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# Analytical Solution of Stokes Flow in a Driven Cavity Using the Natural Boundary Element Method

Peng Weihong1,2, Gao Feng1, Cao Guohua3, Xu Yong2, Cheng Hongmei1
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, China.
School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, China.
School of Mechatronic Engineering, China University of Mining and Technology Xuzhou, China.

Computer Modeling in Engineering & Sciences 2014, 100(2), 133-155. https://doi.org/10.3970/cmes.2014.100.133

### Abstract

In this paper, the natural boundary element method is used to solve two-dimensional steady-state incompressible Stokes flows in a driven cavity. The analytical functions are expressed for the Stokes problem in an exterior circular domain under single value conditions, which satisfy the Stokes equations’ solutions in the form of complex functions. In order to obtain a uniform integral formula, the velocities on the boundary are expanded into Laurent series, and then compared with the analytical solutions obtained as described above. In this manner, the coefficients of the analytical solutions in the form of complex function are further confirmed. According to the formulae of Fourier series and convolutions, the boundary integral formulae related only to boundary velocities are obtained for the Stokes problems in an exterior circular domain. Similarly, the boundary integral formulae are investigated for the interior circular domain. The formulae are applied to the Stokes flows in some circular and annular cavities, and the obtained results are compared with those produced by previous works. The results prove that the current technique is in very good agreement with previous investigations, and that the natural boundary element method is an accurate and flexible method for the solution of Stokes flows.

### Keywords

Natural boundary element method, Stokes flow, driven cavity, Fourier series and convolutions, boundary integral formulae.

Weihong, P., Feng, G., Guohua, C., Yong, X., Hongmei, C. (2014). Analytical Solution of Stokes Flow in a Driven Cavity Using the Natural Boundary Element Method. CMES-Computer Modeling in Engineering & Sciences, 100(2), 133–155.

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