Page 441 - Introduction to Continuum Mechanics

Problems 425

(c) If the velocity field is irrotational, then v/ = -d<p /dxj, where <p is known as the velocity
potential Show that the curves of constant velocity potential <p = constant and the stream line
V = constant are orthogonal to each other.
(d) Obtain the only nonzero vorticity component in terms of ^.

Fig.P6.8

6.45. Show that

represents a two-dimensional irrotational flow of an inviscid fluid. Sketch the stream lines
2
2
in the region x + y ^ a 2
6.46. Referring to Fig.P6.8, compute the maximum possible flow of water. Take the atmos-
pheric pressure to be 93.1 kPa, the specific weight of water 9800 N/m , and the vapor pressure
17.2 kPa. Assume the fluid to be inviscid. Find the length / for this rate of discharge.
6.47. Water flows upward through a vertical pipeline which tapers from 25.4 cm to 15.2 cm
diameter in a distance of 1.83 m. If the pressure at the beginning and end of the constriction
are 207 kPa and 172 kPa respectively. What is the flow rate? Assume the fluid to be inviscid.

6.48. Verify that the equation of conservation of mass is automatically satisfied if the velocity
components in cylindrical coordinates are given by

where the densityp is a constant and V is any function of r and z having continuous second
partial derivatives.
6.49. Derive Eq. (6.25.6).