Overview: Newton's Law of Viscosity

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Overview: Newton's Law of Viscosity

Introduction

If we drop a cup of water over a sloped surface while concurrently pouring a cup of honey, the water will definitely flow faster than the honey. This is due to the fluid property known as viscosity. Viscosity is a measure of the resistance of a fluid to deform under shear stress. It is commonly perceived as resistance to pouring. Following are some important terms of viscosity:

-          Shear Rate: Shear rate is the rate at which layers of fluid move relative to each other due to an applied shear stress. It is often measured in inverse seconds (s^-1).

-          Shear Stress: Shear stress is the stress that acts tangentially to a surface, causing deformation or movement of layers of fluid. It is often measured in units of Pascal (Pa).

-          Viscosity Index: Viscosity index is a measure of how much a fluid's viscosity changes with changes in temperature. It is an important factor to consider in the selection of lubricants and hydraulic fluids for various applications.



 In this article, we will explore an important term related to fluid viscosity that is Newton's law of viscosity.

Newton's Law of Viscosity

It is an essential concept in fluid mechanics that explains the connection between the shear stress and the velocity gradient in a fluid. Sir Isaac Newton introduced the law in the 17th century, and it is still an important topic in the study of today's fluid mechanics.

Newton's law of viscosity can be defined as the velocity gradient impacting shear stress directly. The shear stress between two layers of fluid is directly proportional to the negative velocity gradient between them.

The equation of Newton’s law of viscosity is τ = μ du/dy

Newton’s Law of Viscosity Derivation

Shear stress (τ ) = Force(F)/ Area(A)

Velocity gradient = du/dy. Here, du is the velocity difference, and dy is the distance between the layers.

According to Newton’s law of viscosity, shear stress is proportional to velocity gradient.

τ ∝ du/dy

∴ τ= μ du/dy 

Here, μ is the viscosity or coefficient of viscosity.

The law applies to fluids in a laminar flow regime, which implies the flow is uniform and the layers of fluid move parallel to each other. When a fluid is in turbulent flow, the flow is disorganized, and the layers of fluid move in a disorderly way. In such instances, the connection between shear stress and velocity gradient is not easy and requires more sophisticated equations to express.

Newton's Law of Viscosity has several applications in a variety of domains, including engineering, biology, and medicine. It is used to describe fluid behavior in pipes and channels, blood flow via blood vessels, magma movement in volcanic eruptions, and many other phenomena.

Dynamic and kinematic viscosity

The internal friction or reluctance of a fluid to flow under shear stress is measured by dynamic viscosity, also known as absolute viscosity. It is measured in Pa. s or Ns/m2. A fluid's dynamic viscosity is affected by elements such as temperature, pressure, and the presence of any dissolved compounds.

Dynamic viscosity is a fluid characteristic that is relevant to numerous engineering and scientific endeavours. It is used in the layout of pumps, mixers, and various other fluid-handling equipments, and also in predicting fluid flow through pipes and other conduits.

Kinematic viscosity is defined as the ratio of dynamic viscosity to density and is a measure of a fluid's internal resistance to flow under the effect of gravity. It is denoted by the symbol v and is measured in m²/s. Kinematic viscosity is affected by the same elements that affect dynamic viscosity.

The relation between dynamic viscosity (μ), kinematic viscosity (ν), and density (ρ) is given by:

ν = μ/ρ

The units of dynamic and kinematic viscosity can be converted using the equation:

 

1 Pa·s = 1 N·s/m² = 10 Poise = 1000 centipoise (cP)

 

1 m²/s = 10^6 mm²/s = 100 centistokes (cSt)

Types of fluids



Newtonian fluids are those that obey Newton's viscosity law, which says that the shear stress applied to a fluid is proportional to the rate of deformation or shear rate. In other words, regardless of the shear rate applied, the viscosity of a Newtonian fluid stays constant. Water, air, glycerol, and most gases are examples of Newtonian fluids.

Fluid Newtonian behavior is critical for many applications, including pipeline design, lubrication systems, and hydraulic machinery. Understanding Newtonian fluid properties and behavior is critical for engineers and scientists working in these domains.

Properties and behavior of Newtonian fluids:

Here are some of the key properties and behaviors of Newtonian fluids:

  1. Constant viscosity: The viscosity of Newtonian fluids is constant independent of shear rate. As a result, they are straightforward to model and predict using simple mathematical formulae.
  2. Linear relationship between shear stress and shear rate: Newton's Law of Viscosity describes the linear relationship between shear stress and shear rate in Newtonian fluids. Because of this linear relationship, it is simple to measure and characterise the viscosity of Newtonian fluids.
  3. Incompressible: Most Newtonian fluids, like as water and air, are incompressible, which means that their density remains constant under pressure. This feature is critical in a variety of applications, including hydraulic systems and fluid movement through pipes.
  4. Homogeneous flow: Newtonian fluids flow smoothly and uniformly, having the same shear rate at every location in the fluid. As a result, they are beneficial in applications that involve lubrication and heat transfer.

