Viscosity

Suppose a person has two bowls, one bowl contains water and the other has honey in it, that honey takes more time to form the shape of the bowl than water. This happens because the viscosity of the two fluids is different. Viscosity is the property of the liquids that prevents liquids from spreading. The property of liquids due to which each fluid opposes the relative motion between its different layers is called Viscosity. The force generated due to viscosity is called Viscous Force. Since this force is between the layers of liquids, it is also called internal friction. 

In this article, various different problems on an important property found in liquids will be learned. Flowing fluid has many properties, the flowing fluid carries many of the energies inside it. The flowing fluid has many factors that prevent it from becoming an ideal liquid, not all liquids are ideal liquids, and they exert some resistance to motion. This resistance in liquid motion can be seen as internal friction similar to friction produced by motion on the surface in solids, it is called Viscosity. This force is present when the fluid surfaces have relative motion. 

Viscosity

The property of a liquid by virtue of which an opposing force (internal friction) comes into play between different layers of a liquid, whenever there is a relative motion between these layers of the liquid is called viscosity.

Imagine a fluid moving parallel to it on the horizontal plane. The fluid may be thought to consist of several layers. If the flow of the fluid is sharp, then the layer that is in contact with the horizontal plane will have a velocity of about zero. But as the distance of the layer from the horizontal plane increases, the velocity of the flow of the layer will increase. 

As shown in the figure above, the topmost fluid layer has a maximum velocity therefore, has always a difference in the velocity of the two adjacent addresses of the fluid. One layer is flowing faster than the other layer, just as in solid objects, due to their relative motion, the opposing frictional force acts between their entire planes. Similarly, due to the relative motion between their adjacent layers in the fluid also, a frictional force acts between them, which is relative to the layers, resists the motion. In other words, it can be said that a layer flowing at low speed tries to reduce the velocity of a layer flowing at a fast speed. This property of the fluid is called Viscosity. 

Velocity Gradient: The velocity gradient is defined as the ratio of change of velocity (dv) to the distance (dx). The direction of velocity gradient is normal to the direction of flow, directed in the direction of increasing velocity. Mathematically, it is given by:

Velocity gradient = dv / dx

Co-efficient of Viscosity 

Representation for Co-efficient of Viscosity 

Consider a flow of a liquid over the horizontal solid surface as shown in the figure. Let us consider two layers AB and CD moving with velocities v and (v + dv) respectively from the fixed solid surface. According to Newton’s law of viscosity, the viscous drag or backward force (F) on the solid surface, between these layers is,

• Directly proportional to the area (A) of the layer. 
• Directly proportional to the velocity gradient (dv/dx) between the layers. 

Therefore, it can be written as:

F ∝ A (dv/dx)

Lets remove the proportionality sign by introducing a proportionality constant η.

F = -η A (dv/dx)

Here η is called the coefficient of viscosity. Also, the negative sign in the above expression shows that the direction of viscous drag (F) is just opposite to the direction of the motion of the liquid. 

If A = 1 m² and dv/dx = 1 s^-1 then the above expression becomes: 

η = F

Thus, the coefficient of viscosity of a liquid is defined as the viscous drag or force acting per unit area of the layer having a unit velocity gradient perpendicular to the direction of the flow of the liquid. 

Units of Co-efficient of Viscosity 

• In the CGS system, the unit of coefficient of viscosity is dynes s cm^-2 or Poise.
• While in SI system the unit of coefficient of viscosity N s m^-2 or deca-poise. 
• The dimensional formula for the coefficient of viscosity is [ML^-1 T^-1].

Variation of Viscosity

• Effect of Temperature on Viscosity: The viscosity of liquids decreases with an increase in temperature. The viscosity of gases increases with an increase in temperatures as η ∝ √T.  
• Effect of Pressure on Viscosity: The coefficient of viscosity of liquids rises as pressure increases, although there is no relationship to explain the phenomenon thus far.

