Page 437 - Introduction to Continuum Mechanics

Problems 421

6.10. In astrophysical applications, an atmosphere having the relation between the density p
and pressure/? given by

where p 0 and p 0 are some reference pressure and density, is known as a polytropic atmos-
phere. Find the distribution of pressure and density in a polytropic atmosphere.
6.11. For a steady parallel flow of an incompressible linearly viscous fluid, if we take the flow
direction to be 63,

(a) show that the velocity field is of the form

(b) If v(xi, X2) = k*2 > find the total normal stress on the plane whose normal is in the direction
of e 2 + 63, in terms of the viscosity/* and pressure/?

(c) On what planes are the total normal stresses given by the so-called "pressure"?
6.12. Given the following velocity field (in m/s) for a Newtonian incompressible fluid with a
viscosity ju = 0.96 mPa:

At the point (l,2,l)m and on the plane whose normal is in the direction of e 1?
(a) find the excess of the total normal compressive stress over the pressure/?,
(b) find the magnitude of the shearing stress.

6.13. Do Problem 6.12 except that the plane has a normal in the direction of 3ej + 4^.
6.14. Use the results of Sect. 2D2, Chapter 2 and the constitutive equations for the Newtonian
viscous fluid, verify Eqs. (6.8.1).
6.15. Use the results of Sect.2D3,Chapter 2 and the constitutive equations for the Newtonian
viscous fluid, verify Eqs. (6.8.3).
6.16. Show that for a steady flow, the streamline containing a point P coincides with the pathline
for a particle which passes through the point P at some time t.
6.17. For the two dimensional velocity field

(a) Find the streamline at time t, which passes through the spatial point («i,«2)
(a) find the pathline for the particle which was at (Af lt X 2) aU = 0.
(c) Find the streakline at time t, formed by the particles which passed through the spatial
at
position (a lt « 2) time r<t.