Vectorial version of the general Stokes theorem and the Karman vortex street

F. -H. Li, Y. Kamata, S. Kanemitsu, K. Maruyama
2018 Communications in Numerical Analysis  
Our ultimate goal of our bachelor research is the study on the theory of lubrication, which is indispensable in reducing the friction-the most important challenge of the human being in the 21st century. In our paper [14] which is an outcome of our project research we have shown that by the use of chain rule together with differential forms coupled with the general form of Stokes' theorem, many results in fluid mechanics are made simpler and clearer. In particular, we elucidated the notion of
more » ... ed the notion of divergence and circulation in the 3-dimensional flow case. Further in the case of 2-dimensional flow by the use of complex analysis, we reestablish the results in that theorem. In this bachelor thesis our main purpose is to prove Theorem 3.3 to the effect that the Cauchy equation of motion implies the Navier-Stokes equation and to correct the missing coefficients in [7] . For this purpose we develop the vectorial version of the general Stokes theorem. From it we deduce the most important unicity theorem from which the equation of continuity follows immediately. In the same vein that the Cauchy equation of motion is a consequence of Newton's second law of motion follow immediately. We shall start the project of deriving the deepest results in vector analysis by the theory of generalized functions, which are known mainly as distributions. Our stand point is, however, that of [10] which incorporates the Sato hyper-function as a main tool for interpreting the whirl flow.
doi:10.5899/2018/cna-00331 fatcat:5ll2nql64beavdzxrbqhmfekb4