4 2.5.1 Physical Deductions 1.Streamlines:With respect to the the equator along 0=/2,cos 0 and gr are odd while sin 0 and go are even.Hence the streamlines(velocity vectors)are symmetric fore and aft. 2.Vorticity: 5-SoEo 18(rge)18gr sin r08 >Wa- 3.Pressure From the r-component of momentum equation Op uWa Or cos(=-V×(V×) Integrating with respect to r from r to oo,we get 3 uWa p=p∞一 23c0s9 (2.5.20) 4.Stresses and strains: 1 8gr =W cos0 3a3 Or 2r2 2r4 On the sphere,r=a,err =0 hence orr=0 and Trr =-p+Orr=-Poo+ 3uW cos0 (2.5.21)） 2a On the other hand 9 10g,3Wa -sin0 r00 2r4 Hence at r=a: 3uW T0=00=er0=-2a sin (2.5.22) The resultant stress on the sphere is parallel to the z axis. ∑：=Trr COS9-Tr8sin0=-P Cos0+ 3uw 2a The constant part exerts a net drag in z direction D= de sin :==a -4ra2=6πμWa (2.5.23) This is the celebrated Stokes formula. A drag coefficient can be defined as D 6πuWa 24 24 Co-pWiza=ipWira =eW(2a)= (2.5.24) Rea4 2.5.1 Physical Deductions 1. Streamlines: With respect to the the equator along θ = π/2, cos θ and qr are odd while sin θ and qθ are even. Hence the streamlines (velocity vectors) are symmetric fore and aft. 2. Vorticity: ~ζ = ζφ~eφ Ã 1 r ∂(rqθ) ∂r − 1 r ∂qr ∂θ ! ~eφ = −3 2 W asin θ r2 ~eφ 3. Pressure : From the r-component of momentum equation ∂p ∂r = µW a r3 cos θ(= −µ∇ × (∇ × ~q)) Integrating with respect to r from r to ∞, we get p = p∞ − 3 2 µW a r3 cos θ (2.5.20) 4. Stresses and strains: 1 2 err = ∂qr ∂r = W cos θ Ã 3a 2r2 − 3a3 2r4 ! On the sphere, r = a, err = 0 hence σrr = 0 and τrr = −p + σrr = −p∞ + 3 2 µW a cos θ (2.5.21) On the other hand erθ = r ∂ ∂r µqθ r ¶ + 1 r ∂qr ∂θ = −3 2 W a3 r4 sin θ Hence at r = a: τrθ = σrθ = µerθ = −3 2 µW a sin θ (2.5.22) The resultant stress on the sphere is parallel to the z axis. Σz = τrr cos θ − τrθ sin θ = −p∞ cos θ + 3 2 µW a The constant part exerts a net drag in z direction D = Z 2π o adφ Z π o dθ sin θΣz == 3 2 µW a 4πa2 = 6πµW a (2.5.23) This is the celebrated Stokes formula. A drag coefficient can be defined as CD = D 1 2 ρW2πa2 = 6πµW a 1 2 ρW2πa2 = 24 ρW(2a) µ = 24 Red (2.5.24)