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

Fluid Kinematics

Fluid Kinematics is the branch of fluid mechanics which deals with response of fluids in motion without considering forces and energies in them. The geometry of motion refers to the study of kinematics .

Types of Fluid flow

1. Ideal and Real flow
2. Compressible and incompressible flow
3. Laminar and turbulent flow
4. Steady and Unsteady flow
5. Uniform and non uniform flow
6. Rotational and irrotational flow

Flow Patterns

• Stream line – Tracing of motion of different fluid particles
• Streak line – It is the line traced by series of fluid particles passing through a fixed point
• Path line – It is the actual path traced by a fluid particle over a period of time
• Equipotential line – Location of equal piezometric head

Flow patterns has following trajectory shapes:

Vorticity and stream function parameters exist both in rotational flow and irrotational flow but for irrotational flow its value is zero.

Velocity potential function exists only in ideal flow and irrotational flow.

Free vortex motion

A free vortex motion is that in which the fluid may rotate without any external force being applied on it. A free vortex flow is also called irrotational flow in order to emphasis the irrotationality of flow at all points except at the singular points.

Therefore, The flow of water in a wash hand basin is being emptied through a central opening, is an example of free vortex motion. In a central region of free vortex motion, the viscous effects becomes quite predominant due to which the fluid tends to rotate like a solid body with velocity proportional to the radius.

The motion of the particle is circular i.e. tangential.

Therefore, vortex flow is free vortex motion.

However, In it no external torque rotates the fluid mass.

Forced vortex motion

A motion in which the fluid rotates by the means of some external source of power is a forced vortex motion .

Its surface is parabolic (i.e. the rise of liquid level at ends and fall of liquid at the axis of rotation).

Therefore, Rotational vortex is a forced vortex motion.

However, In it some external torque or force rotates the fluid mass.

Rankine vortex motion

Rankine vortex motion is a combination of Free vortex motion and as well as Forced vortex motion.

The movement of air mass in the case of Tornado is an example of Rankine vortex motion. In other words, it is a Forced vortex motion at the core and Free vortex motion outside.

BERNOULLI’S Equation applicable for

1. along a streamline
2. for steady flow
3. for incompressible flow
4. when the effect of viscosity is neglected

EULER’S Equation is based on

EULER’S Equation is based on momentum conservation but not mass conservation. Thus, integration of differential form of euler’s equation of motion yields to bernoulli’s equation.

1. Elastic Solid Fluid
2. Thixotropic Fluid (n<1, B=/=0)
3. Ideal Plastic Fluid (n=1, B=/=0)
4. Rheopectic/ Rheological Fluid (n>1, B=/=0)
5. Pseudo Plastic Fluid (n>1, B=0)
6. Newtonian Fluid (n=1, B=0)
7. Dilatant Fluid (n<1, B=0)
8. Ideal Fluid

When B=0,

• n=1 Newtonian fluid, Viscosity invariant of shear stress (air, water, gasoline)
• n>1 Shear thickening Fluid/ Pseudo Plastic Fluid, Viscosity increases with increase in deformation (paint, blood, paper pulp, mud slurry)
• n<1 Shear Thinning/ Dilatant Fluid, Viscosity decreases with increase in shear stress (butter, starch, concentrated sugar solution, suspended sand solution)

When B=/=0,

• n=1 Ideal Bingham Fluid/ Ideal Plastic Fluid (tooth paste, drilling mud)
• n>1 Rheological Fluid, Viscosity increases with increase in shear stress
• n<1 Thixotropic Fluid, Viscosity decreases with increase in shear stress (printer’s ink)