Question

Which of the following is not a correct expression regarding the units of coefficient of viscosity? (a) dyne \(\mathrm{cm}^{-2}\) s (b) dyne \(\mathrm{cm}^{2} \mathrm{~s}^{-1}\) (c) \(\mathrm{N} \mathrm{m}^{-2} \mathrm{~s}\) (d) \(\mathrm{Pa} \boldsymbol{s}\)

Short answer

Option (b) dyne \(\mathrm{cm}^{2} \mathrm{~s}^{-1}\) is not a correct expression regarding the units of coefficient of viscosity.

Step-by-step solution

Understanding Coefficient of Viscosity

The coefficient of viscosity, also known as dynamic viscosity, quantifies a fluid's resistance to flow. It has units of force per area per time, which can be expressed as \( \mathrm{N} \cdot \mathrm{s} \cdot \mathrm{m}^{-2} \) or equivalently as \( \mathrm{Pa} \cdot \mathrm{s} \) since \( 1 \mathrm{Pa} = 1 \mathrm{N} \cdot \mathrm{m}^{-2} \) in SI units.

Evaluating Option (a)

Option (a) is \( \text{dyne} \cdot \mathrm{cm}^{-2} \cdot \text{s} \) which reflects the CGS unit system for viscosity (\( \text{dyne} \) is a unit of force in this system). In CGS units, the coefficient of viscosity is represented in \( \text{dyne} \cdot \text{s} \cdot \text{cm}^{-2} \), so this expression is correct.

Evaluating Option (b)

Option (b) shows the expression as \( \text{dyne} \cdot \mathrm{cm}^{2} \cdot \mathrm{s}^{-1} \). This implies the opposite of the expected units of force per area per time, which would have been \( \text{dyne} \cdot \text{s} \cdot \text{cm}^{-2} \). Thus, this option is not a correct expression for the coefficient of viscosity as it incorrectly shows area in the numerator and inverse time.

Evaluating Option (c)

Option (c) is \( \mathrm{N} \cdot \mathrm{m}^{-2} \cdot \mathrm{s} \), which is the correct SI units expression for viscosity. \( \mathrm{N} \cdot \mathrm{m}^{-2} \) can also be written as \( \mathrm{Pa} \), and adding the second part \( \mathrm{s} \) gives us the complete SI unit for dynamic viscosity.

Evaluating Option (d)

Option (d) reads \( \mathrm{Pa} \cdot \mathrm{s} \) which also correctly represents the SI units of dynamic viscosity, as it is equivalent to \( \mathrm{N} \cdot \mathrm{m}^{-2} \cdot \mathrm{s} \).

Key concepts

Dynamic Viscosity

Dynamic viscosity, commonly referred to as absolute viscosity, plays a critical role in describing the flow characteristics of fluids. It represents the internal friction within the fluid, occurring when adjacent layers travel at different speeds. To further grasp this concept, think of how thick syrup flows more sluggishly than water. This difference is due to the higher dynamic viscosity of syrup.
The coefficient of viscosity is essentially a measure of this resistance to flow. If a fluid has high dynamic viscosity, it means there is considerable friction between the fluid layers, requiring more force to move one layer over another. Conversely, a lower dynamic viscosity signifies less internal friction and a more effortless flow.

In a more technical sense, dynamic viscosity is defined as the shear stress—or force per unit area—applied to the fluid divided by the velocity gradient perpendicular to the direction of the force. Mathematically, this is expressed as \( \tau = \mu \frac{dv}{dz} \) where \( \tau \) is shear stress, \( \mu \) is the dynamic viscosity, \( dv \) is the difference in velocity, and \( dz \) is the separation between layers.

Fluid Resistance

Fluid resistance, also known as viscosity, encapsulates how a fluid opposes motion under an applied force. This resistance defines how fluids behave under different conditions; for instance, oil has a higher resistance compared to water when pouring.
Several factors influence fluid resistance, including temperature—usually, heating a fluid lowers its viscosity, leading to less resistance. This phenomenon is vividly observed when heating honey: its viscosity decreases and it flows more readily.

While dynamic viscosity refers to the quantity that measures the resistance, understanding the physical aspects of resistance can help us manipulate fluid movements in various applications. Engineers often consider fluid resistance while designing systems like pipelines or lubrication setups, ensuring efficient operation. Industrial processes, such as painting and food production, also take into account the properties of fluid resistance to ensure quality and consistency.

SI Units

When working within the scientific community or engaging in precise measurements, it is crucial to adopt a consistent system of units—the International System of Units (SI) provides exactly that. For dynamic viscosity, the SI units are the pascal-second (Pa.s). This unit signifies the blended measure of pressure (pascals) with time (seconds).

Understanding SI units is beneficial as it facilitates clear communication among professionals and standardizes equations that describe physical phenomena. In the context of our original problem, knowing that \( 1 \text{Pa} = 1 \text{N.m}^{-2} \) simplifies the process of identifying correct expressions for viscosity. However, note that the dynamics of viscosity aren't fully captured by the SI unit alone. It's the material's response within real-world conditions that determines how that viscosity is observed and measured, and professionals must interpret the values accordingly within the context of their specific fields of work.
Lastly, the commonality of SI units's global usage minimizes conversion errors and enhances the precision and reproducibility of experiments and calculations everywhere.