Magnitudes of Different Forces

Magnitudes of Different Forces

A fluid motion, under all such forces is characterised by

1. Hydrodynamic parameters like pressure, velocity and acceleration due to gravity,
2. Rheological and other physical properties of the fluid involved, and
3. Geometrical dimensions of the system.

It is important to express the magnitudes of different forces in terms of these parameters, to know the extent of their influences on the different forces acting on a flluid element in the course of its flow.

Inertia Force 

• The inertia force acting on a fluid element is equal in magnitude to the mass of the element multiplied by its acceleration.
• The mass of a fluid element is proportional to ρl³ where, ρ is the density of fluid and l is the characteristic geometrical dimension of the system.
• The acceleration of a fluid element in any direction is the rate at which its velocity in that direction changes with time and is therefore proportional in magnitude to some characteristic velocity V divided by some specified interval of time t. The time interval t is proportional to the characteristic length l divided by the characteristic velocity V, so that the acceleration becomes proportional to V²/l.

The magnitude of inertia force is thus proportional to

This can be written as,

(18.1a)

Viscous Force  
The viscous force arises from shear stress in a flow of fluid.

 Therefore, we can write

Magnitude of viscous force  = shear stress  X  surface area over which the shear stress acts

Again, shear stress = µ (viscosity) X rate of shear strain
where, rate of shear strain  velocity gradient     and surface area   

Hence       

	(18.1b)

                                             

Pressure Force 

The pressure force arises due to the difference of pressure in a flow field.

 Hence it can be written as

(18.1c)

(where, Dp is some characteristic pressure difference in the flow.)

Gravity Force 

The gravity force on a fluid element is its weight. Hence,

(18.1d)

(where g is the acceleration due to gravity or weight per unit mass)

Capillary or Surface Tension Force 

The capillary force arises due to the existence of an interface between two fluids.

• The surface tension force acts tangential to a surface .
• It is equal to the coefficient of surface tension σ multiplied by the length of a linear element on the surface perpendicular to which the force acts.

Therefore,

(18.1e)

Compressibility or Elastic Force 

Elastic force arises due to the compressibility of the fluid in course of its flow.

• For a given compression (a decrease in volume), the increase in pressure is proportional to the bulk modulus of elasticity E
• This gives rise to a force known as the elastic force.

Hence, for a given compression       

(18.1f)

The flow of a fluid in practice does not involve all the forces simultaneously.

Therefore, the pertinent dimensionless parameters for dynamic similarity are derived from the ratios of significant forces causing the flow.