Abstract
A method to obtain a time-independent vortex solution of a nonlinear differential equation describing two-dimensional flow is investigated. In the usual way, starting from the Navier–Stokes equation the vortex equation is derived by taking a curl operation. After rearranging the equation of the vortex, we get a continuity equation or a divergence-free equation: \(\partial _1V_1+\partial _2V_2=0\). Additional irrotationality of \(V_1\) and \(V_2\) leads us to the Cauchy–Riemann condition satisfied by a newly introduced stream function \(\Psi\) and velocity potential \(\Phi\). As a result, if we know \(V_1\) and \(V_2\) or a combination of two, the differential equation is mapped to a lower-order partial differential equation. This differential equation is the one satisfied by the stream function \(\psi\) where the vorticity vector \(\omega\) is given by \(-(\partial _1^2+\partial _2^2) \psi\). A simple solution is discussed for the two different limits of viscosity.
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Notes
Note that the pressure, which plays a major role of moving force on fluids, does not appear in the vorticity equation.
When we consider \(\nu \rightarrow \infty\) or a low Reynolds number the Navier–Stokes equation becomes simplified to \(u_j\partial _j u_i=-(\partial _i p)/\rho +\nu \partial _j\partial _j u_i\). In this case we have the vorticity equation \(\partial _j\partial _j \omega _i=0\) which is a Laplace equation. In complex coordinate it can be written as \(\partial {\bar{\partial }}\omega _i=0\). This equation has a solution for any combination of holomorphic and antiholomorphic functions, \(f(z)+\bar{f}(\bar{z})\).
The full solution, if it exists, is a solution that connects the two solutions. Even though the two solutions at the extreme limit of \(\nu\) are singular the full analytic solution may not be singular. For this numerical study may be needed.
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Acknowledgements
This project is supported by the “R&D Center for reduction of Non-CO2 Greenhouse gases (RE2015001690003)” funded by Korea Ministry of Environment (MOE) as “Global Top Environment R&D Program”.
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Lee, S., Ryi, S. & Lim, H. About vortex equations of two dimensional flows. Indian J Phys 91, 1089–1094 (2017). https://doi.org/10.1007/s12648-017-0999-x
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DOI: https://doi.org/10.1007/s12648-017-0999-x