Question

Design an experiment to measure the viscosity of liquids using a vertical funnel with a cylindrical reservoir of height \(h\) and a narrow flow section of diameter \(D\) and length \(L\). Making appropriate assumptions, obtain a relation for viscosity in terms of easily measurable quantities such as density and volume flow rate.

Short answer

Question: Using a vertical funnel with a cylindrical reservoir of height \(h\), and a narrow flow section of diameter \(D\) and length \(L\), design an experiment to measure the viscosity of a liquid. Use the Hagen-Poiseuille equation and consider the flow to be laminar. Calculate the viscosity in terms of easily measurable quantities such as density and volume flow rate. Answer: To measure the viscosity of a liquid using the vertical funnel, follow these steps: 1. Fill the reservoir to height \(h\). 2. Allow the liquid to flow through the narrow flow section of diameter \(D\) and length \(L\). 3. Measure the time \(t\) it takes for a known volume \(V\) of liquid to pass through the narrow flow section. 4. Use the relation for viscosity: \(\eta = \dfrac{\rho g h \pi r^4 t}{8 L V}\), where \(\rho\) is the liquid density, \(g\) is the acceleration due to gravity, and \(r\) is the radius of the narrow flow section (i.e. \(r = D/2\)). With these measurements and the known values for \(L\), \(D\), and \(g\), the viscosity of the liquid can be determined.

Step-by-step solution

Understand the Hagen-Poiseuille equation

The Hagen-Poiseuille equation states that the volume flow rate \(Q\) of an incompressible fluid with viscosity \(\eta\) passing through a cylindrical pipe of length \(L\) and radius \(r\) under a pressure difference \(\Delta P\) is given by: \(\Delta P = \dfrac{8\eta LQ\ }{\pi r^4}\). In our case, the fluid will flow through the narrow section of the funnel with diameter \(D\) and length \(L\). We are given the height \(h\) of the cylindrical reservoir, and we need to find the viscosity of the liquid using measurable quantities.

Define the Pressure difference

The pressure difference \(\Delta P\) is the driving force for the flow and depends on the height \(h\) of the liquid column in the cylindrical reservoir. The pressure difference can be expressed as \(\Delta P = \rho g h\), where \(\rho\) is the liquid density, \(g\) is the acceleration due to gravity, and \(h\) is the height of the liquid column in the reservoir.

Express the volume flow rate in terms of measurable quantities

The volume flow rate \(Q\) can be determined by measuring the time \(t\) it takes for a known volume \(V\) of liquid to pass through the narrow section. Therefore, we have: \(Q = \dfrac{V}{t}\).

Substitute the Pressure difference and volume flow rate into the Hagen-Poiseuille equation

Now we substitute the expressions for the pressure difference and volume flow rate into the Hagen-Poiseuille equation: \(\rho g h = \dfrac{8\eta L\dfrac{V}{t}\ }{\pi r^4}\).

Calculate the viscosity in terms of measurable quantities

Now we solve the equation for the viscosity \(\eta\): \(\eta = \dfrac{\rho g h \pi r^4 t}{8 L V}\). This is the final relation for viscosity in terms of easily measurable quantities such as density and volume flow rate. The experiment can be performed by filling the reservoir to height \(h\), allowing the liquid to flow through the narrow section, and measuring the time \(t\) it takes for a known volume \(V\) of liquid to pass through. With these measurements and the known values for \(L\), \(D\), and \(g\), the viscosity of the liquid can be determined.

Key concepts

Hagen-Poiseuille equation

The Hagen-Poiseuille equation is a fundamental principle used to describe the flow of viscous liquids through a cylindrical pipe. It articulates how different variables such as viscosity, pipe dimensions, and pressure difference influence the volume flow rate of a fluid. The formula derived from this principle is:

• Volume Flow Rate, \( Q = \dfrac{\pi r^4 \Delta P}{8 \eta L} \)
• \( \Delta P \) is the pressure difference across the pipe.
• \( \eta \) represents the dynamic viscosity of the fluid.
• \( L \) is the length of the cylindrical pipe, and \( r \) is the radius.

In the context of the experiment, the diameter of the funnel narrows down the flow to a specific path with given dimensions. This equation is fundamental to predicting how fast and efficiently a fluid will travel through a uniform conduit. Breaking down the equation helps in understanding the reliance of liquid flow on measurable characteristics like radius and pressure difference.

Fluid dynamics experiment

A fluid dynamics experiment centers on the practical application of theoretical fluid flow equations to real-world scenarios. In this context, students can design an experiment using a funnel as a physical model.

• The funnel acts as a vertical channel through which liquid flows, and its dimensions mirror those in the Hagen-Poiseuille equation.
• Key measurable parameters include the diameter (\( D \)) of the funnel's narrow section, its length (\( L \)), and the height (\( h \)) of the reservoir.
• Key tools might include a timer to measure flow time and a graduated cylinder to determine volumes accurately.

The experiment allows students to relate theory with practice by measuring how different liquids behave under similar constraints and verify the accuracy of theoretical predictions about flow rate and viscosity.

Liquid flow rate

Liquid flow rate refers to the volume of fluid that passes through a point in the conduit per unit time. It's an essential measurement in many fluid dynamics experiments. In this exercise, the flow rate \( Q \) can simply be calculated by dividing the known volume \( V \) of liquid by the time \( t \) it takes to flow past a certain point:

• Flow Rate, \( Q = \dfrac{V}{t} \)
• This measurement helps infer properties such as viscosity when combined with other data.
• An understanding of flow rate enables an appreciation of how quickly or slowly different liquids pass through conduits.

Ensuring the measurement of flow rate is precise has a direct impact on the accuracy of the viscosity calculation and, thus, on the experiment's overall success.

Pressure difference in fluids

The pressure difference, or \( \Delta P \), is crucial in determining how fluids move through a system. In the context of the Hagen-Poiseuille equation, the pressure difference is the driving force behind the flow of the liquid through the funnel.

• The pressure difference can be calculated using the equation \( \Delta P = \rho g h \), where \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid column.
• This component of the equation reflects how gravitational force contributes to fluid dynamics.
• In experiments, by adjusting the height of the liquid column, students can observe changes in flow characteristics corresponding to this pressure differential.

A firm grasp of pressure differences and their measurement will greatly aid in understanding why liquids move the way they do in different settings. Sensitivity to changes in this variable is key when analyzing results from such experiments.