Dynamic viscosity is the resistance to the movement of one layer of fluid over another layer of fluid.
Friction within a fluid causes viscosity. It is caused by the intermolecular forces that exist between particles in a fluid.
The dimensions of dynamic viscosity are (force × time) ÷ area. The unit of viscosity, accordingly, is newton-second per square meter, which is usually expressed as pascal-second in SI units.
| Definition | A measure of the resistance of a fluid to flow under an applied force or stress. It is a property of the fluid and is a measure of how thick or resistant to flow the fluid is. |
| Symbol | μ |
| Formula | τ = μ(dv/dy) |
| Variables | τ: Shear stress, μ: Dynamic viscosity, dv/dy: Velocity gradient |
| Units | Pascal-second (Pa·s) – SI unit Poise (P) – CGS unit, equal to 0.1 Pa·s Centipoise (cP) – equal to 0.001 Poise or 0.0001 Pa·s Pound-force second per square inch (lb·s/in²) – FPS unit Poundal second per square foot (lb·s/ft²) – English gravitational unit Reyn (reyn) – used for very low viscosity fluids, defined as 1 dyne·s/cm² or 0.1 Pa·s. |
| Examples | Honey typically has a relatively high dynamic viscosity, ranging from around 2 Pa·s to 10 Pa·s, depending on factors such as temperature and moisture content. Engine oil generally has a dynamic viscosity in the range of 0.05 Pa·s to 0.5 Pa·s, although this can vary depending on the specific type of oil and the temperature at which it is tested. Blood plasma has a lower dynamic viscosity than honey or engine oil, typically ranging from 0.001 Pa·s to 0.003 Pa·s, although this can also vary depending on factors such as hematocrit (the percentage of red blood cells in the plasma) and temperature. |
Table of Contents
Daily Life Examples of Dynamic Viscosity
Here are some examples of dynamic viscosity in daily life:
- Motor oil: The dynamic viscosity of motor oil plays an important role in lubrication of engines. The oil must be able to flow freely through the engine while still providing enough resistance to prevent metal parts from rubbing against each other and causing damage.
- Paint: Dynamic viscosity is an important property of paint, as it determines the paint’s ability to flow and spread smoothly. If the paint has too high a viscosity, it will be too thick and difficult to apply, whereas if the viscosity is too low, the paint may run or drip.
- Syrups and sauces: The dynamic viscosity of syrups and sauces determines their texture and consistency. For example, a high viscosity syrup will be thick and pour slowly, whereas a low viscosity sauce will be thinner and pour more easily.
- Blood: Blood is a fluid with a relatively low dynamic viscosity compared to other fluids like honey or motor oil. The viscosity of blood is an important property in medicine, as it affects blood flow and can be used to diagnose conditions such as anemia or other blood disorders.
- Shampoo: The dynamic viscosity of shampoo determines its ability to flow and spread through the hair, while still providing enough resistance to create suds and clean the hair effectively.
Dynamic Viscosity Formula Derivation
Dynamic viscosity, represented by the symbol μ, is defined as the ratio of the shearing stress τ to the rate of deformation of a fluid (shear rate γ). Mathematically, we can express it as:
μ = τ / γ
To derive the formula for dynamic viscosity, we need to start with some basic assumptions about the behavior of fluids. Specifically, we assume that fluids behave as Newtonian fluids, meaning that the shear stress is directly proportional to the shear rate. This relationship is expressed mathematically as:
τ = μγ
This equation is the defining equation for dynamic viscosity, and it expresses the relationship between shear stress and shear rate in a fluid. To derive this equation, we can start with the definition of shear stress, which is given by:
τ = F / A
where F is the force applied to the fluid, and A is the area over which the force is applied. Next, we can define the shear rate as the rate of change of velocity with respect to distance. Mathematically, we can express this as:
γ = du / dy
where u is the velocity of the fluid in the x direction, and y is the distance from the surface where the shear stress is applied.
We can assume that the velocity profile in the fluid is linear, meaning that the velocity varies linearly with distance from the surface where the shear stress is applied. Mathematically, we can express this as:
u = (τ / μ) y
where μ is the dynamic viscosity of the fluid. Substituting this equation for u into the expression for shear rate, we get:
γ = du / dy = d/dy [(τ / μ) y] = τ / μ
where we have used the fact that the derivative of y with respect to y is 1. Finally, substituting this expression for shear rate into the definition of dynamic viscosity, we get:
μ = τ / γ = τ / (τ / μ) = μ
This shows that the assumption of a linear velocity profile is consistent with the definition of dynamic viscosity, and that the relationship between shear stress and shear rate is indeed given by τ = μγ.
Solved Numerical Problems
Example No 1: A fluid has a shear stress of 4 N/m² and a shear rate of 0.5 s⁻¹. Determine the dynamic viscosity of the fluid.
Solution:
We can use the formula for dynamic viscosity to solve this problem. We have:
μ = τ/γ̇
Plugging in the given values, we get:
μ = 4 N/m² / 0.5 s⁻¹
μ = 8 Pa s
Therefore, the dynamic viscosity of the fluid is 8 Pa s.
Example No 2: A fluid has a shear stress of 4 N/m² and a shear rate of 0.5 s⁻¹. Determine the dynamic viscosity of the fluid.
