Fluid Mechanics IB
Lecturer: Dr Natalia Berloff
Office: CMS G1.02
E-mail: N.G.Berloff@damtp.cam.ac.uk
Class web page: www.damtp.cam.ac.uk/user/ngb23/FD/
0. INTRODUCTION
What is a fluid?
By definition, a fluid cannot support a shear stress while at rest. Ordinary gases (such as
air) and liquids (such as water and milk) are all fluids.
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Motivation
This course applies mathematical techniques, such as vector calculus, integral theorems
and determination of solutions of Laplace’s equation, to describe of a large variety of
physical problems involving fluid motion. The course develops mathematical models of
real practical importance, applying for instance to:
prediction of lift force on aircraft wings
surface wave dynamics and propagation speeds
pressure distribution near collapsing bubbles (with implications for damage to ship propellors, high-speed pumps, and other machinery using high-energy flows)
forces and energy loss in converging or diverging pipes or in impact of jets
flow variations in rivers and over weirs
Beyond this course — fluid dynamics enters many important problems, such as:
industrial flows: hydroelectric power, chemical processing, jet-driven cutting tools
our fluid environment: ozone loss, climate change, ocean currents and eddies, weather and
climate forecasting, volcanic eruptions, flow inside and around buildings
geophysical and astrophysical flows: magma flow and mineral deposition, motion of continents, dynamo action in planets and stars, stellar pulsation, inflow to black holes.
computer graphics: simulations of water, clouds, smoke, tornado etc.
Alternatively, from a mathematical viewpoint, we will study what techniques have been
developed to determine the solutions for the velocity field u(x, t) and the pressure field
p(x, t) of the nonlinear partial differential equations
∂u
+ (u · ∇)u = −∇p + F,
∇·u=0
ρ
∂t
given ρ, F and appropriate boundary conditions.
Books:
• Acheson: Elementary Fluid Dynamics
• Paterson: A First Course in Fluid Dynamics
• Batchelor: Introduction to Fluid Dynamics
• Lighthill: An Informal Introduction to Theoretical Fluid Mechanics
• van Dyke: Album of Fluid Motion
1. KINEMATICS
§1.1 Continuum model; Lagrangian and Eulerian descriptions
Gases and liquids are made up of individual molecules, e.g. air at STP(0◦ C, 1 atmosphere)
−9
−8
has 2.7 × 1025 molecules per m3 , molecules <
∼ 10
m long, intermolecular spacing <
∼ 10
m,
−7
2
−1
mean free path between collisions ∼ 10 m and r.m.s. velocity ∼ 10 m s .
On larger scales of everyday experience, fluid appears as smooth continuum. Quantities
such as density and velocity are continuous functions of position. Define the continuous
fields of velocity u(x, t), density ρ(x, t) and pressure p(x, t) by averaging over small linear
dimension L3 ,
mass in box centred at x
e.g.
ρ(x, t) =
volume of box (∼ L3 )
−5
with 10−8 m L length of continuum variation (e.g. >
∼ 10
m for ordinary turbulent
flow in a teacup during vigorous stirring). The purpose of this averaging is to exchange
an enormous number of ordinary differential equations that describe the motion of each
molecule for a few partial differential equations that describe the averaged field.
This continuum approximation does not always give a good model of fluid motion. An
example is a spacecraft re-entering the atmosphere: intermolecular spacings and free paths
are not small in comparison with spacecraft lengthscales.
For the rest of the course we
shall assume that the continuum model is adequate — which is valid under an enormous
range of conditions.
Lagrangian vs. Eulerian description.
Lagrangian description of the motion follows individual fluid parcels/particles. In this
description, ρ, u, p, etc. are functions not of position vector x and t but of the initial
position of the particle x0 and t; and the position x = X(x0 , t) of each fluid parcel has to
be determined as part of the solution.
Eulerian description: works directly in terms of fields such as density ρ(x, t), velocity
u(x, t) and pressure p(x, t). Notice that if we keep x fixed and let t vary in u(x, t) (and if
u 6= 0), then we are looking at the velocities of a succession of fluid parcels moving past a
given point whose position is x.
§1.2 Flow visualization: pathlines, streaklines and streamlines
There are three distinct types of line or curve commonly used when visualizing fluid motion
for given velocity field u(x, t), as for example in a simple laboratory experiments.
1) Pathline: trajectory of a single fluid parcel or particle, starting at some given point in the
flow. Corresponds to long-exposure photograph of a marked particle or small blob of dye.
2) Streakline: locus of dye released at some fixed point, or of parcels/particles marked as they
emanate from that point during some time interval — as with dye continually released
from the end of a fine tube.
3) Streamline: curve everywhere tangent to the instantaneous flow direction. First imagine
taking a short-exposure photograph of densely spaced marked particles, producing short
streaks, or infinitesimal trajectories, which show the local flow magnitudes and directions,
then draw a curve everywhere tangent to the short streaks.
(Both streamlines and streaklines will generally change with time.)
Each of these types of curves may be defined by a differential equation, as follows.
EXAMPLE: A 2-dimensional velocity field u(x, t) = (yt, 1).
1) : A pathline is a curve x = X(t; x0 ) (X(t) for brevity) defined by
∂
X(t) = u X(t), t
∂t
with X = x0 at t = 0 .
For example, if u(x, t) = (yt, 1)
2) : A streakline is a curve X(s; x0 , t) ([X(s, t) for brevity; where the release time s varies
along the curve at any given instant t], defined by
∂
X(s, t) = u X(s, t), t
∂t
with X = x0 at t = s .
For example, if u(x, t) = (yt, 1)
3) : A streamline is a curve X(s; x0 , t) (X(s) for brevity) defined by
∂
X(s) = u X(s), t
with X = x0 at s = 0 .
∂s
For example, if u(x, t) = (yt, 1)
Note clear differences between pathlines (1), streaklines (2) and streamlines (3). Note
particularly that the curves are coincident only if the flow is steady.