Elastico-viscous boundary-layer flow on the surface of a sphere

Summary The Prandtl boundary-layer theory is extended for an idealized elastico-viscous liquid. The boundary layer equations are solved approximately by Kármán-Pohlhausen technique for the case of a sphere. It is shown that the increase in the elasticity of the liquid causes a shift in the point of separation towards the forward stagnation point.

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Zusammenfassung Die Prandtlsche Grenzschicht-Theorie wird für eine idealisierte viskoelastische Flüssigkeit erweitert. Die Grenzschichtgleichungen werden für den Fall einer angeströmten Kugel näherungsweise mit Hilfe der Kármán-Pohlhausen-Methode gelöst. Es wird gezeigt, daß das Anwachsen der Flüssigkeitselastizität eine Verschiebung des Ablösepunktes auf den vorderen Staupunkt hin zur Folge hat.

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Nomenclature b ^ik arbitrary contravariant tensor - D non-dimensional boundary layer thickness - g _ik metric tensor of a fixed coordinate system - K curvature at any point on the generating curve - K ₀ elastico-viscous parameter - p arbitrary hydrostatic pressure - p _ik stress tensor - p ^ik part of stress tensor associated with the change of shape of material - R radius of the sphere - r radius of any transverse cross-section of the sphere - t time - U potential velocity around the body - U stream-velocity at a large distance from the body - u, w velocity components along (x, z) directions respectively - x distance measured along a generating line from the forward stagnation point - z distance measured along a normal to the surface - non-dimensional elastico-viscous parameter - density of the liquid - boundary layer thickness - convected time derivative - ₀ limiting viscosity for very small changes in deformation velocity - angle measured along the transverse direction - x/R - v kinematic coefficient of viscosity - T _s shearing stress on the surface of the sphere With 2 figures and 1 table