The velocity field of a Newtonian fluid flowing around a sma

The velocity field of a Newtonian fluid flowing around a small sphere is given by: V_r = V_ infinity(a^2/2 r^3 - 3a/2r + 1) cos theta V_theta = V_ infinity(a^3/4 r^3 + 3a/4r - 1) sin theta V_phi = 0, where (r, theta, phi) are the spherical coordinates. V_r, V_theta and V_phi are the spherical physical components of the velocity field and a is the radius of the sphere. Tell whether the flow is steady, incompressible and/or irrotational. Compute the drag over the sphere. Compute the terminal velocity of a small sphere. Determine the updraft that would keep a sand grain afloat in air for spheres of radii 0.1mm and 10/mu n.

Solution

a) The flow is steady since the velocity expression doesnt contain any time variable. That is, velocity is independent of time so the flow must be steady.