MATH0015 Week 9 Written Assignment 3
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MATH0015 Week 9 Written Assignment 3
1. A funnel is in the form of a cone of semi-angle α and is placed with its vertex downwards. It is filled with water to a depth D and then the water is allowed to flow out of the funnel through a small hole, of cross-sectional area A, at the vertex. You may assume that the fluid surface falls sufficiently slowly that at each instant the flow is approximately steady.
(a) Find the velocity, u, of the fluid in the exit stream when the depth of the water in the funnel is h and the rate at which the water level is decreasing is equal to U .
(b) By neglecting terms of order U2 /u2 compared to unity, show further that
=
A 2g 1
π tan2 α h3 .
(1)
(c) Assuming that formula (1) holds for the entire motion, find how long it takes for the cone to empty.
2. The depth of water in a channel is equal to h at a location where the width is d. If the volume rate of flow is equal to Q, show that
= h2 (H − h),
2gd2
where H is a constant and g is gravity. Show that, for a range of values of Q, there are two possible values of h, the larger of which lies between 
H and H. If the width of the channel increases by a small amount and if h lies in the range between 
H and H, does the depth of the water increase or decrease?
3. A small-amplitude wave is progressing in the positive x-direction on the surface of water of constant density ρ and infinite depth, so that the equation of the surface is z = η(x, t) where z is measured vertically upwards from the undisturbed surface (z = 0). The two- dimensional linearised Euler equations governing the flow can be written
∂u 1 ∂p
= −
∂t ρ ∂x ,
∂w 1 ∂p
+ = 0,
p = pa − ρgz + ρΦ ,
where pa is the constant atmospheric pressure.
(a) Derive the governing partial differential equation satisfied by Φ.
(b) By relating w to η , derive the kinematic boundary condition on Φ at z = 0.
(c) By considering the pressure on the free surface, derive the dynamic boundary con- dition on Φ at z = 0.
(d) If η(x, t) = ∈ sin(kx − ωt), find Φ and the dispersion relation relating ω and k .
(e) Surface tension, of constant value σ, causes the pressure at the water surface to be different from that in the atmosphere by an amount proportional to the curvature of the surface by introducing an additional restoring force directed towards the undis- turbed horizontal water surface (z = 0). The fluid pressure at the surface for small
waves is thus
∂2 η
p = pa − σ
i. Show that the phase speed, c = ω/k, of waves when surface tension is present is given by
c2 = g/k + (σ/ρ)k.
ii. Does surface tension cause a given wave to travel faster or slower and why should this be so?
iii. Which waves are most affected by surface tension and why should this be so?
2022-03-25