Slip at the surface of a sphere

University of Warsaw Advanced Hydrodynamics

Faculty of Physics Selected Topics in Fluid Mechanics

Summer Semester 2019/20

Homework 3 Due March 27, 2020

Solutions should be sent togustavo.abade@fuw.edu.pl

1. Slip at the surface of a sphere. Consider the problem of streaming flow past a sta- tionary sphere of radius a. The fluid of viscosity η streams in the negative z direction with uniform velocity U at points infinitely distant from the sphere. It is of interest to examine the possibility that fluid may slip at the surface of the sphere. We assume that the tangential fluid velocity v_θ relative to the solid at a point on the sphere surface is proportional to the stress at that point. This assumption takes the form,

σ_rθ = βv_θ at r = a. (1)

For axisymmetric flows,

v = [v_r(r, θ), v_θ(r, θ), 0], (2)

the stress is σ_rθ = η 1

r

∂v_r

∂θ + r ∂

∂r

v_θ r



. (3)

As a purely kinematic condition, the relative normal velocity v_rvanishes at the surface of the sphere.

(a) Reformulate the condition (1) in terms of the Stokes stream function Ψ(r, θ) such that

v_r = 1 r²sin θ

∂Ψ

∂θ, v_θ = − 1 r sin θ

∂Ψ

∂r. (4)

(b) Show that the general solution has the form Ψ = sin²θ A

r + Br + Cr²+ Dr⁴



, (5)

and determine the constants by applying the appropriate boundary conditions.

(c) Show that the force on the sphere is F = −6πηaU βa + 2η

βa + 3η,

and discuss the particular cases of no slip (β = ∞) and perfect slip (β = 0).

2. Fluid sphere: Hadamard-Rybczy´nski solution. It is of relevance to consider the exis- tence of an internal circulation inside drops falling through another immiscible liquid.

Here we assume that a spherical drop of radius a and viscosity ηiis "macroscopically”

at rest while the external fluid of viscosity η_o streams past it with velocity U (in the direction of z negative). Typical streamlines for both the internal and external motions are shown in Fig. 1. We distinguish between the fluid motions occurring inside and outside the drop by the indices i and o, respectively. Surface tension γ acting at the in- terface between the two fluids tends to maintain a spherical shape of the drop (against the shearing stresses which tend to deform it).

(a) The boundary conditions for velocity are:

i. uniform flow with velocity U at infinity;

ii. kinematic condition of mutual impenetrability at the interface;

iii. the tangential velocity must be continuous across the interface of the two immiscible fluids;

iv. the tangential stress σ_rθmust be continuous across the interface¹.

Assume that the internal and external flows are axisymmetric and reformulate the boundary conditions above in terms of the Stokes stream functions Ψ⁽ⁱ⁾and Ψ^(o). They are related to the velocity components as given by Eq. (4).

(b) Obtain the stream functions Ψ⁽ⁱ⁾and Ψ^(o). They have the form given by Eq. (5).

(c) Show that the drag on the spherical drop is F = −6πη_oaU 1 + ²₃λ

1 + λ ,

where λ ≡ ηo/η_i is the viscosity ratio. Why does the interfacial tension γ does not figure explicitly in the results? Discuss the particular cases where η_i  η_o and ηi  ηo.

1The surface tension γ introduces a discontinuity in the normal stress σrracross the interface. This manifests itself by the pressure difference (given by the Young–Laplace equation),

p⁽ⁱ⁾= p^(o)+2γ

a , (6)

at each point on the spherical interface.

2

Figure 1: Streamlines for liquid droplet showing internal circulation.

3