The viscous friction force acting on a sphere moving in water is:
$$\vec{F} = - 6\pi r \eta \vec{v},$$
where $r$ is the radius of the sphere, $\eta=0.89~\mathrm{mPa} \cdot{s}$ is the viscosity coefficient of water, and $v$ is the velocity of the sphere.
The current flowing through the photodiode is proportional to the intensity of light incident on it.
The true density of sand is $\rho_s = 2.9~\mathrm{g}/\mathrm{см}^3$, and the density of water is $\rho_w=1.00~г/см^3$. Sand grains will be considered as spheres. The acceleration of free fall is $g=9.8~\text{m}/\text{s}^2$. The width $L$ of cuvvete cross-section is $8.4~\mathrm{mm}$
Let the total number of sand grains in a certain volume of suspension be $N_0$. The number of particles $dN$with sizes in the range $[r, r+dr]$ is:
\[ dN = N_0 \cdot f(r) dr,\]
where the function $f(r)$ characterizes the probability of finding a particle of size $r$, if it is taken from this specific volume.
The dynamics of the settling mixture can be studied optically. Due to light scattering by the particles, the intensity of light decreases exponentially:
\[ I= I_0 \exp (- \alpha L) = I_0 \exp \left(-\frac{LN_0}{V}\int\limits_0^{+\infty} \pi r^2 f(r) dr \right), \]
where $\alpha$ is the optical density of the solution.
When the particle suspension is homogeneous, $f(r)$ is the same everywhere. However, after the spontaneous settling process begins, $f(r)$ starts to depend on height and time. For example, fast-settling particles quickly reach the bottom and no longer contribute to the scattering of light passing through the middle of the cuvette.
