CONTINUUM MECHANICS for ENGINEERS

The velocity field of the medium has components
v 1 = x 2 + 2x 3, v 2 = x 3 – x 1, v 3 = x 1 + 3x 2
Determine the material derivative dθ/dt of the temperature field.
Answer: dθ/dt = –e –3t (3x2 + 6x1 x3 + 8x2x3)/x4 4.9 In a certain region of a fluid the flow velocity has components
v1 = A( )e –kt , v2 = A( )e –kt 3 2
2 3
x + x x
xx +
x , v3 = 0
1
1 2
where A and k are constants. Use the (spatial) material derivative
operator to determine the acceleration components at the point (1, 1, 0)
when t = 0.
Answer: a1 = –2A(k – 5A), a2 = –A(k – 5A), a3 = 0
4.10 A displacement field is given in terms of the spatial variables and
time by the equations
u 1 = x 2t 2 , u 2 = x 3t, u 3 = x 1t
Using the (spatial) material derivative operator, determine the velocity
components.
Answer: v1 = (2x2t + x3t2 + x1t3 )/(1 – t4 )
v 2 = (x3 + x1t + 2x2t3 )/(1 – t4 )
v 3 = (x1 + 2x2t2 + x3t3 )/(1 – t4 )
4.11 For the motion given by the equations
x 1 = X 1 cos ωt + X 2 sin ωt
x 2 = –X 1 sin ωt + X 2 cos ωt
x 3 = (1+ kt)X 3
where ω and k are constants, determine the displacement field in
Eulerian form.
Answer: u1 = x1(1 – cos ωt) + x2 sin ωt
u2 = –x1 sin ωt + x2(1 – cos ωt)
u3 = x3kt/(1 + kt)
4.12 Show that the displacement field for the motion analyzed in Problem
4.1 has the Eulerian form
u 1 = x 1 – (x 1 + x 2)e –t /2 – (x 1 – x 2)e t /2
1
2 3