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The lift on a small sphere touching a plane in the presence of a simple shear flow

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Abstract

The contribution from purely viscous forces to the liftL on a sphere of radiusa touching a plane in the presence of a shear flow field of strength\(\dot \gamma \) is zero. An exact integral expression for the lift to leading order in the Reynolds number\(R \equiv \dot \gamma a^2 /v\) is derived using known creeping flow solutions to related problems. The integral is evaluated numerically to obtain the value of the lift\({L \mathord{\left/ {\vphantom {L {\dot \gamma \mu a^2 }}} \right. \kern-\nulldelimiterspace} {\dot \gamma \mu a^2 }}\dot = 9.22R\) orL/F x ≐0.287R whereF x is the lateral viscous force on the sphere.

Zusammenfassung

Der Beitrag der Zähigkeit allein zum AuftriebL an einer Kugel vom Radiusa, welche in einer Scherströmung mit dem Gradienten\(\dot \gamma \) eine unendliche Ebene berührt ist null. Ein exakter Integralausdruck für den Auftrieb wird in erster Ordnung der Reynoldszahl,\(R \equiv \dot \gamma a^2 /v\), hergeleitet unter Benützung bekannter verwandter Lösungen in schleichender Strömung. Der Wert des Integrals wird numerisch bestimmt und gibt für den Auftrieb\({L \mathord{\left/ {\vphantom {L {\dot \gamma \mu a^2 }}} \right. \kern-\nulldelimiterspace} {\dot \gamma \mu a^2 }}\dot = 9.22R\) oderL/F x ≐0.287R; dabei istF x die seitliche Zähigkeitskraft auf die Kugel.

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Leighton, D., Acrivos, A. The lift on a small sphere touching a plane in the presence of a simple shear flow. Z. angew. Math. Phys. 36, 174–178 (1985). https://doi.org/10.1007/BF00949042

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  • DOI: https://doi.org/10.1007/BF00949042

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