Question

We derived the stream function ψ(r,θ) and corresponding velocity compo-

nents vr,θ(r,θ) for a stationary sphere of radius R and a velocity field char-
acterized by a uniform velocity v=Uhat(z) far from the sphere. We showed that
the general solution consistent with the uniform far-field flow was

ψ(r,θ)=1/2Usin²θ[r²+arR+b R³/r],

where the dimensionless constants a and b depend on the boundary conditions
at the surface of the sphere. For the case of no penetration or slip at the
surface of the sphere, a=-32 and b=12, as we showed in class.

(a) For this general form of ψ in Eq. (1) with arbitrary values of a and
b, find the pressure gradient delPdelr from the Stokes equation and use this
to determine P(r,θ) in terms of various parameters, including the con-
stants a and b. Assume that the pressure vanishes far from the sphere.

Answer 1

To find the pressure gradient delPdelr using the Stokes equation, we can calculate the Laplacian of the stream function ψ and substitute it into the equation. By using the given expression for ψ and multiplying by the density, we can determine delPdelr in terms of the constants a and b.

Step-by-step explanation:

To find the pressure gradient deluder using the Stokes equation, we can first calculate the Laplacian of the stream function ψ using the polar coordinate system:

Δψ = 1/r * ∂/∂r(r * ∂ψ/∂r) + 1/r^2 * ∂²ψ/∂θ² = ∂²ψ/∂r² + (1/r^2) * ∂²ψ/∂θ²

Next, we can use the fact that the pressure gradient is given by deluder = -ρ * (∂²ψ/∂r² + (1/r^2) * ∂²ψ/∂θ²), where ρ is the density of the fluid. Since the pressure vanishes far from the sphere, we can set the constant term in ψ to zero:

ψ = 1/2 * U * sin²θ * (r² + Arri + bR³/r)

By substituting this expression into the Laplacian and multiplying by -ρ, we can obtain the pressure gradient deluder in terms of various parameters, including the constants a and b.