Sinner Souter Figure 1: Inner and outer closed surfaces

University of Warsaw Advanced Hydrodynamics

Faculty of Physics Selected Topics in Fluid Mechanics

Summer Semester 2019/20

Homework 5 Due April 13, 2020

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

1. Evaluate the force on a sphere of radiusa translating with velocity U and immersed in (a) an arbitrary linear flow,

u₀(r) = U0+ Γ0· r,

where U₀ and Γ₀ ≡ ∇u₀ are constants;

(b) a two-dimensional Poiseuille flow, u0(y) = G

2η(hy − y²) ˆex,

whereG is the negative of the pressure gradient and h is the distance between the two walls.

2. Transmission of force and torque. The hydrodynamic force, F^h, and torque, N^h, result- ing from the fluid stress on the surfaceS are, respectively,

F^h = Z

S

σ(r) · n(r) dS(r), N^h = Z

S

r × σ(r) · n(r) dS(r),

Show that Stokes flow “transmits” unchanged the hydrodynamic force and torque from an inner closed surface to an outer enclosing surface (shown in Fig.1).

Sinner

Souter

Figure 1: Inner and outer closed surfaces.

Consider one of two possible routes:

(a) use the divergence theorem applied to the Stokes momentum equation ∇ · σ = 0 in the fluid volumeV between the inner and outer closed surfaces;

(b) consider the fluid regionV between the surfaces and two velocity fields, u1and u₂ (with respective tensor stress fields σ₁and σ₂), both satisfing the Stokes equations inV . Then use the Loretz reciprocal theorem, viz.

Z

∂V

u₁· (σ2· n) dS = Z

∂V

u₂· (σ1· n) dS,

and choose for u₂ a rigid-body motion.

2