The generalized Blasius equation revisited. (English) Zbl 1144.76307
Summary: The main subject of this paper is the model \((|f^{\prime\prime}|^{n-1}f^{\prime\prime})^{\prime}+\frac{1}{n+1}ff^{\prime\prime}=0,f(0)=f^{\prime}(0)=0, f^{\prime}(\infty)=1\), arising in the study of a 2D laminar boundary-layer with power-law viscosity, \(f=f(\eta) \) is the non-dimensional stream function and \(n>0\). Besides proving the existence and uniqueness results, we investigate the precise behavior of \(f\) for small and large \(\eta\). In particular, for \(n>1\) we show that \(f(\eta)\) is linear for \(\eta \geq \eta_0 >0\). This means that in original \((x,y)\) coordinates, the domain \(y \geq a_0x^{\frac{1}{n+1}}, a_0 = \text{const.}\), the velocities \(u = \text{const.}, v=0\) and the boundary-layer is the domain \(y < a_0x^{\frac{1}{n+1}}\) (“spatial localization of the layer”).
MSC:
| 76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |
| 34B40 | Boundary value problems on infinite intervals for ordinary differential equations |
| 76A05 | Non-Newtonian fluids |
Keywords:
non-Newtonian fluids; boundary-layer; Blasius equation; degenerate differential equation; Crocco transformationReferences:
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