Non-Newtonian fluids do not obey Newton's Law of Viscosity. Non-Newtonian fluids do not have an even viscosity and have more complex correlations among shear stress and shear rate. In other words, the viscosity of non-Newtonian fluids varies with stress or shear rate. Gels, polymers, suspensions, and various biological fluids are examples of non-Newtonian fluids.

There are several types of non-Newtonian fluids, each with its own unique behaviour:

  • Shear-thinning fluids: Shear-thinning fluids have a decreasing viscosity as the shear rate increases. They're common in things like ketchup, toothpaste, and paint.
  • Shear-thickening fluids: have increasing viscosity as the shear rate increases. They are frequently employed in protective equipment, such as body armour, as well as in some suspension systems.
  • Bingham plastic fluids: These fluids have a yield stress below which they do not flow but behave as a viscous fluid above which they flow. They are frequently found in drilling fluids, food, and cosmetics.
  • Viscoelastic fluids: These fluids have both viscous and elastic qualities, which mean they can flow like a liquid but can deform like a solid. Polymers, gels, and biological fluids such as blood are all examples.
  • Thixotropic fluids: Under steady stress or shear rate, the viscosity of these fluids decreases over time. They are frequently found in coatings and inks.
  • Rheopectic fluids: Rheopectic fluids are liquids or gases whose viscosity of the fluid increases with stress over time. The behaviour of these fluids can be described as a time-dependent dilatant behaviour. Thus, these fluids are a rare class of non-Newtonian fluids. Also, they show an increased viscosity upon agitation. Some common examples of rheopectic fluids include some gypsum pastes, printer ink, lubricants, etc.


The Rheological behaviour of fluids

Fluid rheological behaviour relates to how fluids deform and flow under various conditions. Understanding fluid rheology is crucial in many fields, such as manufacturing, engineering, and medicine. Fluids can display many distinct forms of rheological behaviour, including:

1.      Elastic behaviour: Elastic behaviour is defined as a fluid's capacity to return to its original shape following deformation. Viscoelastic fluids are fluids that show elastic behaviour. Rubber, silicone, and some polymers are examples of viscoelastic fluids. Viscoelastic fluids are used in a variety of sectors, including the manufacture of rubber goods and the creation of medical implants.

2.     Plastic behaviour: Plastic behaviour is characterised by a fluid's capacity to undergo irreversible distortion without reverting to its original shape. Viscoplastic fluids are fluids that display plastic behaviour. Toothpaste, clay, and various types of lubricants are examples of viscoplastic fluids. Viscoplastic fluids are used in a variety of sectors, including cosmetics, food preparation, and manufacturing.

3.      Shear-thinning behaviour: Shear-thinning behaviour is distinguished by a reduction in viscosity as the shear rate rises. Shear-thinning fluids, also known as pseudoplastic fluids, have shear stress related to shear rate to a power less than one. Paints, gels, and some types of lubricants are examples of shear-thinning fluids.

4.      Shear-thickening: Shear-thickening behaviour is defined by an increase in viscosity as the shear rate increases. Shear-thickening fluids, commonly known as dilatant fluids, have shear stress proportional to the shear rate to a power larger than one. Cornflour and water mixtures are two shear-thickening fluids.

5.      Thixotropic behaviour: is distinguished by a decrease in viscosity with time under constant shear force. Thixotropic fluids are frequently employed in industrial processes because they may be easily mixed and poured when subjected to shear stress but thicken again when left undisturbed. Some paints and inks are examples of thixotropic fluids.

     To recap, fluid rheological behaviour is crucial to understand since it can impact fluid performance in a variety of applications. Elastic, plastic, shear-thinning, shear-thickening, and thixotropic behaviour are all examples of rheological behaviour in fluids. Understanding fluid rheology may aid in the creation of new goods and the optimisation of production processes.

Temperature and Pressure dependency of Viscosity

Temperature and pressure can have an effect on viscosity, which is a measure of a fluid's resistance to flow. Let's look at how viscosity is affected by each of these variables:

  1. Temperature Dependence:

In general, viscosity reduces with increasing temperature. Most fluids, including liquids and gases, exhibit this behaviour. The explanation for this association is due to the molecular structure and mobility of the fluid.

For liquids, as the temperature rises, so does the kinetic energy of the molecules. Because of the increased kinetic energy, the molecules move faster, resulting in reduced intermolecular interactions and a drop in viscosity. As a result, at higher temperatures, liquids become less viscous.

In the case of gases: Viscosity in gases is mostly determined by molecular collisions. Higher temperatures cause gas molecules to have more kinetic energy and collide more often, resulting in lower viscosity.

It should be noted that this temperature dependence does not apply to all compounds. Certain substances, such as polymers or molten materials, may exhibit non-Newtonian behaviour, in which viscosity increases with temperature. However, the general trend for most common fluids is a reduction in viscosity with rising temperatures.

