University of Warsaw Advanced Hydrodynamics
Faculty of Physics Selected Topics in Fluid Mechanics
Summer Semester 2019/20
Homework 6 Due April 27, 2020
Solutions should be sent togustavo.abade@fuw.edu.pl
1. Flow in a Hele-Shaw cell. Let viscous fluid be in steady flow between two flat, parallel, and rigid boundaries at z = 0 and z = h = const. (Hele-Shaw cell). The flow is being driven through the cell by a steady horizontal pressure gradient ∇p = G applied between the ends of the cell. Let U be a typical horizontal flow speed and L a typical horizontal length scale of the flow. Assume, in addition, that
h L. (1)
(a) Using the condition (1), show that the incompressible Navier-Stokes equations for the velocity field u = (u, v, w), in the absence of gravity, may be simplified to
∂p
∂x = η∂2u
∂z2, ∂p
∂y = η∂2v
∂z2, ∂p
∂z = η∂2w
∂z2, (2)
∂u
∂x +∂v
∂y + ∂w
∂z = 0. (3)
Show, in particular, that the neglect of the term (u · ∇)u does not require the conventional Reynolds number R = U L/ν to be small.
(b) Show that ∂p/∂z is small compared with the horizontal pressure gradients. Then p = p(x, y) and the first two equations in (2) may be trivially integrated with respect to z. Show that the horizontal velocity,
u = u(z)ex+ v(z)ey, is given by
u = − 1
2ηz(h − z)∇p, ∇ ≡ ∂
∂x ex+ ∂
∂y ey, (4)
(c) Show that in a Hele-Shaw cell, the streamline pattern of the flow in the x − y plane is independent of z and corresponds to a 2-D irrotational flow.
(d) The flow ¯u of a viscous fluid through a porous media is governed by the Darcy’s law
u = −¯ k
η∇p, (5)
where k is the permeability of the medium.
Show that bi-dimensional flows in porous media may be reproduced experimen- tally using a Hele-Shaw cell. To this end, introduce the mean velocities
¯ u = 1
h Z h
0
udz, v =¯ 1 h
Z h 0
vdz, (6)
and determine the condition for dynamical similarity between the flow pattern u = (¯¯ u, ¯v) in a Hele-Shaw cell and a 2-D flow through a porous media.
(e) Conservation of mass dictates that ∇ · ¯u = 0. Show that this implies that the pressure is a harmonic function,
∇2p = 0. (7)
Suppose there is one circular cylinder of radius a contained in the cell as illus- trated in Fig.11.
Figure 1: Circular cylinder in a Hele-Shaw cell.
Find the pressure distribution around the cylinder. Consider that the pressure must be linear in G [by linearity of the Laplace equation (7)] and the corresponding solution to (7) is of the form2
p(r) = p0+ α G · r + β G · r
r2 , (8)
1Some excellent photgraphs of flow past obstacles in a Hele–Shaw cell are contained in An Album of Fluid Motionby M. Van Dyke (Parabolic Press, Stanford, CA, 1982).
2Note that here we use the method of solution (superposition of harmonics) already discussed in the Exercise Sheet 4.
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where α and β are constants, p0 is an arbitrary pressure, and r is the position vector. The appropriate boundary conditions are zero flux of fuid at the surface of the cylinder and ∇p → G at large distances from the cylinder. Apply these conditions to deduce that
p(r) = p0+
1 −a2
r2
G · r. (9)
Show that the resulting velocity profile is u = − 1
2ηz(h − z)
1 − a2 r2
G + 2a2 r4 G · rr
. (10)
2. A rigid sphere of radius a falls under gravity through a Newtonian fluid of viscosity η towards a horizontal rigid plane. Use lubrication theory to show that, when the mini- mum gap h is very small, the speed V of approach of the sphere is
V (h) = h W 6πηa2,
where W is the weight of the sphere corrected for buoyancy.
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