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The Lift On A Small Sphere Touching A Plane In The Presence Of A Simple Shear Flow

D. Leighton, A. Acrivos
Published 1985 · Mathematics

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AbstractThe 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/Fx≐0.287R whereFx is the lateral viscous force on the sphere.ZusammenfassungDer 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/Fx≐0.287R; dabei istFx die seitliche Zähigkeitskraft auf die Kugel.
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