The viscous coefficient, also known as dynamic viscosity ($\mu$), quantifies a fluid's resistance to shear flow or its "thickness." It is calculated by determining the ratio of the shear stress applied to the fluid to the resulting shear rate.
Understanding the Viscous Coefficient
The viscous coefficient represents the internal friction within a fluid. A higher viscous coefficient indicates a greater resistance to flow. To calculate it, you need to understand two primary concepts: shear stress and shear rate.
Key Concepts for Calculation
-
Shear Stress ($\tau$): This is the tangential force per unit area exerted on a fluid layer. When fluid layers move relative to one another, they experience a tangential force.
- It is calculated as the force (F) applied to the fluid divided by the area (A) over which the force is distributed:
- $\tau = \frac{F}{A}$
- Here, F represents the tangential force required to make one layer of liquid skid over another, and A is the area of the liquid layer under consideration.
- It is calculated as the force (F) applied to the fluid divided by the area (A) over which the force is distributed:
-
Shear Rate ($\gammȧ$): Also known as the velocity gradient, this describes how quickly the fluid's velocity changes with distance perpendicular to the flow direction. Imagine two parallel layers of liquid: one stationary and one moving.
- It is calculated as the velocity (v) of the moving layer relative to the stationary layer, divided by the distance (d) between the two layers:
- $\gammȧ = \frac{v}{d}$
- Here, v is the velocity difference between the layers, and d is the distance separating them. The term "layers of liquid skidding over each other" helps visualize this relative motion.
- It is calculated as the velocity (v) of the moving layer relative to the stationary layer, divided by the distance (d) between the two layers:
The Calculation Formula
Based on Newton's Law of Viscosity, the shear stress ($\tau$) in a Newtonian fluid is directly proportional to the shear rate ($\gammȧ$), with the constant of proportionality being the dynamic viscosity ($\mu$):
$\tau = \mu \cdot \gammȧ$
Rearranging this formula to solve for the viscous coefficient ($\mu$):
$\mu = \frac{\tau}{\gammȧ}$
By substituting the definitions of shear stress and shear rate into this equation, you get the fundamental formula for calculating the viscous coefficient:
$\mu = \frac{F/A}{v/d}$
Which simplifies to:
$\mu = \frac{F \cdot d}{A \cdot v}$
This formula is particularly applicable in scenarios involving parallel plate models where the force and dimensions can be directly measured.
Variables in the Formula
| Variable | Description |
|---|---|
| $\mu$ | Viscous Coefficient (Dynamic Viscosity) |
| F | Tangential force applied to the fluid layer (e.g., in Newtons) |
| A | Area over which the force is applied (e.g., in square meters) |
| v | Relative velocity between fluid layers (e.g., in meters per second) |
| d | Distance between the fluid layers (e.g., in meters) |
Units of Viscosity
The standard SI unit for dynamic viscosity is the Pascal-second (Pa·s), which is equivalent to Newton-second per square meter (N·s/m²). Another commonly used unit, especially in the CGS system, is the Poise (P).
- 1 Pa·s = 10 Poise
- 1 Centipoise (cP) = 0.001 Pa·s (Centipoise is often used because water at 20°C has a viscosity of approximately 1 cP)
Factors Influencing Viscosity
The viscous coefficient of a fluid is not constant and can be significantly affected by external factors:
- Temperature: For most liquids, viscosity decreases as temperature increases because the kinetic energy of molecules overcomes intermolecular forces, making them flow more easily. For gases, viscosity generally increases with temperature.
- Pressure: For most liquids, viscosity slightly increases with increasing pressure, though this effect is usually less pronounced than that of temperature.
Practical Measurement Methods
In real-world applications, the viscous coefficient is often measured using specialized instruments called viscometers. These devices employ various principles to determine viscosity without requiring direct measurement of all individual force, area, velocity, and distance parameters:
- Rotational Viscometers: Measure the torque required to rotate a spindle immersed in the fluid at a constant speed.
- Capillary Viscometers: Measure the time it takes for a fixed volume of fluid to flow through a narrow tube under gravity.
- Falling-Ball Viscometers: Measure the time it takes for a sphere of known density and diameter to fall through a fluid under gravity.
- Vibrational Viscometers: Measure the damping of a vibrating element immersed in the fluid.
These methods provide practical ways to determine the viscous coefficient for various fluids in industries ranging from automotive to food processing.
