1)what are the boudary conditions for the flow of a viscous fluid past a sphere ?

1. slip condition all over the surface, if it not

rotate must be applied..

2. pressure on the surface can be simply extrapolated

to ensure dp/pn=0

-I agree that for projection scheme of the simple family the BC for pressure is dp/dn=0.

-But for other projection schemes that use incomplete splitting the BC must be of the form

d(delta p)/dn=(1/dt)vn where vn is the "starred" velocity normal to the boundary thas has to be changed, i.e:

v(t=t^n+1)=vn-dt*d(delta p)/dn

-My point is that you BC depends on the scheme you use.

If you could give us more info, our answer would be more adequate.

For god's sake don't put some vague querry.Give more info about the scheme