Chapter 1: Mathematical modelling of uids
Evy Kersal
e
Introduction
Fluid dynamics (or uid mechanics) is the study of the motion of liquids, gases
and plasmas (e.g. water, air, interstellar plasma) which have no large scale
structure and can be deformed t
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Solutions 4
1.
Hydrostatic equilibrium.
The force on the gate is due to the pressure exerted by uids on both sides. For uids at
rest, u = 0, so that Eulers equation is reduced to p = g wi
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Solutions 2
1.
Kinematics of 2-D ows.
The scalar function u measures the rate of change of volume, hence the compressibility, of
a ow and the vector vorticity = u measures its local rotat
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620) Examples 5
1. Water ows in a long channel of rectangular cross-section with its bottom horizontal
except for a hump extending over some intermediate section. Far upstream of the hump, the
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Solutions 3
1. If a velocity eld u is irrotational, i.e. if its vorticity = u = 0, then u = for
some velocity potential . If, in addition, the uid is incompressible then, from the continu
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Solutions 1
1.
Shear ow.
We consider a shear ow u = (u, v, w) = (ky, 0, 0) where k = 0 is a constant.
i. The position x(t) = (x(t), y(t), z(t) of a particle released in the ow at (x0 , y0
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Solutions 5
1.
Critical ow past a hump.
6h0
g
U
hc
Z(x)
h(x)
Zm
V
h0
Flow rate: From mass conservation, the volume ow rate Q = u(x)h(x) must be constant
throughout the channel. So,
Q = u(
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Examples 4
These examples will help you to understand the material taught in the lectures; it is very
important that you try them before the workshops.
1. A canal consists of a horizontal
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Examples 3
These examples will help you to understand the material taught in the lectures; it is very
important that you try them before the workshops.
1. State the conditions under which
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Examples 2
These examples will help you to understand the material taught in the lectures; it is very
important that you try them before the workshops.
1. For each of the following two-di
School of Mathematics
University of Leeds
Fluid Dynamics (MATH2620)
Examples 1
These examples will help you to understand the material taught in the lectures; it is very
important that you try them before the workshops.
1.
The velocity in a simple shearin