, Z.; Stephen, S.O. The crossflow instability of the boundary layer on a rotating cone. J. Fluid Mech.
**2009**, 622, 209–232. [Google Scholar] [CrossRef][Green Version] - Chen, K.; Mortazavi, A.R. An analytic study of the chemical vapor deposition (CVD) processes in a rotating pedestal reacto. R. J. Cryst. Growth
**1986**, 77, 199–208. [Google Scholar] [CrossRef] - von Kármán, T. Uber laminare und turbulente Reiburg. Z. Agnew. Math.
**1921**, 1, 233–252. [Google Scholar] - Hall, P. An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disk. Proc. R. Soc. Lond. A
**1986**, 406, 93–106. [Google Scholar] - Al-Malki, M.; Garrett, S.J.; Camarri, S.; Hussain, Z. The effects of roughness levels on the instability of the boundary-layer flow over a rotating disk with an enforced axial flow. Phys. Fluids
**2021**, 33, 104109. [Google Scholar] [CrossRef] - Al-Malki, M.; Hussain, Z.; Garrett, S.J.; Calabretto, S.A. The effects of parietal suction and injection on the stability of the Blasius boundary-layer flow over a permeable, heated plate. Phys. Rev. Fluids
**2021**, 6, 113902. [Google Scholar] [CrossRef]

**Figure 2.**Plot showing the mean flow profile U, V and W for $\psi =90$, ${T}_{s}=0.30$ (

**left**) and $0.65$ (

**right**).

**Figure 3.**Plot showing the mean flow profile U, V and W for $\psi =90$, ${T}_{s}=-0.01$ (

**left**) and $-0.04$ (

**right**).

**Figure 4.**Plot showing the mean flow profile U, V and W, ${T}_{s}=0.05$ with ${W}^{*}$ = −0.35 (

**A**), 0 (

**B**) and 0.35 (

**C**).

**Figure 5.**Plot showing the velocity effectiveness $\overline{\overline{U}}$ (lower curve at $\eta =20$) and its second derivative ${\overline{\overline{U}}}^{\u2033}$ (upper curve at $\eta =20$) for $\psi ={90}^{0},{T}_{s}=0.30$ (

**left**) and $0.65$ (

**right**).

**Figure 6.**Plot showing the velocity effectiveness $\overline{\overline{U}}$ (below graph at $\eta =20$) and its second derivative ${\overline{\overline{U}}}^{\u2033}$ (above graph at $\eta =20$) for $\psi ={90}^{0},{T}_{s}=-0.01$ (

**left**) and $-0.04$ (

**right**).

**Figure 7.**Plot showing the velocity effectiveness $\overline{\overline{U}}$ (below graph at $\eta =20$) and its second derivative ${\overline{\overline{U}}}^{\u2033}$ (above graph at $\eta =20$) for $\psi ={90}^{0}$, ${T}_{s}=0.05$ with ${W}^{*}$ = −0.35 (

**A**), 0 (

**B**) and 0.35 (

**C**).

**Figure 8.**Plot showing the inviscid motion eigenfunction ${w}_{0}$ against $\eta $ for $\psi =90$, ${T}_{s}=0.30$ (

**left**) and $0.65$ (

**right**).

**Figure 9.**Plot showing the inviscid motion eigenfunction ${w}_{0}$ against $\eta $ for $\psi =90$, ${T}_{s}=-0.01$ (

**left**) and $-0.04$ (

**right**).

**Figure 10.**Plot showing the inviscid motion eigenfunction ${w}_{0}$ against $\eta $ for $\psi =90$, ${T}_{s}=0.05$ with ${W}^{*}$ = −0.35 (

**A**), 0 (

**B**) and 0.35 (

**C**).

**Figure 11.**Plot illustrating predictions of asymptotic neutral wavenumber ${\gamma}_{\delta *}$ against ${R}_{\delta *}$ for type I modes for ${T}_{s}=0.30-0.65$. Larger ${T}_{s}$ values move the plots up as shown by the arrow.

**Figure 12.**Plot illustrating predictions of asymptotic neutral waveangle $\varphi $ against ${R}_{\delta *}$ for type I modes for ${T}_{s}=0.30-0.65$. Larger ${T}_{s}$ values move the plots up as shown by the arrow.

**Figure 13.**Plot illustrating predictions of asymptotic neutral wavenumber ${\gamma}_{\delta *}$ against ${R}_{\delta *}$ for type I modes for ${T}_{s}=(-0.01)-(-0.04)$. Smaller ${T}_{s}$ values move the plots up as shown by the arrow.

**Figure 14.**Plot illustrating predictions of asymptotic neutral waveangle $\varphi $ against ${R}_{\delta *}$ for type I modes for ${T}_{s}=(-0.01)-(-0.04)$. Smaller ${T}_{s}$ values move the plots up as shown by the arrow.

**Figure 15.**Plot illustrating predictions of asymptotic neutral wavenumber ${\gamma}_{\delta *}$ against ${R}_{\delta *}$ for type I modes for ${T}_{s}=0.05$ with ${W}^{*}=(-0.35)$ to $\left(0.55\right)$. Increasing ${W}^{*}$ shifts the curves vertically downwards.

