Test: Newton's Law of Viscosity - Civil Engineering (CE) MCQ

The mathematical model for describing the behavior of fluids is the Power-Law model, which is given as:

If
n = 1; ⇒ fluid is Newtonian Fluid.
n ≠ 1; ⇒ fluid is non-Newtonian
n < 1; ⇒ fluid is called Pseudoplastic fluids.  
n > 1; ⇒ fluids called dilatants fluids.  
The mathematical model for describing the ideal Plastic fluid is given as:

From the given table:
If du/dy = 0 then τ =  10 i.e. non – zero ⇒ Fluid can not be Newtonian and non-Newtonian

Further Shear stress is a linear function of du/dy, so the fluid is ideal plastic. 

Concept:
Newtonian fluids always follow Newton's law of Viscosity. Mathematically Newton's law of viscosity is represented as

Force equation from the above equation can be obtained as

where, μ = Coefficient of Viscosity, τ = Shear Stress
From the given question we can observe that Area(A), Viscosity(μ) and clearance(dy) is constant, hence Force is directly proportional to velocity.

Calculation:
Given:
F₁ = 40 N, F₂ = 300 N, V₁ = 80 cm/s
40/300 = 80/V₂
V₂ = 600 cm/s

Kinematic viscosity:
It is defined as the ratio between the dynamic viscosity and density of the fluid.
v = μ/ρ
Units of kinematic viscosity:


∴ the SI unit of kinematic viscosity is m²/s and the C.G.S unit of kinematic viscosity is cm²/s or 'Stoke'.
1 Stoke = 10^-4 m²/s.

Newtonian fluids: fluids for which the shear stress is linearly proportional to the shear strain rate and follow Newton's law of viscosity.

•  where μ is shear viscosity of the fluid
• Newtonian fluids are analogous to elastic solids (Hooke’s law: stress proportional to strain)
• Any common fluids, such as air and other gases, water, kerosene, gasoline, and other oil-based liquids, are Newtonian fluids
• A fluid whose viscosity does not change with the rate of deformation or shear strain is known as a Newtonian fluid.

Non-Newtonian fluid: fluids for which the shear stress is not linearly related to the shear strain rate are called non-Newtonian fluids 

• In non-Newtonian, the viscosity is dependent on the shear rate (Shear Thinning or Thickening)
• examples include slurries and colloidal suspensions, polymer solutions, blood, paste, and cake batter 

Ideal fluid: fluids which don’t have viscosity and are incompressible are termed as an ideal fluid

• The ideal fluid does not offer shear resistance i.e no resistance is encountered as the fluid moves.

Real fluid: fluids which do possess viscosity are termed as real fluids.

• These fluids always offer shear resistance i.e. Certain amount of resistance is always offered by these fluids as they move.
  

Concept:
Shear stress in the oil

where: μ = Dynamic viscosity, τ = Shear stress, V = velocity of plate, y = distance between the plates.

Calculation:
Given:
Thickness = t = 1.2 cm = 0.012 m, μ = 15 poise = 1.5 Ns/m², ΔV = 3.25 m/s

For a Newtonian fluid,
shear stress, 
or,  where τ = shear stress,
du/dy = dθ/dt = rate of shear strains,
θ = displacement in fluid layer

Concept:
Shear stress in flow over a flat plate is given as:

where, μ = dynamic viscosity, y = distance in meter above the plate.
1 P = 0.1 Pa-sec

Calculation:
Given:
μ = 8.5 poise = 8.5 × 10^-1 Pa-sec, y = 0.15 m, 


∴ τ = 8.5 × 10^-1 × 0.45 
τ = 0.3825 N/m² or Pa

Concept:

For a Newtonian fluid,
share stress (τ)  = dynamic visocity (μ) × rate of share strain (du/dy)
For the above situation, 

where, A = Area in contact of a Newtonian fluid, V = Terminal velocity of the block
Calculation:
Given:
Edge of the block, a = 100 mm = 0.1 m , weight, W = 1 kN, θ = 30°, y = 0.02 mm, 

Shear stress acting at the face of the cube in contact with fluid,

V = 5 m/s

According to Newton's law of viscosity, shear stress is directly proportional to the rate of angular deformation (shear strain) or velocity gradient across the flow.


where, τ = shear stress, μ = absolute or dynamic viscosity, du/dy = velocity gradient ⇒ dα/dt = rate of angular deformation (shear-strain).
Units of Dynamic viscosity:

∴ the unit of dynamic viscosity in the SI unit is Ns/m² or Pa-s.
The dimension of dynamic viscosity is ML^-1T^-1.
1 Pa-s = 10 Poise
∴ 1 Poise = 0.1 Pa-s or Ns/m²
Kinematic viscosity: 
It is defined as the ratio between the dynamic viscosity and density of the fluid.
v = μ/ρ
Units of kinematic viscosity:

∴ the SI unit of kinematic viscosity is m²/s and the CGS unit of kinematic viscosity is cm2/s or 'Stoke'.
1 Stoke = 10^-4 m²/s = 0.0001 m²/s.
The dimensional formula for Kinematic viscosity (ν) is ν = L²T^-1.

Concept:
From Newton's law of viscosity

We know that 
F = τ × A

Calculation:
Given: F₂ = 3 kN, F₁ = 0.9 kN, u₁ =1.25 cm/s

∴ u₂ = 4.166 cm/s