U r ∈ ∂V, (3) where∂V denotes the surface of the sphere

University of Warsaw Advanced Hydrodynamics

Faculty of Physics Selected Topics in Fluid Mechanics

Summer Semester 2019/20

Exercise Sheet 3

Low Reynolds number flow past a translating sphere

Consider a sphere of radius a moving with a constant velocity U in an unbounded qu- iescent fluid of viscosity η. The problem is to determine the fluid flow u induced by the motion of the sphere and the friction force F^h that the fluid exerts on the sphere. We assume low-Reynolds number flow, so that the velocity and pressure distribution are governed by the stationary Stokes equations

0 = −∇p + η∇²u, ∇ · u = 0, (1)

with boundary conditions

u→ 0 as r → ∞, (2)

and

u(r) = U r ∈ ∂V, (3)

where∂V denotes the surface of the sphere.

We consider three different methods of solution:

1. A solution using the Stokes stream function may be found in [Acheson, ch. 7.2, p.

223–226].

2. An alternative solution is described in [Landau–Lifshitz, p. 58–61].

3. Below we describe in more detail the method based on the so-called force density approach. The Stokes equations take the form

0 = −∇p + η∇²u+ F(r), ∇ · u = 0, (4)

where F(r) is the force density exerted by the translating sphere on the fluid, so that the boundary condition (3) on∂V is satisfied.

The formal solution (obtained e.g. by Fourier transform technique) reads u(r) =

Z

T(r − r^′) · F(r^′)dr^′, (5)

p(r) = Z

g(r − r^′) · F(r^′)dr^′, (6)

where

T(r) = 1

8πηr[1 + ˆrˆr], (7)

g(r) = 1

4πr²ˆr, (8)

are, respectively, the Green’s functions for velocity and pressure, with ˆr = r/r [see https://arxiv.org/abs/1312.6231for details on the derivation of the hydrodynamic Green’s functions].

We need a closure or constitutive equation for the force density F(r), which is concen- trated on the surface∂V of the sphere (assumed to be centered at the origin),

F(r) = f(θ, ϕ)δ(r − a). (9)

The simplest guess (considering the linearity of the equations) is to assume that f(ˆr) is proportional to the local velocity of the surface element on the sphere,

f(ˆr) = c

4πa²U, (10)

with the constantc to be determined from the boundary conditions.

Then from Eq. (5),

u(r) = c

4πa² 1 8πη

Z

∂V

dS^′ 1

|r − r^′|



1+(r − r^′)(r − r^′)

|r − r^′|²



· U

≡ c

4πa² 1

8πηJ(r) · U, (11)

where

J(r) = 4πa a r +1

3

a r

³

1+ a r −a

r

³ ˆrˆr



, (12)

and ˆr = r/r. Explicit calculation of the integral J(r) is in the Appendix. It was taken from [J. K. G. Dhont, An introduction to dynamics of colloids. Elsevier, 1996].

Forr = a

J(r = a) = 16

3 πa1, (13)

2

and

u(r = a) = c 4πa²

1 8πη

16

3 πa1 · U = U, (14)

yieldingc = 6πηa.

Then the solution (for a sphere with center at R) is

u(r) = M(r − R) · U, (15)

where

M(r) = 3 4

a

r [1 + ˆrˆr] +1 4

a r

³

[1 − 3ˆrˆr] . (16)

The solution u is linear in U. Reversing the direction of U merely leads to a change of the sign of u everywhere (kinematic reversibility). From the form of M(r), the distur- bance due to the moving sphere extends to a considerable distance from the sphere, the velocity decaying as1/r at large values of r.

The hydrodynamic force that the fluid exerts on the sphere is F^h = −

Z

∂V

f(r)dS = −6πηaU. (17)

This is the Stokes’ friction law for the translational motion of a sphere.

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Appendix

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