Question

The one dimensional flow part of governing consolidation equation of three dimensional consolidation having radial symmetry is _______

(a) \(\frac{∂\overline{u}}{∂t}=C_{vr} \frac{∂\overline{u}}{∂r^2}\)

(b) \(\frac{∂\overline{u}}{∂t}=C_{vr} (\frac{∂\overline{u}}{∂r^2}+\frac{1}{r}\frac{∂\overline{u}}{∂r})+C_{vz}\frac{∂^2 \overline{u}}{∂z^2}\)

(c) \(\frac{∂\overline{u}}{∂t}=C_{vz} (\frac{∂\overline{u}}{∂r^2}+\frac{1}{r}\frac{∂\overline{u}}{∂r})+C_{vz}\frac{∂^2 \overline{u}}{∂z^2}\)

(d) \(\frac{∂\overline{u}}{∂t}=C_{vr} (\frac{∂\overline{u}}{∂r^2}+\frac{1}{r}\frac{∂\overline{u}}{∂r})\)

This question was addressed to me in an online quiz.

My doubt stems from Consolidation Equation in Polar Coordinates topic in section One Dimensional Consolidation of Geotechnical Engineering I

Answer 1

The correct choice is (a) \(\frac{∂\overline{u}}{∂t}=C_{vr} \frac{∂\overline{u}}{∂r^2}\)

To explain I would say: In three dimensional consolidation of sand drain, having radial symmetry, the governing consolidation equation is,

\(\frac{∂\overline{u}}{∂t}=C_{vr} (\frac{∂\overline{u}}{∂r^2}+\frac{1}{r}\frac{∂\overline{u}}{∂r})+C_{vz}\frac{∂^2 \overline{u}}{∂z^2}\), in this the one dimensional flow part of equation is,

\(\frac{∂\overline{u}}{∂t}=C_{vr} \frac{∂\overline{u}}{∂r^2}\).