In SI units, what is the derived unit used for measuring the coefficient of viscosity of a fluid?

Introduction / Context:
Viscosity is a fundamental property of fluids that measures their resistance to flow. Thick fluids such as honey have high viscosity, while thin fluids such as water and alcohol have low viscosity. In physics and engineering, we often work with the coefficient of dynamic viscosity and need to express it in standard SI units. This question tests whether you can correctly identify the SI unit associated with viscosity.

Given Data / Assumptions:

• The quantity of interest is the coefficient of dynamic viscosity.
• We are using the SI (International System of Units).
• Candidate units include coulomb, newton second per square metre, watt per metre per degree Celsius and joule per kilogram per kelvin.
• We assume standard definitions of force, energy and thermal quantities.

Concept / Approach:
Dynamic viscosity is defined through the relationship between shear stress and velocity gradient in a fluid. In simple form, shear stress tau is proportional to viscosity eta times velocity gradient, so tau = eta * (du/dy). Shear stress has units of force per area (newton per square metre), while velocity gradient has units of per second. Therefore viscosity has units of shear stress divided by velocity gradient, which gives (newton per square metre) / (per second) = newton second per square metre. This is equivalent to the pascal second, because 1 pascal is 1 newton per square metre.

Step-by-Step Solution:
Step 1: Recall the definition tau = eta * (du/dy), where tau is shear stress and eta is viscosity. Step 2: Identify the units of tau as newton per square metre (N·m^-2), also called the pascal (Pa). Step 3: Identify the units of velocity gradient du/dy as (metre per second) / metre = per second (s^-1). Step 4: Solve for viscosity units: eta has units of tau divided by velocity gradient. Step 5: Divide units: (N·m^-2) / (s^-1) = N·s·m^-2, which is newton second per square metre. Step 6: Conclude that the SI unit of viscosity is newton second per square metre, also written as pascal second.

Verification / Alternative check:
Standard tables of physical constants list the viscosity of water at room temperature as around 1.0 * 10^-3 Pa·s. Since 1 Pa is 1 N·m^-2, this is the same as 1.0 * 10^-3 N·s·m^-2. No other units in the options match this standard description. Coulomb is the SI unit of electric charge, watt per metre per degree Celsius is a unit of thermal conductivity and joule per kilogram per kelvin is a unit of specific heat capacity, not viscosity.

Why Other Options Are Wrong:
Coulomb: This is the unit of electric charge and is unrelated to fluid flow and shear stress.

Watt per metre per degree Celsius: This unit measures thermal conductivity, describing heat flow rather than resistance to fluid motion.

Joule per kilogram per kelvin: This unit is used for specific heat capacity, describing how much energy is needed to raise the temperature of a unit mass.

Common Pitfalls:
A common confusion arises because many material properties, such as thermal conductivity and specific heat, also have compound SI units. Students may remember the form of one property and incorrectly apply it to another. To avoid this, always start from the defining equation. For viscosity, use tau = eta * (du/dy) and work through the unit analysis from shear stress and velocity gradient. This approach clearly leads to newton second per square metre as the correct unit.

Final Answer:
In SI units, the coefficient of viscosity is measured in newton second per square metre (N·s·m^-2).