A hybrid asymptotic-numerical method for calculating drag coefficients in 2-D low Reynolds number flows

Steady-state incompressible low-Reynolds-number fluid flow past a cylindrical body in an unbounded two-dimensional domain is a singular perturbation problem involving an infinite logarithmic expansion in the Reynolds number \({\displaystyle \varepsilon }\) as \({\displaystyle \varepsilon }\rightarrow 0\). The central difficulty with applying a conventional matched asymptotic approach to this problem is that only the first few terms in the infinite logarithmic expansion of the drag coefficient and of the flow field can be calculated analytically. To overcome this difficulty, a hybrid asymptotic-numerical method that incorporates all logarithmic correction terms is implemented for three low-Reynolds-number flow problems. In particular, for a nanocylinder of circular cross section with surface roughness, modeled by a Navier boundary condition involving a sliplength parameter, a hybrid asymptotic-numerical method is formulated and implemented to determine an approximation to the drag coefficient that is accurate to all powers of \({-1/\log {\displaystyle \varepsilon }}\). A similar analysis is done to determine a corresponding approximation of the drag coefficient for a porous cylinder, where the flow inside the cylinder is modeled by the Brinkman equation. For both the nano- and porous-cylinder problems, the hybrid asymptotic-numerical method is extended to calculate the first transcendentally small correction term to the Stokes flow near a body. This term, which governs weak upstream/downstream asymmetry in the Stokes flow, is extrapolated to finite \({\displaystyle \varepsilon }\) to predict the formation of any eddies near the body. Finally, the hybrid method is used to determine the drag coefficient, valid to within all logarithmic terms, for two identical cylinders of circular cross section in tandem alignment with the free stream. An extension of the theoretical framework to more general slow viscous flow problems is discussed.