Verify that in the case of a simple shear flow, u = [u(y), 0, 0], Eq

University of Warsaw Advanced Hydrodynamics

Faculty of Physics Selected Topics in Fluid Mechanics

Summer Semester 2019/20

Homework 1 Due 9:15, March 3, 2020

1. The stress vector t on a surface element with normal n may be written as t = Σ · n,

where Σ is the stress tensor.

For an incompressible, Newtonian viscous fluid of viscosityµ, Σ = −p1 + µ(∇u + ∇u^T).

Show that the stress vector may be written as

t = −pn + µ[2(n · ∇)u + n × (∇ × u)]. (1)

2. Verify that in the case of a simple shear flow, u = [u(y), 0, 0], Eq. (1) reduces, when n = (0, 1, 0), to

t =

 µdu

dy, −p, 0

 .

3. Derive expressions which describe (a) the velocity distribution and (b) the shear stress (τ = Σxy) distribution across the two fixed parallel plates for a steady flow of two layers of fluid under a pressure gradient∂p/∂x = −k, as shown in the figure below.

ρ₁, µ₁ ρ₂, µ₂

y x

h

h

Additional problem

1. Show that in the case of purely rotary flow, u =u(r)ˆeθ, Eq. (1) reduces, when n = ˆe_r, to

t = −pˆer+µr d dr

u r

ˆe_θ,

and note that the second term vanishes in the case of uniform rotation,u ∝ r, for there is then no deformation of fluid elements.

2