Which of the following represents the unit of kinematic viscosity?
cm2/s
Understanding Kinematic Viscosity and Its Units
Viscosity is a measure of a fluid's resistance to flow. There are two main types of viscosity: dynamic viscosity and kinematic viscosity.
Dynamic viscosity (also known as absolute viscosity) measures the fluid's internal resistance to shear flow. Its dimensions are typically represented as $\left[ \text{Mass} \cdot \text{Length}^{-1} \cdot \text{Time}^{-1} \right]$. Common units include Pascal-second (Pa·s) in the SI system and Poise (P) in the CGS system. One Poise is equal to 1 dyne-second per square centimeter ($\text{dyne} \cdot \text{s/cm}^2$) or 1 gram per centimeter-second ($\text{g/cm} \cdot \text{s}$).
Kinematic viscosity is related to dynamic viscosity and is calculated by dividing the dynamic viscosity by the fluid's density. It represents the fluid's resistance to flow under the influence of gravity. Its dimensions are typically represented as $\left[ \text{Length}^{2} \cdot \text{Time}^{-1} \right]$.
Calculating the Unit of Kinematic Viscosity
The relationship between kinematic viscosity ($\nu$) and dynamic viscosity ($\mu$) is given by the formula:
$\nu = \frac{\mu}{\rho}$
where $\rho$ is the density of the fluid.
Let's determine the unit of kinematic viscosity by looking at the units of dynamic viscosity and density in common systems:
| Quantity | Symbol | SI Unit | CGS Unit | Dimensions |
|---|---|---|---|---|
| Dynamic Viscosity | $\mu$ | Pa·s (or N·s/m² or kg/(m·s)) | Poise (P) (or dyne·s/cm² or g/(cm·s)) | $\left[ \text{M L}^{-1} \text{T}^{-1} \right]$ |
| Density | $\rho$ | kg/m³ | g/cm³ | $\left[ \text{M L}^{-3} \right]$ |
| Kinematic Viscosity | $\nu$ | m²/s | Stokes (St) (or cm²/s) | $\left[ \text{L}^{2} \text{T}^{-1} \right]$ |
Using the formula $\nu = \mu/\rho$, we can find the unit of $\nu$ in the CGS system:
- Unit of $\mu$ in CGS = g/(cm·s) or dyne·s/cm²
- Unit of $\rho$ in CGS = g/cm³
- Unit of $\nu$ in CGS = $\frac{\text{Unit of } \mu}{\text{Unit of } \rho} = \frac{\text{g/(cm} \cdot \text{s)}}{\text{g/cm}^3}$
Let's simplify the unit calculation:
$\frac{\text{g}}{\text{cm} \cdot \text{s}} \div \frac{\text{g}}{\text{cm}^3} = \frac{\text{g}}{\text{cm} \cdot \text{s}} \times \frac{\text{cm}^3}{\text{g}} = \frac{\text{cm}^3}{\text{cm} \cdot \text{s}} = \frac{\text{cm}^2}{\text{s}}$
Thus, the unit of kinematic viscosity in the CGS system is cm²/s. This unit is also known as Stokes (St). A common sub-unit is the centistokes (cSt), where 1 cSt = 1 mm²/s = 10⁻² St.
Analyzing the Given Options
Let's examine each option to see which one represents the unit of kinematic viscosity:
- Option 1: cm²/s
This is the unit of kinematic viscosity in the CGS system (Stokes).
- Option 2: dyne-sec/cm²
The dyne is a unit of force in the CGS system ($\text{g} \cdot \text{cm/s}^2$). So, $\text{dyne} \cdot \text{s/cm}^2 = (\text{g} \cdot \text{cm/s}^2) \cdot \text{s/cm}^2 = \text{g/cm} \cdot \text{s}$. This is the unit of dynamic viscosity in the CGS system (Poise).
- Option 3: gm/cm-sec
This unit is $\text{g/(cm} \cdot \text{s})$. As derived above, this is also the unit of dynamic viscosity in the CGS system (Poise). Note that Option 2 and Option 3 represent the same physical unit.
- Option 4: gm/cm²-sec
This unit is $\text{g/(cm}^2 \cdot \text{s})$. Let's check its dimensions: $\frac{\text{M}}{\text{L}^2 \cdot \text{T}} = \left[ \text{M L}^{-2} \text{T}^{-1} \right]$. This does not match the dimensions of either dynamic viscosity $\left[ \text{M L}^{-1} \text{T}^{-1} \right]$ or kinematic viscosity $\left[ \text{L}^{2} \text{T}^{-1} \right]$. Therefore, this is not a standard unit of viscosity.
Based on the analysis, the unit representing kinematic viscosity among the options is cm²/s.
Revision Table: Viscosity Units
| Property | Unit System | Standard Unit | Common Alternative Unit(s) | Related Formula |
|---|---|---|---|---|
| Dynamic Viscosity ($\mu$) | SI | Pa·s (Pascal-second) | N·s/m², kg/(m·s) | $\tau = \mu \frac{du}{dy}$ (Newton's law of viscosity) |
| Dynamic Viscosity ($\mu$) | CGS | Poise (P) | dyne·s/cm², g/(cm·s) | $\tau = \mu \frac{du}{dy}$ |
| Kinematic Viscosity ($\nu$) | SI | m²/s | $\nu = \frac{\mu}{\rho}$ | |
| Kinematic Viscosity ($\nu$) | CGS | Stokes (St) | cm²/s, 100 cSt | $\nu = \frac{\mu}{\rho}$ |
Additional Information about Viscosity
- Viscosity is a crucial property for fluid mechanics and engineering applications, affecting how fluids flow in pipes, lubrication, and mixing processes.
- Temperature significantly affects viscosity. For most liquids, viscosity decreases as temperature increases. For most gases, viscosity increases as temperature increases.
- Kinematic viscosity is often used in calculations involving fluid flow where gravitational forces are important, such as determining the Reynolds number, which predicts flow patterns (laminar or turbulent).
- The dimension of kinematic viscosity $\left[ \text{L}^{2} \text{T}^{-1} \right]$ implies that it relates to how quickly momentum diffuses through a fluid, similar to how thermal diffusivity relates to heat diffusion.
Important Questions from Properties of Fluids
Surface tension of water _____.
Which of the following forces does NOT act on fluid which is at rest?
Flow occurring in a pipeline when a valve is being opened is
In which of the following unit kinematic viscosity of fluid is measured?
If no resistance is encountered by displacement, such a substance is known as _____.
