UNIVERSITY OF STAVANGER
Department of Mathematics and Natural Science
MPT 130 – Fluid Dynamics
Exercises spring 2013.
Exercises for 23. January 2013
Problem 1:Tritton, problem 2.
We note from Tritton, sect. 2.3, that in cylindrical coordinates (r, ϕ, x) the net pressure
force on a fluid element can be written:
δF P=−∂p
∂x r δ r δφ δx Tritton 2.13
The net viscous force is:
δF V=µ∂
∂r r∂u
∂r δr δφ δx Tritton 2.11
The weight of such an element can be written (g= 9.8m/s2is the acceleration of gravity,
ρis the density):
δF G=−ρgr δr δφ δx
Problem 2:
Water is flowing horizontally at velocity u(r)in the gap between two co-axial pipes, the
inner having an outer radius a= 10 cm, the outer having an inner radius b= 15 cm.
The water is flowing in the direction of the pipes, which we take as our x-axis. The
water has viscosity 1.0·10−3Pa s and density ρ= 1 000 kg/m3.
a) Write down the equation for u(r)(the equation of motion). [Hint: Use FVand FP
from the previous problem.]
b) State the boundary conditions for u(r).
c) Show that the equation of motion has the solution:
u(r) = −
Gr2
4µ+Aln r+B , Tritton 2.15
where Gis the pressure gradient, and determine the constants Aand B.
d) Find an expression for the volume flow ˙
V, and find its numerical value if G=
10 Pa/m.
The following integral may prove useful:
Zrln r
cdr=r2
4h2 ln r
c−1i