An Application of Homotopy Analysis to the Viscous Flow Past a Circular Cylinder

Abstract

We consider the application of a new analytic method based on homotopy analysis to the solution of the steady flow of a viscous incompressible fluid past a fixed circular cylinder. The solutions obtained using this method produce some interesting results. For instance, an analytic verification of the critical Reynolds number $R_d$ for which a standing vortex first appears behind the cylinder is given for the first time and found to be $R_d\preccurlyeq 2.4$. Since these values of the critical Reynolds number are beyond the range of validity of Oseen theory, no analytic verification of this value had previously been given. As a check on the accuracy of the solutions, the calculated drag coefficients at 6th-order approximation are found to agree reasonably well with experimental measurements for $R_d\simeq 30$ which is considerably larger than the $R_d\simeq 1$ results currently available using other analytic techniques. This buttresses the usefulness of the homotopy analysis method (HAM) as an important tool in solving highly nonlinear problems.