Obtaining data on the viscosity of a certain substance assists producers in predicting how the material will behave in the real world. Knowing a material’s viscosity influences how production and transportation operations are built. Therefore, the following is the list of some fluids and their coefficient of viscosity at different temperatures:

Fluid	Temperature (in °C)	η (deca-poise)
Air	20	0.018 × 10-3
Water	0	1.8 × 10-3
	20	1.0 × 10-3
Blood	100	0.3 × 10-3
	37	2.7 × 10-3
Engine Oil	30	250 × 10-3
Glycerine	0	10
	20	1.5

Applications of Viscosity 

Knowledge of the viscosity of various liquids and gases has been put to use in daily life. Some applications of its knowledge are discussed as under:

1. The coefficient of viscosity of organic liquids is used to calculate their molecular weights.
2. Knowing the coefficient of viscosity and how it varies with temperature allows us to select the best lubricant for each machine. Thin oils with low viscosity (for example, lubricating oil used in clocks) are utilised in light machinery. Highly viscous oils (for example, grease) are employed in heavy machinery. Viscosity is the most important quality of lubricating oils in lubrication, and it is also highly important in greases, which is frequently overlooked. The resistance to movement is defined as viscosity. Water has a low viscosity because it flows quickly, but honey has a high viscosity. 
3. The viscosity of a few drugs, such as the numerous solutions used to eradicate moles, has also been reduced to make application simpler. To coat the throat, drug firms provide treatments with a high viscosity yet are still drinkable, such as cough syrup. 
4. The viscosity of paints, varnishes, and other home items is tightly controlled so that they may be applied smoothly and uniformly with a brush roller.
5. Viscosity is an important factor in food preparation and serving. Cooking oils’ viscosity may or may not vary as they heat, but many become considerably more viscous when they cool.When fats are cold, they become solid because they are viscous when heated.
6. To function properly, manufacturing equipment need the use of appropriate lubricant. Too viscous lubricants can block and block pipes. Lubricants that are excessively thin provide insufficient protection for moving components.
7. Coating viscosity is one of the important characteristics that determines the success of the coating technique. Because the uniformity and repeatability of the coating operation are frequently connected to the viscosity of the coating, it is an important parameter to regulate.

Sample Problems 

Problem 1: There is a 3 mm thick layer of glycerin between a flat plate and a large plate. If the viscosity coefficient of glycerin is 2 N s/m² and the area of the plane plate is 48 cm. How much force is required to move the plate at a speed of 6 cm/s? 

Solution:

Given that, 

The thickness of the layer, dx = 3 mm = 3 × 10^-3 m.

The coefficient of viscosity, η = 2 N s/m²

The change in speed, dv = 6 cm/s = 6 × 10^-2 m/s.

The area of the plate, A = 48 cm² = 48 × 10^-4 m²

The formula to calculate the force required to move the plate is, 

F = ηA × (dv/dx)

Substitute the given values in the above expression as:

F = 2 N s/m² × 48 × 10^-4 m² × (6 × 10^-2 m/s / 3 × 10^-3 m)

  = 192 × 10^-3 N

  = 0.192 N

Problem 2: The diameter of a pipe is 2 cm. what will be the maximum average trick of water for level flow? The viscosity coefficient for water is 0.001 N-s/m².

Solution: 

Given that, 

The diameter of the pipe, D = 2 cm = 0.02 m.

The viscosity coefficient, η = 0.001 N-s/m².

The density of water, ρ = 1000 kg/m³.

Since, the maximum value of K for level flow is 2000.

Therefore, the formula to calculate the maximum speed of water is,    

v = Kη / ρD

Substitute the given values in the above expression to calculate v as,

v = 2000 × 0.001 N-s/m² / 1000 kg/m³ × 0.02 m

   = 0.1 m/s 

Problem 3: How the viscosity of the liquids and gases will change with temperature?

Solution: 

The momentum transfer between the molecules of gases caused viscosity. With the increase in the temperature of the liquid, the molecular speed increases, reducing the viscosity. And the viscosity of a liquid is due to cohesion between its molecules. With an increase in the temperature of the liquid, cohesion increases, leading to an increase in viscosity.

Problem 4: The shear stress at a point in a liquid is found to be 0.03 N/m². The velocity gradient at the point is 0.21 s^-1. What will be its viscosity?