  1. Pressure Dependence:

The viscosity of fluids is also affected by pressure; however the relationship is often not as strong as the temperature dependency. For various liquids and gases, a rise in pressure causes a modest increase in viscosity.

For liquids, higher pressures compress the molecules in the liquid, leading to a reduction in intermolecular spacing. This compression increases the intermolecular forces, causing the viscosity to rise.

For gases, rising pressure brings gas molecules closer together, increasing the frequency of molecular collisions. This increased collision frequency causes a modest rise in viscosity in gases.

It is crucial to note, however, that the effect of pressure on viscosity is typically far smaller than the effect of temperature. When dealing with extremely high pressures or specialised fluids, the influence of pressure on viscosity is generally overlooked in most practical applications.

Significance of Newton’s law of Viscosity

Newton's law of viscosity is significant because it may describe the behaviour of fluids under shear stress, which is a fundamental feature of fluids. The rule is widely utilised in numerous areas, including engineering, materials science, and medicine, and it gives fundamental knowledge of fluid flow.

Here are some of the most important implications of Newton's law of viscosity:

  1. Predicting fluid behaviour: Newton's rule of viscosity gives a mathematical foundation for predicting fluid behaviour under various variables, such as temperature, pressure, and shear rate variations. This is critical in many sectors, including pipeline, pump, and other fluid-handling handling system designs.
  2. Creating new goods: Predicting the behaviour of fluids under various situations is critical in the creation of new products such as paints, coatings, and lubricants. Manufacturers may develop items that function effectively in a specific application by understanding the viscosity of a fluid.
  3. Understanding the viscosity of physiological fluids such as blood and synovial fluid is vital in the diagnosis and treatment of a wide range of medical diseases. A high viscosity of blood, for example, can cause cardiovascular illness, whereas a low viscosity of synovial fluid might cause joint difficulties.
  4. The rule is especially important in numerical simulations of fluid flow, such as computational fluid dynamics (CFD). Numerical simulations may be used to anticipate the behavior of fluids in complicated geometries and under diverse situations by applying the law to the equations of motion.

To summarise, Newton's rule of viscosity is a key principle in the study of fluid mechanics, with several applications. The law establishes a mathematical foundation for predicting fluid behavior under various settings, allowing for the development of novel goods, medicinal treatments, and materials.

Examples

  1. A metal plate 0.07 m2 in area rests horizontally on a layer of oil 1 μm = 1 x 10-6 m thick. A force of 30 N applied to the plane horizontally keeps it moving with a uniform speed of 15 cm/s. Find the viscosity of oil.

Given: Area of plate = 0.07 m2, Thickness of layer= dy = 1 μm = 1 x 10-6 m, Applied force = F = 30 N, Velocity of the plate = dv = 15 cm s-1 = 15 x 10-2 m s-1.

Sol:

By Newton’s law of viscosity

F=ƞ. A. (dv/dy)

Ƞ= F.dy/A.dv

Ƞ= (30*1*10-6)/(0.07*15 * 10-2 )

Ƞ=0.00285 Pascal second

 

2.      A Newtonian fluid fills the clearance between a shaft and a sleeve. When a force of 0.8kN is applied to the shaft parallel to the sleeve, the shaft attains a speed of 1.25 cm/s. What will be the speed of the shaft if a force of 4kN is applied?

Concept:

From Newton’s Law of Viscosity

τ = μ du/dy

We know that, F= τ * A

F = ƞ. A. (dv/dy)

So, F2/F1 = V2/V1

Sol:

F2 = 4kN, F1 = 0.8kN, V1 = 1.25cm/s

V2 = 6.25 cm/s

References

  • Newton’s law of viscosity, Numerical Problems on Viscosity, DOI: https://thefactfactor.com/tag/newtons-law-of-viscosity/
  • Engineering ToolBox, (2003). Viscosity - Absolute (Dynamic) vs. Kinematic. [online] Available at: https://www.engineeringtoolbox.com/dynamic-absolute-kinematic-viscosity-d_412.html 
  • https://wiki.anton-paar.com/en/basic-of-viscometry/
  • George, H.F., Qureshi, F. (2013). Newton’s Law of Viscosity, Newtonian and Non-Newtonian Fluids. In: Wang, Q.J., Chung, YW. (eds) Encyclopedia of Tribology. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-92897-5_143
  • Newtonian fluids, Roberto Gauntlett, https://slideplayer.com/slide/4117308/
  • Difference between Thixotropic and Rheopectic Fluids, May 13, 2020 Posted by Madhu, https://www.differencebetween.com/difference-between-thixotropic-and-rheopectic-fluids/.
  • https://testbook.com/objective-questions/mcq-on-newton-law-of-viscosity--5eea6a0c39140f30f369e157#:~:text=According%20to%20Newton's%20law%20of,velocity%20gradient%20across%20the%20flow.&text=where%2C%20%CF%84%20%3D%20shear%20stress%2C,deformation%20(shear%2Dstrain).

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