Suppose I have some vector field equations $(f(\theta,\phi), g(\theta,\phi))$. The StreamPlot can be created easily in 2D, but I would like to visualize the stream line plot in a 3D spherical surface $(\theta,\phi)$, so how can it be done? As an example:
StreamPlot[{Cot[θ]Cos[ϕ],- Sin[ϕ]}, {ϕ,-π,π},{θ,0,π}, StreamColorFunction->Hue] 
Ideally, the front of spherical surface should be partially transparent so that the flow line at the rear end can be visualized at the same time.
An alternative way to post-process the StreamPlot output into a Graphics3D object using @user18792's trick:
sp = StreamPlot[{Cot[θ] Cos[ϕ], -Sin[ϕ]}, {ϕ, -π, π}, {θ, 0, π}, StreamColorFunction -> Hue, ImageSize -> 400]; sp3d = Graphics3D[sp[[1]] /. Arrow[z_] :> Arrow[z /. {x_Real, y_Real} :> {Cos[x] Sin[y], Sin[y] Sin[x], Cos[y]}], ImageSize -> 400]; Row[{sp, sp3d}, Spacer[5]] 
To add a semi-transparent Sphere:
Graphics3D[{sp[[1]] /. Arrow[z_] :> Arrow[z /. {x_Real, y_Real} :> {Cos[x] Sin[y], Sin[y] Sin[x], Cos[y]}], Opacity[.5], Sphere[]}, ImageSize -> 400] 
gr = Normal@StreamPlot[{Cot[θ] Cos[ϕ], -Sin[ϕ]}, {ϕ, -π, π}, {θ, 0, π}, StreamColorFunction -> Hue]; Graphics3D[Cases[gr, _Arrow, Infinity] /. {x_Real, y_Real} :> {Cos[x] Sin[y], Sin[y] Sin[x], Cos[y]}] 
