Interdisciplinary Applied Mathematics

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6-4-1 External Flow


For external flow past a sphere in the Stokes flow regime (Re ^ 0), the drag on the sphere is given by the Stokes drag:


Fd = 6npUR,    (6.19)


where p is the absolute viscosity of the fluid, U is the external flow velocity, and R is the sphere radius. Nondimensionalizing the drag by the dynamic head and the cross-sectional area of the sphere gives the drag coefficient Cd:


_ FD _ 12 p _ 12 D ^pU2TrR2 pUR Re’


where p is the fluid density, and Re = ELF This classical result shows that the drag coefficient for the sphere is inversely proportional to the Reynolds number in the Stokes flow regime (Re ^ 0). For Re « 1, corrections to the Stokes formula due to increased inertial effects are necessary. A well-known approximation for this case is given by Oseen (Batchelor, 1998).


In the slip flow regime (Kn < 0.1), the total drag on the sphere is given by (Barber and Emerson, 2002)





The total drag can be decomposed into three components due to the skin-friction, viscous normal stresses and the pressure drag.


Skin-friction drag:









1 + 4ff Kn

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