Solution:
We can use the formula for dynamic viscosity to solve this problem. We have:
μ = τ/γ̇
Plugging in the given values, we get:
μ = 4 N/m² / 0.5 s⁻¹
μ = 8 Pa s
Therefore, the dynamic viscosity of the fluid is 8 Pa s.
Frequently Asked Questions
1. What is the kinematic viscosity of air?
The viscosity of air depends mostly on the temperature. At 15 °C, the viscosity of air is 1.81 × 10-5 kg/(m·s), 18.1 μPa·s or 1.81 × 10-5 Pa·s. The kinematic viscosity of air at 15 °C is 1.48 × 10-5 m2 /s or 14.8 CST.
2. What is viscous flow?
A viscous fluid is defined as a fluid with high flow resistance. The degree of resistance between the fluid layers is measured by viscosity, which is a fluid factor. Fluids that have no or minimal internal friction resistance are categorized as non-viscous fluids.
3. Laminar flow definition
Laminar flow is a form of fluid (gas or liquid) flow in which the fluid flows smoothly and in predictable patterns. In contrast, turbulent flow is a type of fluid in which the fluid fluctuates and mixes irregularly.
4. What is the SI unit of viscosity?
The unit of viscosity is newton-second per square meter, which is usually expressed as pascal-second in SI units.
5. Why does oil float on water?
Oil floats on water because its density is lower than that of water. Density in liquids is defined as the amount of mass that may be filled into a cubic meter of volume. Water has a density of roughly 1000 kg/cubic meter, while oil has a density ranging from 800 to 960 kg/cubic meter.
6. What is air?
Air is a homogeneous mixture of different gases. The air in the atmosphere is composed of nitrogen, oxygen (which is required for animal and human life), carbon dioxide, water vapor, and trace amounts of other elements (argon, neon, etc.). At higher elevations, the air contains ozone, helium, and hydrogen.
Multiple Choice Questions
- Which of the following units is used to measure dynamic viscosity? A) m/s B) kg/m³ C) N/m² D) Pa s
Answer: D) Pa s - A fluid with a dynamic viscosity of 0.01 Pa s is known as a: A) Newtonian fluid B) Non-Newtonian fluid C) Viscous fluid D) Incompressible fluid
Answer: A) Newtonian fluid - Which of the following relationships defines dynamic viscosity? A) τ = μ/γ̇ B) τ = μγ̇ C) τ = γ̇/μ D) γ̇ = μτ
Answer: B) τ = μ*γ̇ - The dynamic viscosity of a fluid: A) increases with increasing temperature B) decreases with increasing temperature C) is independent of temperature D) is not affected by temperature
Answer: B) decreases with increasing temperature - Which of the following statements is true of a fluid with a high dynamic viscosity? A) It flows easily B) It flows slowly C) It has low resistance to flow D) It has a low shear stress
Answer: B) It flows slowly.
Exam Questions with Hints
- A fluid with a dynamic viscosity of 0.005 Pa s is subjected to a shear rate of 100 s⁻¹. What is the shear stress in the fluid?
Hint: Use the formula for dynamic viscosity to solve for the shear stress: τ = μ*γ̇. Plug in the given values and solve for τ.
- The dynamic viscosity of a fluid is measured to be 0.002 N s/m². What is the kinematic viscosity of the fluid if its density is 800 kg/m³?
Hint: Use the formula for kinematic viscosity, ν = μ/ρ, where ν is the kinematic viscosity, μ is the dynamic viscosity, and ρ is the density of the fluid. Plug in the given values and solve for ν.
- The dynamic viscosity of a fluid is 0.01 Pa s, and its density is 1000 kg/m³. If the fluid is flowing through a circular pipe with a radius of 0.02 m at a velocity of 2 m/s, what is the pressure drop per unit length of the pipe?
Hint: Use the formula for pressure drop in a pipe due to viscous flow: ΔP = 32μLQ/(πr⁴), where ΔP is the pressure drop, μ is the dynamic viscosity, L is the length of the pipe, Q is the volumetric flow rate, and r is the radius of the pipe. Rearrange the equation to solve for ΔP and plug in the given values.
- A non-Newtonian fluid is observed to have a dynamic viscosity that decreases with increasing shear rate. What type of non-Newtonian fluid is this?
Hint: The behavior described is characteristic of a shear-thinning fluid, which has a decreasing viscosity with increasing shear rate. Examples include certain types of polymers and some food products.
- A fluid is observed to have a dynamic viscosity of 0.002 Pa s and a density of 900 kg/m³. What is the Reynolds number of the fluid if it is flowing through a circular pipe with a diameter of 0.03 m at a velocity of 2 m/s?
Hint: Use the formula for Reynolds number, Re = ρvd/μ, where v is the velocity of the fluid, d is the diameter of the pipe, and μ is the dynamic viscosity of the fluid. Convert the given diameter to radius and plug in the given values to solve for Re.
Conclusion
Dynamic viscosity is a measure of a fluid’s resistance to deformation under shear stress. It is denoted by the symbol mu (μ) and is expressed in units of Pascal-seconds (Pa s) or Newton-seconds per square meter (N s/m²). The article explains that Newtonian fluids obey a linear relationship between shear stress and shear rate, while non-Newtonian fluids exhibit more complex behavior.
Dynamic viscosity plays an important role in many areas of engineering and science, such as fluid mechanics, rheology, and materials science. The article discusses the various methods used to measure dynamic viscosity, such as capillary viscometers and rotational viscometers.
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