Vol. 37 No. 3 (2026): July - September
2026-09-30
Open Access

A Spectral-Grid Method for the Stability Analysis of the Blasius Boundary Layer and the Dynamics of the Velocity Vector

Chori Normurodov
Muhriddin Tilovov
Sardorbek Murodov
Abdulla Raimov
Nasiba Djurayeva

Received: 2026-10-13

Published: 2026-09-30

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Abstract

This study investigates the longitudinal and transverse components of linear perturbations in the Blasius boundary layer within the framework of local hydrodynamic stability analysis. The base-flow profile is determined from the Falkner--Skan equation, after which the linearized equations of motion are reduced to the Orr--Sommerfeld spectral problem. For its numerical solution, a spectral-grid method is employed, based on dividing the computational domain into elements and representing the solution on each element by a finite series of Chebyshev polynomials. The resulting generalized eigenvalue problem is reduced to a lower-dimensional system, followed by reconstruction of the eigenvector of the original system and the corresponding eigenfunction. Based on a selected physically relevant mode, the components of the perturbed velocity field are determined. For the test case, the computed eigenvalue is compared with a published reference numerical result; the influence of the distribution of the number of polynomials among the elements of the computational domain is also investigated. The obtained results confirm the applicability of the proposed computational procedure to the analysis of linear perturbations in the Blasius boundary layer.

Keywords

List of references

  1. Shoukat Z, Siddiqui AA, Yaseen MH, et al. An approximate solution of the Blasius problem using spectral method. Partial Differ Equ Appl Math. 2024. 11:100896. https://doi.org/10.1016/j.padiff.2024.100896.

  2. Rahman MM, Khan S, Akbar MA. Numerical and analytical solutions of new Blasius equation for turbulent flow. Heliyon. 2023. 9(3):e14319. https://doi.org/10.1016/j.heliyon.2023.e14319. ISSN 2010-7250 (Print) | Uzbek Journal “Problems of Mechanics” | 2026; 37(3); Article 7 72 of 73

  3. Van Stijn TL, Van De Vooren AI. An accurate method for solving the Orr–Sommerfeld equation. J Eng Math. 1980. 14:17–26. https://doi.org/10.1007/BF00042862.

  4. Dholey S. Unsteady boundary layer flow and heat transfer of a viscous fluid over a semi-infinite stationary flat plate: Blasius flow problem. Phys Scr. 2021. 96(2):025204. https://doi.org/10.1088/1402-4896/abcd69.

  5. Fazio R. Free boundary formulations for two extended Blasius problems. Int J Non-Linear Mech. 2023. 153:104423. https://doi.org/10.1016/j.ijnonlinmec.2023.104423.

  6. Al-Khaled K, Ajeel MS, Darweesh A, Al-Khalid H. A reliable algorithm for solving Blasius boundary value problem. Results Nonlinear Anal. 2024. 7(4):1–8. https://doi.org/10.31838/rna/2024.07.04.001.

  7. Zubarev VM. Analytical approximation of solutions to the Blasius problem in the boundary layer on a flat plate. Uchen Zap Kazan Univ Ser Fiz-Mat Nauki. 2025. 167(2):227–243. (In Russian). https://doi.org/10.26907/2541-7746.2025.2.227-243.

  8. Zubarev VM. Accurate analytical solutions of the velocity profile in the boundary layer on a plate (Blasius problem). J Eng Phys Thermophys. 2025. 98(3):644–654. https://doi.org/10.1007/s10891-025-03144-9.

  9. Grosch CE, Orszag SA. Numerical solution of problems in unbounded regions: coordinate transforms. J Comput Phys. 1977. 25(3):273–295. https://doi.org/10.1016/0021-9991(77)90102-4.

  10. Normurodov C, Tursunova BA. Numerical modeling of the boundary value problem of an ordinary differential equation with a small parameter at the highest derivative by Chebyshev polynomials of the second kind. Results Appl Math. 2023. 19:100388. https://doi.org/10.1016/j.rinam.2023.100388.

  11. Normuradov C, Djurayeva N, Deraman F, Mat Safar A, Mohd Asi S. One effective method for solving singularly perturbed equations. Malays J Sci. 2025. 44(1):63–69. https://doi.org/10.22452/mjs.vol44no1.8.

  12. Narmuratov C, Solov’ev AS. Stability of two-phase gas-solid particle flow in a boundary layer. Fluid Dyn. 1987. 22(2):217–221. https://doi.org/10.1007/BF01052251.

  13. Narmuratov C, Solov’ev AS. Effect of suspended particles on the stability of plane Poiseuille flow. Fluid Dyn. 1986. 21(1):38–44. https://doi.org/10.1007/BF01051098.

  14. Normurodov C, Ziyakulova S, Murodov SK. On one highly accurate and efficient method for solving the biharmonic equation. Int J Appl Math. 2025. 38(4):437–453. https://doi.org/10.12732/ijam.v38i4.1.

  15. Normurodov C, Djuraeva NT, Normatova MM. A highly accurate and efficient method for studying the dynamics of derivatives of different orders of a singularly perturbed equation. Chebyshevskii Sbornik. 2025. 26(4):357–369. (In Russ.). https://doi.org/10.22405/2226-8383-2025-26-4-357-369.

About the Authors

Chori Normurodov
ch.normurodov@gmail.com
Termez State University
Muhriddin Tilovov
mtilovov21@gmail.com
Termez State University
Sardorbek Murodov
smurodov870@gmail.com
Termez State University
Abdulla Raimov
raimovabdulla13@gmail.com
Termez State University
Nasiba Djurayeva
nasibajt@mail.ru
Surkhandarya Academic Lyceum of the Ministry of Internal Affairs

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How to Cite

A Spectral-Grid Method for the Stability Analysis of the Blasius Boundary Layer and the Dynamics of the Velocity Vector. (2026). Uzbek Journal “Problems of Mechanics”, 37(3). https://pmjournal.uz/jour/article/view/25

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ISSN 2010-7250 (Print)