Understanding Dynamic Viscosity and the Poise Unit

This question asks about the equivalent value of one poise, which is a unit used to measure dynamic viscosity. Let's break down what dynamic viscosity is and how its units are defined.

What is Dynamic Viscosity?

Dynamic viscosity, often denoted by the Greek letter \(\mu\) (mu), is a measure of a fluid's resistance to flow when subjected to a shear stress. A fluid with high dynamic viscosity, like honey, resists flow more strongly than a fluid with low dynamic viscosity, like water.

Units of Dynamic Viscosity: Poise and Pascal-second

Dynamic viscosity has units derived from its definition related to shear stress and velocity gradient. The standard units come from two main systems:

• CGS System (Centimetre-Gram-Second): The unit is the poise (P). It is defined based on fundamental units of mass, length, and time.
• SI System (International System of Units): The unit is the pascal-second (Pa.s). It is derived from the SI units of force (Newton), area (square meter), and time (second).

Defining One Poise

One poise (1 P) is defined in the CGS system. It corresponds to a shear stress of one dyne per square centimetre (1 dyne/cm²) required to maintain a velocity gradient of one centimetre per second per centimetre (1 (cm/s)/cm = 1 s⁻¹). The fundamental units work out as follows:

Viscosity (\(\mu\)) = Shear Stress / Velocity Gradient

In CGS units:

1 Poise = \(\frac{1 \text{ dyne/cm}^2}{1 \text{ cm/(s} \cdot \text{cm})} = \frac{1 \text{ dyne}}{1 \text{ s}^{-1} \cdot \text{cm}^2}\)

Since 1 dyne = 1 gm \(\cdot\) cm / s², we can substitute this into the equation:

1 Poise = \(\frac{(1 \text{ gm} \cdot \text{cm/s}^2)}{\text{cm}^2} \cdot \text{s} = \frac{1 \text{ gm}}{\text{cm} \cdot \text{s}}\)

So, one poise is equivalent to 1 gram per centimetre-second (1 gm/cm-sec).

Relationship between Poise and Pascal-second

The SI unit, pascal-second (Pa.s), is related to the poise. 1 Pa.s is equivalent to 10 poise.

1 Pa.s = 1 N \(\cdot\) s/m²

Let's convert this to CGS units:

• 1 N = \(10^5\) dynes
• 1 m² = \((10^2 \text{ cm})^2 = 10^4\) cm²

1 Pa.s = \(\frac{10^5 \text{ dynes} \cdot \text{s}}{10^4 \text{ cm}^2} = 10 \frac{\text{dyne} \cdot \text{s}}{\text{cm}^2}\)

Since 1 Poise = 1 dyne \(\cdot\) s/cm², we get:

1 Pa.s = 10 Poise

Conversely, 1 Poise = 0.1 Pa.s.

Analyzing the Given Options

Let's evaluate each option provided to see which one matches the definition of one poise:

1. 1 kg/m-hr: This unit has the dimensions of mass/(length \(\cdot\) time), which is correct for viscosity. However, the units are in kilograms, meters, and hours, not grams, centimetres, and seconds. Let's convert it:

   • 1 kg = 1000 gm
   • 1 m = 100 cm
   • 1 hr = 3600 sec

   1 kg/m-hr = \(\frac{1000 \text{ gm}}{100 \text{ cm} \cdot 3600 \text{ sec}} = \frac{10 \text{ gm}}{3600 \text{ cm} \cdot \text{sec}} = \frac{1}{360} \text{ gm/cm-sec}\)

   Since 1 gm/cm-sec = 1 Poise, 1 kg/m-hr = \(\frac{1}{360}\) Poise. This option is not equivalent to 1 Poise.

2. 1 gm/cm-sec: As derived from the definition of poise in the CGS system, 1 Poise is exactly equal to 1 gm/cm-sec. This option matches the fundamental definition of the unit poise.

3. 98 dyne/sec: This unit has dimensions of force/time. The dimensions of viscosity are (force \(\cdot\) time)/area or mass/(length \(\cdot\) time). This option does not have the correct dimensions for viscosity and is therefore not equivalent to 1 Poise.

4. 68 kgf-sec/m\(^2\): This unit has dimensions of (force \(\cdot\) time)/area, which is correct for viscosity. kgf is kilogram-force, a unit of force. Let's relate it to standard units:

   • 1 kgf is approximately 9.80665 Newtons (N).
   • 1 m\(^2\) = \((10^2 \text{ cm})^2 = 10^4\) cm\(^2\).
   • 1 sec = 1 sec.

   Let's convert 1 Poise (0.1 Pa.s) to kgf-sec/m\(^2\).

   1 Poise = 0.1 N \(\cdot\) s / m\(^2\)

   Since 1 kgf \(\approx\) 9.80665 N, then 1 N \(\approx \frac{1}{9.80665}\) kgf.

   1 Poise \(\approx 0.1 \cdot \frac{1}{9.80665} \text{ kgf} \cdot \text{s / m}^2 \approx 0.010197 \text{ kgf} \cdot \text{s / m}^2\)

   The option is 68 kgf-sec/m\(^2\). This value is vastly different from the calculated equivalence of 1 Poise. Therefore, this option is not equivalent to 1 Poise.

Based on the analysis, the only option that is directly equivalent to one poise is 1 gm/cm-sec, which is the fundamental definition of the poise unit in the CGS system.

Conclusion on One Poise Equivalence

The analysis confirms that one poise is directly defined as 1 gm/cm-sec in the CGS system of units for dynamic viscosity. Other options represent different values or incorrect units.

Revision Table: Key Viscosity Concepts

Additional Information on Fluid Viscosity

Understanding viscosity is crucial in fluid mechanics. Here are a few more points:

• Newtonian vs. Non-Newtonian Fluids: Fluids are classified based on how their viscosity behaves under stress. Newtonian fluids have constant viscosity regardless of the shear rate (like water or air). Non-Newtonian fluids have viscosity that changes with the shear rate (like ketchup or paint).
• Effect of Temperature: Viscosity is highly dependent on temperature. For most liquids, viscosity decreases as temperature increases (they become more 'runny'). For most gases, viscosity increases as temperature increases.
• Stokes: While poise is the CGS unit for dynamic viscosity, the CGS unit for kinematic viscosity is the stokes (St), defined as 1 cm\(^2\)/s. Kinematic viscosity is related to dynamic viscosity by the fluid's density: \(\nu = \mu / \rho\).

Being familiar with both the CGS (Poise, Stokes) and SI (Pascal-second, m\(^2\)/s) units for viscosity and their conversions is essential for solving problems in fluid mechanics.