Page 474 - Introduction to Continuum Mechanics

458 Problems

PROBLEMS
7.1. Verify the divergence theorem for the vector field v = 2rej + z 62 , by considering the
region bounded by x = 0,* = 2,y = 0,y = 2,2 = 0,z = 2.
7.2. Show that

where Fis the volume enclosed by the boundary S.

73. (a) Consider the vector field v = <pa, where <p is a given scalar field and a is an arbitrary
constant vector (independent of position). Using the divergence theorem, prove that

(b) Show that for any closed surface S that

7.4. A stress field T is in equilibrium with a body force pB. Using the divergence theorem,
show that for any volume F, and boundary surface 5, that

That is, the total resultant force is equipollent to zero.

equilibrium with a body force p B and a surface traction t . Using the divergence theorem,
verity the identity (theorem of virtual work)

7.6. Using the equations of motion and the divergence theorem, verify the following rate of
work identity

7.7. Consider the velocity and density fields

(a) Check the equation of mass conservation.
(b) Compute the mass and rate of increase of mass in the cylindrical control volume of
cross-section A and bounded byjq = 0 andjc^ = 3.