**Figure 16.**Plot illustrating predictions of asymptotic neutral waveangle predictions $\varphi $ against ${R}_{\delta *}$ for type I modes for ${T}_{s}=0.05$ with ${W}^{*}=(-0.35)$ to $\left(0.55\right)$. Increasing ${W}^{*}$ shifts the curves vertically upwards.

${\mathit{T}}_{\mathit{s}}$ | ${\mathit{U}}^{\prime}$ | ${\mathit{V}}^{\prime}$ | $\mathit{\mu}$ | $\overline{\mathit{\eta}}$ | $\mathit{\gamma}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ |
---|---|---|---|---|---|---|---|

0.3 | 0.6219 | −0.7359 | 0.4254 | 1.0878 | 1.5995 | 0.0446 | 0.3056 + 0.1685i |

0.35 | 0.6615 | −0.7628 | 0.4675 | 1.0399 | 1.6759 | 0.0392 | 0.3801 + 0.2153i |

0.4 | 0.7066 | −0.7904 | 0.5112 | 0.9964 | 1.7514 | 0.0346 | 0.4654 + 0.2707i |

0.45 | 0.7567 | −0.8183 | 0.556 | 0.9569 | 1.8257 | 0.0308 | 0.5584 + 0.3351i |

0.5 | 0.8116 | −0.8462 | 0.6019 | 0.9209 | 1.8985 | 0.0275 | 0.6612 + 0.4086i |

0.55 | 0.871 | −0.8741 | 0.6486 | 0.8882 | 1.9698 | 0.0247 | 0.7711 + 0.4907i |

0.6 | 0.9347 | −0.9017 | 0.696 | 0.8582 | 2.0397 | 0.0223 | 0.8861 + 0.5821i |

0.65 | 1.0024 | −0.9291 | 0.744 | 0.8307 | 2.1081 | 0.0203 | 1.0106 + 0.6810i |

${\mathit{T}}_{\mathit{s}}$ | ${\mathit{U}}^{\prime}$ | ${\mathit{V}}^{\prime}$ | $\mathit{\mu}$ | $\overline{\mathit{\eta}}$ | $\mathit{\gamma}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ |
---|---|---|---|---|---|---|---|

−0.01 | 0.5104 | −0.6145 | 0.2329 | 1.9695 | 1.1516 | 0.0982 | 0.0618 + 0.0302i |

−0.02 | 0.5018 | −0.6138 | 0.2305 | 1.479 | 1.1426 | 0.0996 | 0.0604 + 0.0292i |

−0.03 | 0.5115 | −0.6131 | 0.2307 | 1.4827 | 1.1389 | 0.1 | 0.0602 + 0.0289i |

−0.04 | 0.5121 | −0.612 | 0.2278 | 1.4959 | 1.1267 | 0.1019 | 0.0582 + 0.0275i |

${\mathit{T}}_{\mathit{s}}$ | ${\mathit{W}}^{*}$ | ${\mathit{U}}^{\prime}$ | ${\mathit{V}}^{\prime}$ | $\mathit{\mu}$ | $\overline{\mathit{\eta}}$ | $\mathit{\gamma}$ | ${\mathit{I}}_{1}$ | ${\mathit{I}}_{2}$ |
---|---|---|---|---|---|---|---|---|

0.05 | −0.55 | 0.4725 | −0.9094 | 0.1886 | 1.0753 | 1.5614 | 0.0368 | 0.0392 + 0.0228i |

0.05 | −0.45 | 0.4825 | −0.8509 | 0.1997 | 1.1305 | 1.4885 | 0.0432 | 0.0438 + 0.0253i |

0.05 | −0.35 | 0.4914 | −0.7958 | 0.2116 | 1.1872 | 1.4212 | 0.0506 | 0.0488 + 0.0281i |

0.05 | 0 | 0.5132 | −0.6270 | 0.2543 | 1.3944 | 1.2238 | 0.0868 | 0.6830 + 0.0408i |

0.05 | 0.35 | 0.5185 | −0.4907 | 0.2956 | 1.6131 | 1.0731 | 0.1458 | 0.1165 + 0.594i |

0.05 | 0.452 | 0.5171 | −0.4561 | 0.3071 | 1.6789 | 1.0358 | 0.1689 | 0.1287 + 0.0663i |

0.05 | 0.55 | 0.5145 | −0.4249 | 0.3177 | 1.7430 | 1.0023 | 0.1943 | 0.1414 + 0.0598i |

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

© 2021 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**

Al Saeedi, B.; Hussain, Z. Inviscid Modes within the Boundary-Layer Flow of a Rotating Disk with Wall Suction and in an External Free-Stream. *Mathematics* **2021**, *9*, 2967.
https://doi.org/10.3390/math9222967

**AMA Style**

Al Saeedi B, Hussain Z. Inviscid Modes within the Boundary-Layer Flow of a Rotating Disk with Wall Suction and in an External Free-Stream. *Mathematics*. 2021; 9(22):2967.
https://doi.org/10.3390/math9222967

**Chicago/Turabian Style**

Al Saeedi, Bashar, and Zahir Hussain. 2021. "Inviscid Modes within the Boundary-Layer Flow of a Rotating Disk with Wall Suction and in an External Free-Stream" *Mathematics* 9, no. 22: 2967.
https://doi.org/10.3390/math9222967