Viscosity is the property of fluids (liquids and gases) to resist the displacement of one part of the fluid relative to another.
Let us consider a liquid flowing through a tube. The tube exerts a retarding force (the viscous friction force) on the liquid. It is known that the viscous friction force $F_{fric}$ acting on the liquid is proportional to the average flow velocity $v$ and the viscosity coefficient $\eta$:
\[
F_{fric} = \alpha v \eta,
\]
where $\alpha$ is a constant depending on the geometrical dimensions of the setup.
The viscosity coefficient is a characteristic of a liquid. As follows from the formula, the greater the viscosity coefficient $\eta$, the greater the viscous friction force. For comparison, the table presents the viscosity coefficients of some liquids. The objective of this experiment is to determine the viscosity coefficient of water.
Liquid $\eta,~\text{mPa}\cdot\text{s}$ Petrol $0.53$ Olive oil $84$ Honey $\sim10000$ Water ?
A1 Measure the inner radius $R_0$ of the flexible plastic tube and the inner radius $R_\text{s}$ of the $10~\text{ml}$ syringes. Provide a sketch illustrating the measurement method.
Note. Direct measurement of the radii with a ruler will not provide sufficient accuracy.
Hint 1. If a liquid column has a height $h$ and a cross-sectional area $S$, its volume is calculated using the formula $V = Sh$. If the volume of the liquid $V$ and the height of the column $h$ are known, then the cross-sectional area can be determined by the formula $S = V/h$.
Hint 2. The area of a circle $S$ and its radius $r$ are related by the formula $S = \pi r^2$.
The ends of the plastic tube are attached to the tips of the $10~\text{ml}$ syringes so that together they form a system of two communicating vessels.
By pouring water and adjusting the position of the syringes, make sure that one syringe is filled up to the $10~\text{ml}$ mark, the other syringe is empty, and the connecting tube is completely filled with water. Close the valve on the tube.
Lower and fix the empty syringe so that the distance between its bottom and the bottom of the upper syringe is $H \approx 5~\text{cm}$ (see figure). If the valve is opened, the water will begin to flow from the upper syringe into the lower one.
A3
Open the valve and simultaneously start the stopwatch. Measure the dependence of the liquid volume in the lower syringe $V$ on time $t$ (write down the values of the volume in the lower syringe and the time at which they are reached). Measure at least 7 data points. After the liquid volumes in the syringes stop changing, record the value of the volume in the lower syringe $V_\infty$. Repeat the measurements several times if necessary. Make sure that $H$ remains the same during all measurements!
From the equations of hydrodynamics, it follows that the volume of water in the lower syringe $V$ depends on time $t$ according to the formula:
\[
V(t) = V_\infty \left( 1 - e^{-\gamma t} \right),
\]
where $V_\infty$ is the volume in the lower syringe after a long time (measured in the previous task), and $\gamma > 0$ is a certain coefficient determined by the parameters of the liquid.
Using the definition of the natural logarithm $\ln$, the formula can be rewritten as:
\[
-\ln\left(1 - \frac{V(t)}{V_\infty}\right) = \gamma t.
\]
If we plot the dependence of $-\ln\left(1 - \frac{V(t)}{V_\infty}\right)$ on $t$, it will be linear with a slope equal to $\gamma$.
From the solution of the hydrodynamic equations it is known that $\gamma$ is related to the viscosity coefficient $\eta$ as follows:
\[
\gamma = \frac{\rho g \pi R_0^4}{4S_0L \eta},
\]
where $\rho = 1000~\text{kg}/\text{m}^3$ is the density of water, $g= 9.81~\text{m}/\text{s}^2$ is the free-fall acceleration, $R_0$ is the inner radius of the tube measured in item A1, $S_s$ is the inner cross-sectional area of the syringe, and $L$ is the length of the tube connecting the syringes.