# Viscosity in Fluid Mechanics | Newtonian and Non-Newtonian Fluids

## Viscosity Meaning:

Viscosity in fluid mechanics is a property of the fluids by virtue of which they offer resistance to shear or angular deformation. This is mainly due to cohesion and molecular motion exchange between fluid layers, and as flow occurs, these effects manifest as shear stresses between the moving layers.

It is a measure of the resistance of the fluid to deformation It is duo lo cohesion and molecular momentum exchange between fluid layers and as flow occurs, these effects appear as shoring stressors between the moving layers

Suppose one layer of fluid is moving with respect to the other layer by a velocity du and the vertical gap between the two layers be dy.

The upper layer which is moving faster tries to draw the lower slowly moving layer along with it.

Similarly, as a reaction to this, the lower layer tries to retard the upper one.

In time dt, the top layer will move with respect to the bottom layer by a distance ds= du.dt

Hense,    ds/dy = dθ = shear strain

du.dt/dy = dθ

dθ/dt = du/dy ( viscosity equation )

Rate of change od shear strain (dθ/dt) = Velocity gradient (du/dy)

the rate of change of angular deformation is equal to the velocity gradient across the flow.

Based on the relationship between the applied shear stresses and the rate of flow or deformation, fluids can be classified as Newtonian and non-Newtonian fluids.

## Newtonian Fluids:

The fluid that obeys newton’s law of viscosity is known as a Newtonian fluid.

### Newton’s law of viscosity:

The fluid for which the rate of deformation is linearly proportional to shear stress.

Thus, for a Newtonian fluid

τ ∝ dθ/dt

τ = shear stress opposing the movement of fluid

τ ∝ du/dy

τ = µ. du/dy

µ = absolute viscosity, “or” coefficient of viscosity “or” dynamic viscosity

water, air, and gasoline are Newtonian under normal conditions.

#### Dynamic Viscosity:

• Dynamic viscosity is denoted by the symbol “µ”
• SI unit of dynamic viscosity is Pa-sec, or N-sec/m² or kg/m-sec
• Dynamic viscosity unit in CGS system-

1 poise = Dyne-sec/cm² {Dyne = gm-cm/sec²}

1 poise = gm/cm-sec

• Dimentions of dynamic viscosity is [ML-1T-1]

Note: Water is nearly 55 times viscous than air.

#### Kinematic Viscosity:

• Kinematic viscosity is denoted by the symbol “ν”
• Kinematic viscosity (ν) is = dynamic viscosity/mass density = μ/ρ
• SI unit of kinematic viscosity is m²/sec
• Kinematic viscosity unit in CGS system-

cm²/sec or stoke

1 stoke = cm²/sec

1 centipoise = 1/100 stoke

• Dimension of kinematic viscosity is [L2T-1]
• At 20°C and at standard atmospheric pressure,

value of kinematic viscosity for water is 1 × 10-6 m²/sec

value of kinematic viscosity for air is 15 ×  10-6 m²/sec

Note: kinematic viscosity of air is about 15 times greater than the corresponding value of water.

Difference Between Dynamic Viscosity And Kinematic Viscosity in Fluid Mechanics:

 Dynamic Viscosity Kinematic Viscosity It is defined as absolute viscosity. It gives more information about the force required to make a liquid flow at a specific rate. It is defined as the diffusivity of motion. To be precise, it describes how fast a liquid is moving when a certain amount of external force is applied. Whereas; dynamic viscosity, represents the viscous force of the liquid. It represents the inertia, as well as the viscous force of fluid. The symbol of dynamic viscosity is µ The symbol of the kinematic viscosity is ν This represents the ratio between shear stress to shear strain. This represents the ratio between dynamic viscosities to density. Dynamic force is utilized only when viscous force is dominant It is utilized when inertia and viscous force are dominant Dynamic viscosity is a derived property. Kinematic viscosity is a more fundamental property. Unit of Dynamic viscosity is Ns/m² Unit of kinematic viscosity is m²/s

### Effect of Temperature on Viscosity:

The viscosity of liquids decreases with an increase in temperature, whereas the viscosity of gases increases with an increase in temperature.

The reason for the above phenomena is that; in liquids; viscosity is primarily due to molecular cohesion which decreases with an increase in volume due to temperature increment, while In gases, viscosity is due to molecular momentum transfer that increases with an increase in the number of collisions between gas molecules.

Effect of temperature on dynamic viscosity

In liquid: In the case of liquids temperature increases dynamic viscosity decreases.

In Gases: In the case of gases temperature increases dynamic viscosity increases.

Effect of temperature on kinematic viscosity

In liquid: In the case of liquids temperature increases kinematic viscosity decreases.

In Gases: In the case of gases temperature increases kinematic viscosity increases.

### Effect of Pressure on Viscosity:

For liquid, viscosity is practically independent of pressure except at extremely high pressure. so when increases pressure dynamic viscosity and kinematic viscosity will remain the same for liquid.

For gases, dynamic viscosity is generally independent of pressure particularly ( at low to moderate pressure ) but kinematic viscosity decreases as density are proportional to pressure. so when increasing in pressure dynamic for gases it remains the same but kinematic viscosity decreases.

## Non-Newtonian Fluids:

Fluids for which shear stress is not directly proportional to deformation rate are non-Newtonian. Non-Newtonian fluids commonly are classified as having time-independent time-dependent behavior.

They do not obey Newton’s law of viscosity.

Toothpaste and paint are examples of non-Newtonian fluids.

Relation between shear stress and rate of deformation for non-Newtonian fluid can be represented as ( τ )

$\huge τ\ =\ A\left(\frac{du}{dy}\right)^n+B$

Where, n = flow behavior index; k = consistency index

Apparent Viscosity:

• apparent viscosity denoted by “η”
• dynamic viscosity is constant ( except for temperature effects ) while apparent viscosity depends on the shear rate.
• Equation of apparent viscosity is:

### Various Types of Non-Newtonian Fluids Are:

1. Pseudoplastic: Fluids in which the apparent viscosity decreases with increasing deformation rate ( n< 1) are called pseudoplastic fluids (or shear thinning) Most non-Newtonian fluids fall into this group.

n<1 and B=0

Ex. paper pulp, Polymer solutions, colloidal suspensions, milk, blood, and paper pulp in water.

2. Dilatant: If the apparent viscosity increases with increasing deformation rate (n>1), the fluid is termed as dilatant (or shear-thickening)

n>1 and B=0

Ex. butter, Suspensions of starch, saturated sugar solution

3. Bingham Plastic: Fluids that behave as a solid until minimum yield stress, and flow after crossing this limit are known as ideal plastic or Bingham plastic. The corresponding shear stress model is τ = τ y + du/dy

n=1 and B≠0

Ex. sewage sludge, Clay suspensions, drilling muds, creams, and toothpaste

4 Thixotropic: Apparent viscosity (η) for thixotropic fluids decreases with time under constant applied shear stress.

Ex. Paints, Printer inks

n<1 and B≠0

5. Rheopectic: Apparent viscosity (η) for rheopectic fluids increases with time under constant shear stress.

n>1 and B≠0

Ex. Gypsum pastes and bentonite solution.

Note:
(i) There is no relative movement between fluid attached to the solid boundary and solid boundary i.e. the fluid layer just adjacent to the solid surface will have the same velocity as the solid surface.
(ii) Viscoelastic: Fluids which after some deformation partially return to their original shape when the applied stress is released such fluids are called viscoelastic.
(iii) Rheology: Branch of science that deals with the studies of different types of fluid behaviors.