Page 475 - Introduction to Continuum Mechanics

Integral Formulation of General Principles 459

(c) Compute the net mass inflow into the control volume of part (b). Does the net mass inflow
equal the rate of mass increase?
7 J. (a) Check that the motion

corresponds to the velocity field of Prob. 7.7.
l
(b) For a density fieldp - p 0 e~^~ °', verify that the mass contained in the material volume
that was coincident with the control volume of Prob. 7.7 at time t 0, remain a constant.
(c) Compute the total linear momentum for the material volume of part (b).

7.9. Do Problem 7.7 for the velocity field v = xi ej and the density field p - — and for the

cylindrical control volume bounded by x\ = 1 and x\ = 3.
7.10. The center of mass \ m of a material volume is defined by the equation

Demonstrate that the linear momentum principle may be written in the form

where a cjn is the acceleration of the mass center.
7.11. Consider the following velocity field and density field

(a) Compute the total linear momentum and rate of increase of linear momentum in a
cylindrical control volume of cross-sectional area A and bounded by the plane KI = I and
*!-3.

(b) Compute the net rate of outflow of linear momentum from the control volume of part (a).
(c) Compute the total force on the material in the control volume.
(d) Compute the total kinetic energy and rate of increase of kinetic energy for the control
volume of part (a).
(e) Compute the net rate of outflow of kinetic energy from the control volume.
7.12. Consider the velocity and density fields

For an arbitrary time t, consider the material contained in the cylindrical control volume
bounded by x± = 0 and jci = 3.