The flow of a fluid when each particle of the fluid follows a smooth path, the paths which never interfere with one another. Therefore, the laminar flow is that flow in which the velocity of the fluid is constant at any point in the fluid.

In conclusion, the flow of a viscous fluid in which particles of the fluid move in parallel layers, each of which has a constant velocity but is in motion relative to its neighboring layers is termed as Laminar Flow.

#### Laminar flow between two parallel plates

**a) When both plates are fixed**

- U max = 1.5 U avg
- Kinetic energy correction factor = 1.5
- Momentum correction factor = 1.2

**b) When one plate is moving and other at rest**

Laminar flow between two parallel plates located at distance apart such that the lower is at rest and upper plate moves uniformly with velocity V.

When pressure gradient is zero, it indicates that velocity distribution is linear. This particular case is known as **simple COUETTE flow**.

In this simple couette flow

- Kinetic energy correction factor = 2
- Momentum correction factor = 4/3

#### No-slip condition

Inviscid flow is the flow of an inviscid fluid, in which the viscosity of the fluid is zero.

While the inviscid flow theory allows the fluid to “slip” past a solid surface, real fluids will adhere to the surface because of inter-molecular interactions, that is, a real fluid satisfies the condition of zero relative velocity at a solid surface. This is the so-called No-slip condition.

When viscous forces are neglected, such as the case of inviscid flow, the Navier-Stokes equation can be simplified to a form known as the Euler equation.

#### Steady laminar flow

a) In steady laminar flow of a liquid through a circular pipe of internal diameter D, carrying a constant discharge,

The hydraulic gradient i.e. hf/L

Therefore, hydraulic gradient is inversely proportional to D^4.

b) In steady laminar flow of a liquid through a circular pipe shear stress is** maximum at the wall of pipe** and **zero at the centre line**.

Maximum velocity is two times the average velocity.

V max = 2 (V avg)

#### Reynolds Number *(Re)* of laminar flow

For laminar flow in pipe, **0 <= Re <= 2000**

Let us know how some parameters vary with the increase in Reynolds Number (Re) in laminar flow.

#### Darcy Weisbach friction factor

Darcy Weisbach friction factor in the case of pipe flow for laminar flow is:

f = (64/Re)

So, when for a given discharge, the height of the surface roughness will change, the Darcy Weisbach friction factor will remained unchanged. Because it is independent of height.

#### Coefficient of friction

Friction factor (f) = (8*shear stress) / (mass density*mean velocity^2)

Coefficient of friction (f ‘) = f / 4 = (2*shear stress) / (mass density*mean velocity^2)

#### Rouse criteria for stability of laminar flow

Rouse developed a criterion for stability of laminar flow at any point

Local instability occur first at a point where rouse criterion is maximum. Therefore,

#### Effect of laminar flow on power consumption

If the **laminar flow** of oil between 2 points of a given pipeline is **doubled**, then** power consumption (P)** is increased to **four times** the original power. This is because in the laminar flow through the circular pipe, **head loss (hf) varies directly as the discharge** **(Q)**.

P ∝ Q.hf , hf ∝ Q

Therefore,** P ∝ Q^2**

#### Profile of laminar flow

We know, shear stress = µ(dv/dy)

So, in laminar flow if we consider **shear stress profile as a straight line** then the **velocity profile is parabolic** in nature.

#### Principles and its effects

**Hele Shaw flow**– Laminar flow between parallel lines/ plates**Stokes law**– Settling of fine particles**Hagen Poiseuille flow**– laminar flow in tubes/ pipes**Pascal’s Law**– surface of equal pressure

#### Navier-Stokes equation

Newton second law of motion, F = ma

If only gravity, pressure, viscosity forces are taken into account, then

**Fp + Fg + Fv = ma**

This equation is known as Navier-Stokes equation and it is applicable in all cases of laminar flow.

Therefore, **Navier-Stokes equation** is applicable in following cases of laminar flow

- In circular pipes
- Between concentric rotating cylinders
- between parallel plates (both in stationery and in relative motion)

#### Flow change from laminar to turbulent depends on

The critical value of reynolds number at which the boundary layer changes from laminar to turbulent depends upon

- Roughness : More roughness leads to more instability
- Pressure gradient
- Turbulence in ambient flow
- Intensity and scale of turbulence

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