Keywords
collocation method
boundary value problem
MATLAB bvp4c
numerical simulation
How to Cite
Abstract
we performed a numerical study of steady thermocapillary flow of a viscous incompressible fluid in a flat channel with a stationary solid bottom and a free surface. We considered the creeping-flow regime, in which the only driving force was temperature-dependent surface tension. We used a quasi-one-dimensional expansion of the velocity, temperature, and pressure fields in the transverse coordinate, retaining terms up to second order in the longitudinal coordinate. We solved the resulting boundary-value problem for a system of ordinary differential equations using a fourth-order collocation method implemented in MATLAB with the bvp4c solver. We applied the computational algorithm for five values of the Marangoni number. We obtained profiles of the vertical and horizontal velocity components and the modal temperature and pressure, and constructed two-dimensional distributions of the stream function, pressure, and temperature. We found that two counter-flowing streams formed over the range of parameters considered, separated by a level of zero horizontal velocity, and determined the positions of the corresponding zeros. Analysis of the residuals of the governing equations confirmed the high accuracy of the numerical solution. We investigated the dependence of the flow structure on the governing parameter. The results provide a benchmark for testing numerical schemes for thermocapillary convection.
References
Ferziger J. H., Perić M., Street R. L. Computational Methods for Fluid Dynamics. Berlin: Springer; 2002. 596 p. DOI: 10.1007/978-3-319-99693-6.
Temam R. Navier–Stokes Equations: Theory and Numerical Analysis. Amsterdam: North-Holland; 2001. 500 p.
Canuto C., Hussaini M. Y., Quarteroni A., Zang T. A. Spectral Methods in Fluid Dynamics. Berlin: Springer; 1988. 567 p. DOI: 10.1007/978-3-642-84108-8.
Anderson J. D. Computational Fluid Dynamics: The Basics with Applications. New York: McGrawHill; 1995. 547 p.
Davis S. H. Thermocapillary Instabilities. Annual Review of Fluid Mechanics. 1987;19:403–435. DOI: 10.1146/annurev.fl.19.010187.002155.
Schatz M. F., Neitzel G. P. Experiments on Thermocapillary Instabilities. Annual Review of Fluid Mechanics. 2001;33:93–127. DOI: 10.1146/annurev.fluid.33.1.93.
Kuhlmann H. C. Thermocapillary Convection in Models of Crystal Growth. Berlin: Springer; 1999. 226 p. DOI: 10.1007/BFb0109562.
Smith M. K., Davis S. H. Instabilities of Dynamic Thermocapillary Liquid Layers. Part 1. Convective Instabilities. Journal of Fluid Mechanics. 1983;132:119–144. DOI: 10.1017/S0022112083001512.
Wanschura M., Kuhlmann H. C., Rath H. J. Three-Dimensional Instability of Axisymmetric Buoyant Convection in Cylinders Heated from Below. Journal of Fluid Mechanics. 1996;326:399– 415. DOI: 10.1017/S0022112096008373.
Peyret R. Spectral Methods for Incompressible Viscous Flow. New York: Springer; 2002. 432 p. DOI: 10.1007/978-1-4757-6557-1.
Ascher U. M., Mattheij R. M. M., Russell R. D. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Philadelphia: SIAM; 1995. 595 p. DOI: 10.1137/1.9781611971231.
Kierzenka J., Shampine L. F. A BVP Solver Based on Residual Control and the MATLAB PSE. ACM Transactions on Mathematical Software. 2001;27(3):299–316. DOI: 10.1145/502800.502801.
De Boor C., Swartz B. Collocation at Gaussian Points. SIAM Journal on Numerical Analysis. 1973;10(4):582–606. DOI: 10.1137/0710052.
Gladwell I., Shampine L. F., Thompson S. Solving ODEs with MATLAB. Cambridge: Cambridge University Press; 2003. 263 p. DOI: 10.1017/CBO9780511615542.
Shampine L. F., Gladwell I., Thompson S. Solving ODEs with MATLAB. New York: Cambridge University Press; 2003.
Trefethen L. N. Spectral Methods in MATLAB. Philadelphia: SIAM; 2000. 165 p. DOI: 10.1137/1.9780898719598.
Boyd J. P. Chebyshev and Fourier Spectral Methods. 2nd ed. New York: Dover Publications; 2001. 688 p.
Shampine L. F., Reichelt M. W., Kierzenka J. Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c. 2000. Режим доступа: https://www.mathworks.com/help/pdf_doc/matlab/index.html.
Kierzenka J., Shampine L. F. A BVP Solver That Controls Residual and Error. Journal of Numerical Analysis, Industrial and Applied Mathematics. 2008;3(1–2):27–41.
Ascher U. M., Petzold L. R. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Philadelphia: SIAM; 1998. 314 p. DOI: 10.1137/1.9781611971392.
Gubareva K. V., Eremin A. V. Numerical Solution of the Heat Conduction Problem in a Porous Plate with Topology of Triply Periodic Minimal Surfaces. Advanced Engineering Research (Rostovon-Don). 2025;25(1):23–31. DOI: 10.23947/2687-1653-2025-25-1-23-31.
Gubareva K. V., Prosviryakov E. Y., Eremin A. V. Inhomogeneous Couette–Poiseuille Flow of a Viscous Incompressible Fluid in an Infinite Horizontal Layer with Permeable Boundaries. Diagnostics, Resource and Mechanics of Materials and Structures. 2025;(5):6–28. DOI: 10.17804/2410-9908.2025.5.006-028.
Бурмашева Н. В., Просвиряков Е. Ю. Крупномасштабная слоистая стационарная конвекция вязкой несжимаемой жидкости под действием касательных напряжений на верхней границе. Исследование полей температуры и давления. Вестник Самарского государственного технического университета. Серия: Физико-математические науки. 2017;21(4):736–751. DOI: 10.14498/vsgtu1568.
Aristov S. N., Knyazev D. V., Polyanin A. D. Exact Solutions of the Navier–Stokes Equations with the Linear Dependence of Velocity Components on Two Space Variables. Theoretical Foundations of Chemical Engineering. 2009;43(5):642–662. DOI: 10.1134/S0040579509050066.
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