Stokes Flow: In class we saw that flow around a sphere of radius, a, at very...

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Stokes Flow: In class we saw that flow around a sphere of radius, a, at very low Reynolds number (Re << 1) can be computed with a streamfunction, &, in spherical coordinates with v = 0 and no o dependence: Remembering that: a Show that (r, 0) = Ua² 4 2 ² sin²(0) [2 (1) ² - 3 ur (r, 0) = up (r, 0) = 1 მს r² sin(0) 20 1 მს r sin(0) Ər 3 a ur(r, 0) = U cos(0) [1- [¹ - 2 + 1/ (9) ³] +#+0] 3 a ue(r, 0) Usin(0) -1 + = b Give a brief explanation why u is independent of viscosity. c Is the flow symmetric? + ()] P(r, 0) = P - (1) 3μaU 2r2 Cos (0) (2) (3) d Estimate to how many sphere radius the effect of the sphere is felt. Take the distance at which the velocity deficit is still 10% of free stream velocity U. e Show that the pressure field is: (4) (5) (6) where Poo is the free stream pressure f What can you conclude about the distribution of pressure in the front and back of the sphere? What does this imply for the drag on the sphere? g Show that the total drag, FD, found by integrating the pressure and shear around the sphere, is: Fn = 6πμUa The shear stress distribution in the fluid in spherical coordinates with symmetry is: Tre (r, 0) = μ ¹ 1 Jur ????ue ug + 20 Ər r h Which fraction of the total drag is contributed by pressure and shear stress. (7) (8) Stokes Flow: In class we saw that flow around a sphere of radius, a, at very low Reynolds number (Re << 1) can be computed with a streamfunction, &, in spherical coordinates with v = 0 and no o dependence: Remembering that: a Show that (r, 0) = Ua² 4 2 ² sin²(0) [2 (1) ² - 3 ur (r, 0) = up (r, 0) = 1 მს r² sin(0) 20 1 მს r sin(0) Ər 3 a ur(r, 0) = U cos(0) [1- [¹ - 2 + 1/ (9) ³] +#+0] 3 a ue(r, 0) Usin(0) -1 + = b Give a brief explanation why u is independent of viscosity. c Is the flow symmetric? + ()] P(r, 0) = P - (1) 3μaU 2r2 Cos (0) (2) (3) d Estimate to how many sphere radius the effect of the sphere is felt. Take the distance at which the velocity deficit is still 10% of free stream velocity U. e Show that the pressure field is: (4) (5) (6) where Poo is the free stream pressure f What can you conclude about the distribution of pressure in the front and back of the sphere? What does this imply for the drag on the sphere? g Show that the total drag, FD, found by integrating the pressure and shear around the sphere, is: Fn = 6πμUa The shear stress distribution in the fluid in spherical coordinates with symmetry is: Tre (r, 0) = μ ¹ 1 Jur ????ue ug + 20 Ər r h Which fraction of the total drag is contributed by pressure and shear stress. (7) (8)

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a Show that urr 0 U cos0 3 a 2 r uer 0 U sin0 1 12 3 a 4r Solution Using Equation 1 urr0 Uasin02r 3r r4 Substituting for Uasin0 we get urr0 Ucos03ar2r 3r r4 Using Equation 2 uer0 Usin03ar4 Substitutin... View the full answer