Differential Analysis of Fluid Flow_B

Conservation of Linear Momentum or Navier-Stokes Equation

Applying Reynold transport theorem,

Navier-Stokes Equation is for:

• Unsteady
• Non-Linear 
• 2nd Order Partial Differential Equation
• Incompressible Flow

Problem in solving NSE:

• Non-linearity due to Convective Acceleration terms.
• No general schemes for solving non-linear partial DE.

Four unknowns (3 velocity components, 1 pressure) 

• Use above 3 equations.
• Use Incompressible Continuity equation.

Application of NSE:

• Solved for several simple flow conditions.
• Used to describe complex flows.

Applications of Navier-Stokes Equation

1. Steady, Laminar Flow between Fixed Parallel Plates

2. Couette Flow

It is defined as, "Laminar flow of viscous fluid in the space between two parallel plates (one of which is moving relative to the other)".

Flow is driven by:

1. Viscous drag force acting at the fluid.
2. Applied pressure gradient parallel to the plates.

Application are:

• Hydro-static lubrication
• Viscosity pumps
• Turbine
• Applied to Journal bearings (used in mechanical systems).

3. Steady, Incompressible & Laminar Flow in a Circular Tube

It is defined as, "Steady, incompressible and laminar flow through a straight circular tube of constant cross-section". It is also called Hagen-Poiseuille Flow.

Lubrication

Navier-stokes equation used to find out the velocity profile inside a bearing. There are 3 types of lubrication:

1. Full Film Lubrication   ↦   Bearing surfaces are fully separated by a film of lubricant.
2. Boundary Lubrication   ↦   Bearing surfaces physically come into contact.
3. Mixed Film Lubrication   ↦   Combination of partial lubricant films and some asperity (roughness/difficult to endure) contact between the surfaces.

Full film lubrication is further divided into the following:

1. Hydrostatic   ↦   Surfaces are separated by an externally pressurized lubricant.
2. Hydrodynamics   ↦   Surfaces are separated by a dynamic film of lubricant.
3. Elastohydrodynamic   ↦   Characterized by elastic deflections of the surfaces under very high pressures within the contact zone with an increase in viscosity of the fluid. 

Hydrodynamic Lubrication

It is governed by Reynold's equation of Classical Lubrication Theory which came into existence by assuming:

• Liquid flowing through thin region separated by 2 moving surfaces.
• Pressure does not vary across the film.
• Appropriate boundary conditions required for Reynold's equation solution.

Reynold's Equation

It is defined as, "Analysis of Momentum Transport and Continuity equations leads to a single elliptic DE". 

• Analysis served as a basis of designing hydrodynamically lubricated bearings.

Assumptions of Reynold's equation:

1. Fluid is Newtonian & Incompressible.
2. Flow in the gap is Laminar.
3. Fluid inertia is negligible.
4. Body force may be neglected.
5. Fluid viscosity is constant.
6. Pressure is constant across the film.
7. Speed of moving surface and load are constant.

Journal Bearings

It is defined as, "a kind of bearing that contains a journal or shaft that freely rotates in a support with a shell or metal sleeves".

There are two cases:

1. Infinitely Long Bearing
2. Short Bearing

1. Infinitely Long Bearing

2. Short Bearing

• Navier Stokes Questions.
• Hydrodynamic Bearings.

References:

• Material from Class Lectures + Book named Fundamentals of Fluid Mechanics by Munson, Young & Okiishi's (8th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube (Engineering Sights).