Question

The rate of change of velocity along the depth of layer is ___________

(a) \(\frac{∂v}{∂z}=\frac{1}{γ_w}\frac{∂^2 \overline{u}}{∂z^2} \)

(b) \(\frac{∂v}{∂z}=\frac{k}{γ_w}\frac{\overline{u}}{∂z^2} \)

(c) \(\frac{∂v}{∂z}=\frac{∂^2 \overline{u}}{∂z^2} \)

(d) \(\frac{∂v}{∂z}=\frac{k}{γ_w} \frac{∂^2 \overline{u}}{∂z^2} \)

This question was addressed to me in quiz.

Asked question is from Terzaghi’s Theory of One Dimensional Consolidation topic in chapter One Dimensional Consolidation of Geotechnical Engineering I

Answer 1

Correct answer is (d) \(\frac{∂v}{∂z}=\frac{k}{γ_w} \frac{∂^2 \overline{u}}{∂z^2} \)

For explanation I would say: The velocity with which the excess pore water flows is given by Darcy’s law,

\(v=ki=\frac{k}{γ_w}\frac{∂\overline{u}}{∂z}\) ——————————-(1)

Differentiating the equation (1) with respect to depth z, we get,

∴ \(\frac{∂v}{∂z}=\frac{k}{γ_w}\frac{∂^2 \overline{u}}{∂z^2}.\)