2 and ν = 1 . 0 × 10 − 6 m 2 / s. Hence, Re = ( a / 6 × 10 − 5 ) 3 , where a is measured in meters. Thus, expression (9.128), which is strictly speaking only valid when Re ≪ 1, but which turns out to be approximately valid for all Reynolds numbers less than unity, only holds for sand grains whose radii are less than about 60 microns. Such grains fall through water at approximately 8 × 10 − 3 m / s. For the case of a droplet of water falling through air at 20 ◦ C and atmospheric pressure, we have ρ/ρ = 780 and ν = 1 . 5 × 10 − 5 m 2 / s. Hence, Re = ( a / 4 × 10 − 5 ) 3 , where a is measured in meters. Thus, expression (9.128) only holds for water droplets whose radii are less than about 40 microns. Such droplets fall through air at approximately 0 . 2 m / s. At large values of r / a , Equations (9.105), (9.106), and (9.112) yield v r ( r ,θ ) = − V cos θ + 3 2 V cos θ a r + O parenleftbigg a r parenrightbigg 2 , (9.130) v θ ( r ,θ ) = V sin θ − 3 4 V sin θ a r + O parenleftbigg a r parenrightbigg 2 . (9.131) It follows that [ ρ ( v · ∇ ) v ] r = ρ v r ∂v r ∂ r + v θ r ∂v r ∂θ − v 2 θ r ∼ ρ V 2 a r 2 , (9.132) and ( μ ∇ 2 v ) r ∼ μ ∂ 2 v r ∂ r 2 ∼ μ V a r 3 . (9.133) Hence, [ ρ ( v · ∇ ) v ] r ( μ ∇ 2 v ) r ∼ ρ V r μ ∼ Re r a , (9.134) where Re is the Reynolds number of the flow in the immediate vicinity of the sphere. [See Equation (9.94).] Now, our analysis is based on the assumption that advective inertia is negligible with respect to viscosity. However, as is clear from the above expression for the ratio of inertia to viscosity within the fluid, even if this ratio is much less than unity close to the sphere—in other words, if Re ≪ 1—it inevitably becomes much greater than unity far from the sphere: i.e. , for r ≫ a / Re. In other words, inertia always dominates viscosity, and our Stokes flow solution therefore breaks down, at su ffi ciently large r / a . 9.11 Axisymmetric Stokes Flow In and Around a Fluid Sphere Suppose that the solid sphere discussed in the previous section is replaced by a spherical fluid drop of radius a . Let the drop move through the surrounding fluid at the constant velocity V e z . Obviously, the fluid from which the drop is composed must be immiscible with the surrounding fluid. Let us transform to a frame of reference in which the drop is stationary, and centered at the origin. Assuming that the Reynolds numbers immediately outside and inside the drop are both much less than unity, and making use of the previous analysis, the most general expressions for the stream function outside and inside the drop are ψ ( r ,θ ) = sin 2 θ parenleftbigg A r + B r + C r 2 + D r 4 parenrightbigg , (9.135) and ψ ( r ,θ ) = sin 2 θ A r + Br + C r 2 + D r 4 , (9.136) respectively. Here, A , B , C , etc. are arbitrary constants. Likewise, the previous analysis also allows us to deduce that v r ( r ,θ ) = − 2 cos θ parenleftbigg A r 3 + B r + C + D r 2 parenrightbigg , (9.137) v θ ( r ,θ ) = sin θ parenleftbigg − A r 3 + B r + 2 C + 4 D r 2 parenrightbigg , (9.138)
186 FLUID MECHANICS − 2 − 1 0 1 2 z/a − 2 − 1 0 1 2 x/a Figure 9.7: Contours of the stream function in the x-z plane for Stokes flow in and around a fluid sphere.