Newton’s Law of Viscosity and its Coefficients

Definition

Newton’s Law of Viscosity states that for a fluid, the shear stress acting on any layer of the fluid is directly proportional to the rate of shear strain (velocity gradient) perpendicular to the direction of motion, provided the fluid temperature remains constant.

Main Content

1. The Concept of Shear Stress

Shear stress ($\tau$) is the force applied parallel to the surface of a fluid layer, causing the layers to slide over one another.
It represents the internal resistance of a fluid to flow or deformation.

2. Velocity Gradient

In a flowing fluid, different layers move at different velocities. The variation of velocity ($du$) over a perpendicular distance ($dy$) is known as the velocity gradient ($du/dy$).
It represents the "rate of shear" or how quickly the fluid layers are changing speed as you move away from a stationary boundary.

3. Dynamic Viscosity Coefficient ($\mu$)

The coefficient of proportionality in Newton’s Law is called the "Dynamic Viscosity" or "Coefficient of Viscosity" ($\mu$).
It is a measure of a fluid's internal friction. High viscosity (like honey) means high resistance to flow, while low viscosity (like water) flows easily.

       Velocity Profile
      u + du   -------->
      u        ------>
      0        ------> (Fixed plate)
               | dy |

Working / Process

1. Establishing the Velocity Gradient

Consider two parallel plates separated by a distance $dy$, with the bottom plate fixed and the top plate moving at a velocity $u$.
The velocity change across the fluid gap is linear, defining the rate at which fluid layers shift past each other.

2. Formulating the Relationship

Mathematically, the law is expressed as: $\tau = \mu (du/dy)$.
Here, $\tau$ (tau) is the shear stress, $\mu$ (mu) is the dynamic viscosity, and $du/dy$ is the velocity gradient.

3. Calculating the Coefficient

To find the coefficient $\mu$, the formula is rearranged: $\mu = \tau / (du/dy)$.
By measuring the force per unit area ($\tau$) required to move a plate and dividing it by the velocity gradient, we determine the fluid's specific viscosity value.

Advantages / Applications

Lubrication Engineering: Essential for choosing the right oil for car engines to reduce friction between moving metal parts.
Pipeline Design: Helps engineers calculate the pumping power required to transport crude oil or water through pipes based on its viscosity.
Fluid Selection: Used in industries like food processing (e.g., controlling the flow of syrups) and medicine to determine the behavior of different liquid formulations.

Summary

Newton’s Law of Viscosity links shear stress directly to the velocity gradient in a fluid.
The coefficient of dynamic viscosity ($\mu$) quantifies the fluid's resistance to deformation.
Newtonian fluids follow this linear relationship, whereas non-Newtonian fluids do not.
Important terms: Shear stress ($\tau$), Velocity gradient ($du/dy$), Dynamic Viscosity ($\mu$), and Newtonian fluid.

Newton’s Law of Viscosity is the fundamental principle used to categorize fluids based on their flow resistance, where the shear stress is proportional to the rate of angular deformation.