Solution:

Given that, 

The shear stress, F/A is 0.03 N/m².

The velocity gradient, dv/dx is 0.21 s^-1.

The formula to calculate the viscous force is,

 F = -ηA (dv/dx)

where F is viscous force, A is area, dv/dx is the velocity gradient and η is the coefficient of viscosity.

Rearrange the above expression for η as,

η = (F/A) / (dv/dx)

Substitute the given values in the above expression to calculate η.

η = 0.03 N/m² / 0.21 s^-1

   = 0.14 N s/m²

Problem 5: Water is flowing slowly on a horizontal plane, viscosity coefficient of water 0.01 poise and its surface area is 100 cm². What is the external force required to maintain the velocity gradient of the flow 1 s^-1?

Solution: 

Given that, 

The viscosity coefficient of water, η = 0.01 poise = 0.001 kg/ms. 

The surface area, A = 100 cm² = 10^-2 m².

The velocity gradient of the flow, dv/dx = 1 s^-1.

The formula to calculate the viscous force, 

F = -ηA (dv/dx)

Substitute the given values in the above expression to calculate F.

F = 0.001 kg/ms × 10^-2 m² × 1 s^-1

   = 10^-5 N 

Problem 6: On a horizontal frictionless floor, a block of 0.10 m² surface area is placed on a layer of fluid 0.03 mm thick. The horizontal force on the block is 0.010 kg when applied, it starts moving at a fixed speed of 0.085 m/s. Find the viscosity coefficient of the fluid. 

Solution: 

Given that, 

The thickness, dx = 0.30 mm = 0.3 × 10^-3 m.

The change in the speed, dv = 0.085 m/s. 

The area, A = 0.10 m².

The force acting, F = 0.010 kg = 0.010 × 10 N = 0.10 N.

The formula to calculate the viscous force is, 

F = -ηA (dv/dx)

Rearrange the above expression for η.

η = F/A × (dv/dx) 

Substitute the given values in the above expression as,

η = (0.10 N / 0.10 m²) × (0.085 m/s / 0.3 × 10^-3 m)

   = 3.53 × 10^-3 Pa s          

Problem 7: What difference we can see in kinematic viscosity, If a fluid, which has a constant specific gravity, is taken to a planet where the acceleration due to gravity is 3 times compared to its value on earth?

Solution: 

Kinematic viscosity depends on the density and dynamic viscosity. Both, density and dynamic viscosity, are independent of acceleration due to gravity. Therefore, kinematic viscosity is independent of acceleration due to gravity.

Problem 8: Explain Bernoulli’s Theorem.

Solution:

When an incompressible and non-volatile fluid i.e. an ideal fluid flows in a torrent stream in a tube, the total energy of its unit volume or unit mass fixed at each point of its path. This is called Bernoulli’s Theorem. The above theorem is written as an equation as follows,

For unit volume: 

P + 1/2dv² + dgh = Constant

For unit mass:

P/d + 1/2 dv² + gh = Constant

where P is the Pressure, d is the Density, v is the velocity of flowing fluid, g is the gravitational acceleration and h is the height of water from earth.

Problem 9: Discuss some of the main similarities and differences between viscosity and solid friction.

Solution:

The following are some of the main similarities between viscosity and solid friction:

1. Both the viscosity and solid friction oppose the relative motion. The viscosity opposes the relative motion between two adjacent liquid layers, whereas solid friction opposes the relative motion between two solid bodies. 
2. Both viscosity and solid friction come into play. Whenever there is relative motion between layers of liquid or solid surfaces as the case may be. 
3. Both viscosity and solid friction are due to intermolecular interactions. 

The following are some of the main differences between viscosity and solid friction:

Viscosity	Solid Friction
Viscosity (or viscous drag) between layers of liquid is directly proportional to the area of the liquid layers.	Friction between two solids is independent of the solid surfaces in contact.
Viscous drag is independent of normal reaction between two layers of liquid.	Friction between two solids is directly proportional to the normal reaction between two surfaces in contact.