Dynamic Viscosity Formula
Last Updated : 04 Feb, 2024
The resistance to motion that most fluids provide is referred to as "viscosity." When there is relative motion between layers of fluid, viscosity develops. It precisely measures the resistance to flow caused by internal friction between fluid layers as they pass through one another during fluid flow. Viscosity may also be defined as a measure of a fluid's thickness or barrier to passing items through it.
Because of its strong intermolecular interactions, a fluid with a high viscosity resists motion by creating a lot of internal friction, which prevents layers from moving past one another. A fluid with low viscosity, on the other hand, flows easily because its molecular composition causes very little friction when it is in motion. Gases have viscosity as well, although it's less noticeable in everyday life.
What is Viscosity?
The viscosity of a fluid is a measurement of its resistance to flow. The ratio of shearing stress to velocity gradient in a fluid is used to calculate viscosity.
- Viscosity is measured in Poiseuille, a SI unit of measurement (PI).
- The newton-second per square meter— (N s m-2) and the pascal-second are the other units (Pa s.)
- [ML-1T-1] is the dimensional formula for viscosity.
The viscosity of liquids reduces fast as the temperature rises, but the viscosity of gases rises as the temperature rises. As a result, liquids flow more freely when heated, and gases flow more slowly. Viscosity is also an intensive property since it does not vary when the amount of matter changes.
Formula for the Coefficient of Viscosity
η = F . dx / A . dv
where,
- η is the coefficient of viscosity,
- dv/dx is the velocity gradient between two layers of liquid,
- F is the viscous force, and
- A is the surface area.
Types of Viscosity
There are two types of viscosity of a fluid:
- Dynamic Viscosity (Absolute Viscosity): This type of viscosity is used to measure the fluid's resistance to flow when a force is applied to it. The term for this is Dynamic Viscosity.
- Kinematic Viscosity: This type of viscosity is used to measure the fluid's resistive flow under gravity's weight. Kinematic viscosity is the name given to this measure of fluid viscosity.
Many people misunderstand the two viscosity measurements and believe they are one and the same. In actuality, they are rather different from one another. Kinematic viscosity is more beneficial than absolute or dynamic viscosity in a few instances.
Dynamic Viscosity
Dynamic viscosity is a method of measuring a fluid's resistance to flow when an external force is applied.
Dynamic viscosityThe viscosity of a fluid is a crucial attribute to know in order to understand its behavior. Also, when it comes into touch with solid limits, how it will move. The viscosity of a fluid is a measurement of its resistance to progressive deformation under tensile or shear stress. The intermolecular friction that occurs when layers of fluids attempt to glide over each other can cause shear stress in the fluid.
A rotational viscometer is a useful tool for determining dynamic viscosity. The probe in the liquid sample will be rotated by these instruments. The force, or torque, required to turn the probe is used to determine viscosity.
Formula for Dynamic Viscosity
Dynamic Viscosity Formula for the fluid will specify its internal resistance to the flow due to a certain shearing force. This is a type of tangential force that occurs when two horizontal planes contact. During the analysis of liquid behavior and fluid motion near solid boundaries, viscosity is an essential fluid characteristic.
As a result, dynamic viscosity is the force required by a fluid to overcome internal molecular friction and allow it to flow. So, dynamic viscosity may be defined as the tangential force per unit area necessary to move a fluid in one horizontal plane relative to another plane at a velocity of unit value while the fluid's molecules remain a unit distance apart.
The tangential force required to shift one horizontal plane of a fluid relative to another is known as dynamic viscosity. As a result, we may write it as:
Dynamic viscosity = Shearing stress / Shearing rate change
or
η = T/γ
where,
- η is the Dynamic Viscosity,
- T is the shearing stress, and
- γ is the shear rate.
The SI unit for Dynamic Viscosity is Pa.s or Ns/m2
Sample Problems
Problem 1: Shearing stress of 0.76 N per m2 in a fluid with a shear rate of 0.5 per second. Which of these fluids does it match to based on its dynamic viscosity? (Water dynamic viscosity = 1 Pa s, Air dynamic viscosity = 0.018 Pa s and Mercury dynamic viscosity = 1.526 Pa s)
Solution:
Given,
T = 0.76 N per m2
γ = 0.5 per second
So the formula is,
η = T / γ
= 0.76 / 0.5
= 1.52 Pa s
As a result, it is obvious that Mercury fluid will be compatible with this fluid.
Problem 2: With a shear rate of 0.35 s-1 and dynamic viscosity of 0.018 Pa s, what pressure is required to move a plane of fluid?
Solution:
Given,
Shear rate = 0.35 s-1
Dynamic viscosity = 0.018 Pa s
From the dynamic velocity formula,
T = η × γ
Substituting the values,
T = (0.018 × 0.35)
T = 0.0063 Pa
= 0.0063 Pa
Problem 3: A 2.5 × 10-4 m2 metal plate is placed over a 0.25 × 10-3m thick layer of castor oil. Calculate the coefficient of viscosity of castor oil if a force of 2.5 N is required to move the plate at a velocity of 3 × 10-2m s-1.
Solution:
Given:
A = 2.5 × 10-4 m2,
dx = 0.25×10-3m,
dv = 3×10-2 m s-1
F = 2.5 N
Formula is,
η = F.dx / A .dv
Substitute the values in the formula,
η = (2.5)(0.25 × 10-3) / (2.5 × 10-4)(3 × 10-2)
= 0.083 × 103 Nm-2s
Problem 4: Water is flowing slowly on a horizontal plane, with a viscosity coefficient of 0.01 poise and a 100 cm2 surface area. What external force is necessary to keep the flow's velocity gradient at 1 s-1?
Solution:
Given,
dv/dx = 1s-1.
A = 100 cm2 = 10-2 m2.
η = 0.01 poise = 0.001 kg/ms.
From the formula:
F = -η A (dv/dx)
Substitute the given values in the above, to calculate F,
F = 0.001 × 10-2 × 1
= 10-5 N
Problem 5: 0.04 N/m2 is found to be the shear stress at a point in a liquid. At this point, the velocity gradient is 0.22 s-1. What would the viscosity be?
Solution:
Given,
F/A = 0.04 N/m2 ( shear stress)
dv/dx = 0.22 s-1
Formula for the viscous force is:
F = -ηA (dv/dx)
By Rearranging the formula:
η = (F/A) / (dv/dx)
Substitute the values to calculate η,
η = 0.04 N/m2 / 0.22 s-1
= 0.181 N s/m2