Question

In SI system units of coefficient of viscosity \(\eta\) are (1) \(\mathrm{kg} \mathrm{s}^{-2} \mathrm{~m}^{-2}\) (2) \(\mathrm{kg} \mathrm{m}^{-1} \mathrm{~s}^{-1}\) (3) \(\mathrm{kg} \mathrm{cm}^{-1} \mathrm{~s}^{-1}\) (4) \(\mathrm{g} \mathrm{m}^{-1} \mathrm{~s}^{-1}\)

Short answer

Option (2): \( \mathrm{kg} \cdot \mathrm{m}^{-1} \cdot \mathrm{s}^{-1} \).

Step-by-step solution

Understand the Quantity

Viscosity is a measure of a fluid's resistance to flow. It is commonly denoted by the Greek letter \( \eta \). The SI unit for this quantity needs to be identified.

Identify the Correct Formula

The dynamic viscosity \( \eta \) can be expressed using the relation \( F = \eta \left( \frac{dv}{dz} \right) A \), where \( F \) is the force applied, \( A \) is the area, and \( \frac{dv}{dz} \) is the velocity gradient.

Determine Units for Each Component

The SI units for each component are: \( F \) (force) in \( \mathrm{N} \) (newton), \( A \) (area) in \( \mathrm{m}^2 \) (square meters), and \( \frac{dv}{dz} \) (velocity gradient) in \( \mathrm{s}^{-1} \) (per second). Newton is defined as \( \mathrm{N} = kg \cdot \mathrm{m/s}^2 \).

Combine Units

From the formula and SI units, \( \eta = \frac{F}{A \cdot \frac{dv}{dz}} = \frac{kg \cdot \mathrm{m/s}^2}{\mathrm{m}^2 \cdot \mathrm{s}^{-1}} = \mathrm{kg} \cdot \mathrm{m}^{-1} \cdot \mathrm{s}^{-1} \).

Choose Correct Option

Compare the result from Step 4 with the given options. The correct unit for the coefficient of viscosity \( \eta \) in the SI system is \( \mathrm{kg \cdot m}^{-1} \mathrm{ \cdot s}^{-1} \). This matches option (2).

Key concepts

dynamic viscosity

Dynamic viscosity, also simply called viscosity, is a measure of a fluid's resistance to flow. This resistance is due to the internal friction within the fluid caused by its molecular composition. Imagine trying to stir honey compared to water; honey's high viscosity means it resists the stirring motion much more than water.

The dynamic viscosity is often represented by the Greek letter \( \eta \). It essentially describes how much force per unit area is needed to move one layer of fluid in relation to another. The formula used to calculate dynamic viscosity is:

\[ F = \eta \left( \frac{dv}{dz} \right) A \]

where:

• \( F \): Force applied
• \( \frac{dv}{dz} \): Velocity gradient (change in velocity over distance)
• \( A \): Area over which the force is applied

This relationship helps us understand the factors affecting a fluid's internal resistance to flow.

SI unit conversion

Understanding how to convert and utilize SI units is key in physics and engineering. The International System of Units or SI units is the standard for measurement globally.

In the context of dynamic viscosity, the challenge is converting the physical quantities into the correct SI units to find viscosity's proper units. Let's break down the units step-by-step:

• Force (\( F \)) is measured in newtons (\( N \)). According to SI units, 1 newton is equal to \( 1 \ \text{kg} \cdot \ \text{m/s}^2 \).
• Area (\( A \)) is measured in square meters (\( \text{m}^2 \)).
• Velocity gradient (\( \frac{dv}{dz} \)) has units of \( \text{s}^{-1} \), meaning per second.

By substituting these into the viscosity formula, we get:

\[ \eta = \frac{F}{A \cdot \frac{dv}{dz}} = \frac{ \left( \text{kg} \cdot \text{m/s}^2 \right)}{ \left( \text{m}^2 \right) \cdot \left( \text{s}^{-1} \right)} = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-1} \]

This breakdown makes it clear why the SI unit for dynamic viscosity is \( \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-1} \).

fluid mechanics

Fluid mechanics is the branch of physics concerned with the behavior of fluids (liquids and gases) and the forces acting on them. It includes the study of fluid dynamics and fluid statics.

Key points in fluid mechanics include:

• Viscosity: A measure of a fluid's resistance to flow. Both dynamic and kinematic viscosities are essential here.
• Reynolds Number: A dimensionless number used to predict flow patterns in different fluid flow situations. It helps distinguish between laminar (smooth) and turbulent (chaotic) flow.
• Bernoulli's Equation: A principle that describes the conservation of energy in a flowing fluid. It is crucial in understanding how fluid pressure varies with speed and height.
• Continuity Equation: States that the mass flow rate of a fluid must remain constant from one cross-section to another.

Fluid mechanics provides the foundation for understanding phenomena such as the lifting force on airplane wings, the flow of water through pipes, and even blood circulation in the body.

Understanding viscosity within fluid mechanics is vital as it influences how fluids interact with their environment and with other materials.