On the Asymptotic Solution of Laminar Channel Flow with Large Suction
Chunqing Lu · SIAM Journal on Mathematical Analysis · 1997
The equation considered is \[ \epsilon f^{iv}=ff^{\prime \prime \prime }-f^{\prime }f^{\prime \prime }, \]with boundary conditions \[ f(0)=f^{\prime \prime }(0)=0,f(1)=1,f^{\prime }(1)=0. \] When $0 < \epsilon \ll 1$, the boundary value problem corresponds to the laminar flow of a viscous fluid through a porous channel under large suction. It is known that there are three solutions in this case: two of them are monotone increasing (types I and II), and the third is nonmonotone (type III). Let $(1-\Delta )$ be the turning point of $f(\eta )$ in (0, 1). This paper presents a rigorous proof of the asymptotic behavior of type III solutions, which is \[ f(\eta )\sim \kappa \sin \frac{\pi \eta }{1-\Delta },\ \ \mbox{where }\ \kappa \sim \frac{1-\Delta }{\pi \Delta }\ \mbox{and } \frac{\Delta }{\epsilon }e^{\Delta /\epsilon }\sim \frac{1}{2e\pi ^{9}\epsilon ^{8}}, \] uniformly on $[0,1-\Delta ]$ as $\epsilon \rightarrow 0^{+}$, and provides detailed information at the turning point.