Newton's Law of Viscosity
Viscosity is a measurement of the resistance of a fluid to flow or deformation. It measures how much fluid resists the motion of sliding layers past one another. There is a mathematical approach under Newton's Law of Viscosity to describe this resistance as involving force and flow.
1.0Statement and Explanation of Newton’s Law of Viscosity
Newton's Law of Viscosity states that shear stress () is directly proportional to the related shear strain rate, which is defined as the velocity gradient between two layers. Another way to say this is the force that must be used to move one layer of fluid over another is proportional to the speed with which the velocity changes in the fluid. Mathematically, it can be written as:
Here:
- F = shear stress (force per unit area (A) applied tangentially to the fluid),
- η = dynamic viscosity of the fluid,
- dudy = velocity gradient, which represents the rate at which the velocity of fluid changes with respect to the distance from the surface (i.e., the rate of shear strain).
- The given formula is known as Newton’s law of viscosity Formula, which shows that for the Newtonian type of fluid, the relationship between shear stress and rate is linear.
- The negative sign indicates the direction of the shear force relative to the flow.
What is the Formula for Viscosity?
The above-mentioned formula can also be used for deriving the formula for viscosity. It shows how the viscosity or internal resistance of fluids is related to the shear stress and the shear rate:
- The SI unit of dynamic viscosity - is pascal-seconds (Pa–s) while the CGS unit of viscosity is poise (P).
- 1 poise (P) = 0.1 Pascal – second
2.0Types of Fluids:
Newtonian Fluids:
These fluids follow Newton's Law of Viscosity. The viscosity is constant and it is independent of the shear rate. The shear stress is always directly proportional to the shear rate. Examples of Newtonian fluids comprise water, air, and any other gas.
Non-Newtonian Fluids:
These fluids do not obey Newton's Law of Viscosity. Their viscosity varies with changes in the shear rate or external factors like temperature. Such compounds include ketchup, blood, paints, and some polymer solutions.
3.0Solved Examples
Problem 1: Two fluids, A and B, are flowing between parallel plates. Fluid A has a dynamic viscosity of 0.4 Pa.s, and fluid B has a viscosity of 0.8 Pa.s. If the shear rate in fluid A is 10 s−1, calculate the shear rate for fluid B when the shear stress is the same in both fluids.
Solution: Let the Dynamic viscosity of fluid A A=0.4Pa.s
Let the Dynamic viscosity of fluid B B\eta_{B}=0.8Pa.s
The shear rate of fluid
According to the question, the shear stress is the same for both fluids. Hence,
Problem 2: A fluid flows through a pipe with a radius of 0.01 m. The pressure difference between the two ends of the pipe is 500 Pa. The velocity gradient in the pipe is 150 s−1. Find the viscosity of the fluid.
Solution: Let the radius of the pipe be r = 0.01m
Pressure difference be P = 500Pa
Velocity Gradient
Here, we need shear stress for using Newton’s law of viscosity formula, which for pipe flow is given by:
Now, using the formula of viscosity:
Problem 3: A fluid flows between two parallel plates, with a distance of 0.02 m between them. The velocity at the lower plate is 0.3 m/s, and the velocity at the top plate is 0.5 m/s. If the dynamic viscosity of the fluid is 0.04 Pa.s, find the shear stress on the fluid.
Solution: Let the distance between plates by dy = 0.02m
Velocity at the lower plate u1,= 0.3m/s
Velocity at the top plate u2 = 0.5m/s
Dynamic viscosity of fluid = 0.04 Pa.s
Velocity gradient can be calculated as:
Now, by using Newton’s law, let’s calculate shear stress:
Table of Contents
- 1.0Statement and Explanation of Newton’s Law of Viscosity
- 1.1What is the Formula for Viscosity?
- 2.0Types of Fluids:
- 2.1Newtonian Fluids:
- 2.2Non-Newtonian Fluids:
- 3.0Solved Examples
Frequently Asked Questions
Shear rate is the velocity gradient in a fluid, indicating how fast the velocity varies with the distance between the fluid layers.
Dynamic viscosity can be defined as the measure of the resistance of a fluid to flow under shear stress. The higher the value of dynamic viscosity, the thicker and more resistant the fluid.
Non-Newtonian fluids exhibit a change in viscosity as a function of shear rate, whereas Newtonian fluids have constant viscosity regardless of the rate of flow.
In gases, viscosity increases with pressure. In liquids, however, the effect of pressure on viscosity is usually negligible except in highly compressed liquids.
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