What is Newton’s law of viscosity?

Short Answer:

Newton’s law of viscosity states that the shear stress between adjacent layers of a fluid is directly proportional to the rate of change of velocity with respect to distance between the layers, provided the temperature and pressure remain constant.

In simple words, it means that when one layer of fluid moves faster than another, a frictional force (shear stress) develops between them, which depends on how fast the velocity changes between the layers. The constant of proportionality is called dynamic viscosity (μ).

Mathematically,

where, τ = shear stress, μ = viscosity, du/dy = velocity gradient.

Detailed Explanation :

Newton’s Law of Viscosity

Newton’s law of viscosity describes the relationship between the shear stress applied to a fluid and the rate of deformation (velocity gradient) of the fluid layers. It explains how fluids resist motion due to internal friction between their layers when subjected to shear forces.

In a flowing fluid, such as oil between two parallel plates, one layer moves faster than another. The faster layer tends to drag the slower one along due to intermolecular friction. This internal resistance is known as viscosity, and its behavior is governed by Newton’s law of viscosity.

The law is applicable to Newtonian fluids, where the viscosity remains constant under varying shear stress and shear rate.

Statement of the Law

According to Newton’s law of viscosity:

“The shear stress (τ) acting between two adjacent fluid layers is directly proportional to the rate of change of velocity (du/dy) between those layers.”

Mathematically,

Where,

  • τ = shear stress (N/m² or Pa)
  • μ = coefficient of dynamic (or absolute) viscosity (N·s/m² or Pa·s)
  • du/dy = velocity gradient or rate of shear strain (s⁻¹)

This equation forms the fundamental relationship used in fluid mechanics to study the flow of fluids under shear forces.

Explanation of Terms

  1. Shear Stress (τ):
    It is the tangential force per unit area acting between adjacent layers of fluid in motion. It represents the internal frictional force resisting relative motion.
  2. Velocity Gradient (du/dy):
    It is the rate at which velocity changes with respect to distance between two fluid layers. If the difference in velocity between layers is large, the velocity gradient is high.
  3. Dynamic Viscosity (μ):
    It is the constant of proportionality in Newton’s law. It represents the fluid’s resistance to flow or deformation. Higher viscosity means greater internal friction.

Derivation Concept

Consider two parallel layers of a fluid separated by a distance dy. The lower layer is stationary, and the upper layer moves with velocity u when a tangential force F acts on it.

The shear stress on the fluid is given by:

where A is the area of the layer.

Experimentally, it is observed that shear stress is proportional to the velocity gradient:

Introducing the constant of proportionality, μ, we get:

Hence, the shear stress increases linearly with the rate of shear strain in Newtonian fluids.

Newtonian and Non-Newtonian Fluids

  1. Newtonian Fluids:
    These fluids obey Newton’s law of viscosity, meaning the ratio of shear stress to velocity gradient remains constant.
    Examples: Water, air, alcohol, kerosene, light oils.
  2. Non-Newtonian Fluids:
    These fluids do not follow Newton’s law of viscosity. The relationship between shear stress and velocity gradient is nonlinear.
    Examples: Blood, toothpaste, paints, mud, and polymer solutions.

For Newtonian fluids, the viscosity μ remains constant regardless of shear rate, while for non-Newtonian fluids, μ changes with flow conditions.

Units and Dimensions

  • SI Unit of μ: Pascal-second (Pa·s) or N·s/m²
  • CGS Unit of μ: Poise (1 Poise = 0.1 Pa·s)
  • Dimensional Formula: M¹L⁻¹T⁻¹

Graphical Representation

When shear stress (τ) is plotted against velocity gradient (du/dy):

  • For Newtonian fluids, the graph is a straight line passing through the origin, indicating a constant viscosity.
  • For Non-Newtonian fluids, the graph is curved, showing variable viscosity.

This simple linear relationship helps engineers model fluid flow in pipes, channels, and machines.

Importance of Newton’s Law of Viscosity

  1. Foundation of Fluid Mechanics:
    The law provides the basic relationship used to analyze fluid flow, especially for laminar flow conditions.
  2. Determination of Shear Force:
    It helps calculate shear stress distribution in fluids moving between surfaces, such as in lubricated bearings or oil films.
  3. Viscosity Measurement:
    The dynamic viscosity of fluids can be determined experimentally using viscometers that apply Newton’s law.
  4. Design of Hydraulic Systems:
    Engineers use viscosity values in the design of pumps, pipes, and turbines to predict energy losses and efficiency.
  5. Lubrication and Material Flow:
    The law explains how lubricants behave between moving machine parts, ensuring proper film thickness and reduced wear.

Applications in Engineering

  1. Flow in Pipes and Channels:
    Newton’s law helps in calculating pressure drops and velocity profiles for laminar flow in circular pipes (Hagen–Poiseuille equation).
  2. Hydraulic Machines:
    It aids in understanding how viscous losses affect performance in pumps, turbines, and fluid couplings.
  3. Lubrication Systems:
    Engineers use it to design oil films in bearings and gears for efficient operation and reduced friction.
  4. Aerospace and Automotive Engineering:
    In aerodynamics, it is used to study boundary layer formation and skin friction on aircraft and vehicles.
  5. Chemical and Food Industries:
    The law is applied to predict how liquids like syrups, oils, and solutions flow through pipes and processing equipment.

Limitations of Newton’s Law of Viscosity

  1. The law applies only to Newtonian fluids with constant viscosity.
  2. It does not hold for non-Newtonian fluids, where viscosity depends on shear rate.
  3. It assumes steady flow conditions and fails under turbulent flow.
  4. Temperature and pressure changes may alter viscosity, violating the law’s assumptions.
Conclusion

In conclusion, Newton’s law of viscosity explains how shear stress in a fluid is proportional to the velocity gradient between its layers. The proportionality constant, viscosity (μ), indicates the fluid’s internal resistance to motion. This law forms the foundation for studying fluid mechanics and analyzing flow behavior in engineering applications. Newtonian fluids follow this law precisely, while non-Newtonian fluids do not. Understanding this law helps engineers design efficient hydraulic systems, lubrication mechanisms, and flow equipment.