Magnitudes of Different Forces - 1 | Fluid Mechanics for Mechanical Engineering PDF Download

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.

 

Dynamic Similarity of Flows governed by Viscous, Pressure and Inertia Forces
The criterion of dynamic similarity for the flows controlled by viscous, pressure and inertia forces are derived from the ratios of the representative magnitudes of these forces with the help of Eq. (18.1a) to (18.1c) as follows:
   (18.2a)
    (18.2b)

The term   is known as Reynolds number, Re after the name of the scientist who first developed it and is thus proportional to the magnitude ratio of inertia force to viscous force .(Reynolds number plays a vital role in the analysis of fluid flow)

The term Δp / _pV² is known as Euler number, Eu after the name of the scientist who first derived it. The dimensionless terms Re and Eu represent the critieria of dynamic similarity for the flows which are affected only by viscous, pressure and inertia forces. Such instances, for example, are

1. the full flow of fluid in a completely closed conduit,
2. flow of air past a low-speed aircraft and
3. the flow of water past a submarine deeply submerged to produce no waves on the surface.

Hence, for a complete dynamic similarity to exist between the prototype and the model for this class of flows, the Reynolds number, Re and Euler number, Eu have to be same for the two (prototype and model). Thus
   (18.2c)
   (18.2d)

where, the suffix p and suffix m refer to the parameters for prototype and model respectively.

In practice, the pressure drop is the dependent variable, and hence it is compared for the two systems with the help of Eq. (18.2d), while the equality of Reynolds number (Eq. (18.2c)) along with the equalities of other parameters in relation to kinematic and geometric similarities are maintained.

• The characteristic geometrical dimension l and the reference velocity V in the expression of the Reynolds number may be any geometrical dimension and any velocity which are significant in determining the pattern of flow.

• For internal flows through a closed duct, the hydraulic diameter of the duct D_h and the average flow velocity at a section are invariably used for l and V respectively.

• The hydraulic diameter D_h is defined as D_h= 4A/P where A and P are the cross-sectional area and wetted perimeter respectively

Dynamic Similarity of Flows with Gravity, Pressure and Inertia Forces

A flow of the type in which significant forces are gravity force, pressure force and inertia force, is found when a free surface is present.

  Examples can be

1.   the flow of a liquid in an open channel.

2.  the wave motion caused by the passage of a ship through water.

3.   the flows over weirs and spillways.

The condition for dynamic similarity of such flows requires

•  the equality of the Euler number Eu (the magnitude ratio of pressure to inertia force),

                                                                                   and

• the equality of the magnitude ratio of gravity to inertia force at corresponding points in the systems being compared.
         Thus ,
      (18.2e)

• In practice, it is often convenient to use the square root of this ratio  so to deal with the first power of the velocity.

• From a physical point of view, equality of  (1g)^1/2 /v implies equality of 1g /v² as regard to the concept of dynamic similarity.

The reciprocal of the term  (1g)^1/2 /v is known as Froude number ( after William Froude who first suggested the use of this number in the study of naval architecture. )

Hence Froude number, Fr = V /(1g)^1/2.

Therefore, the primary requirement for dynamic similarity between the prototype and the model involving flow of fluid with gravity as the significant force, is the equality of Froude number, Fr, i.e..,
   (18.2f)


Dynamic Similarity of Flows with Surface Tension as the Dominant Force

Surface tension forces are important in certain classes of practical problems such as ,

1. flows in which capillary waves appear
2. flows of small jets and thin sheets of liquid injected by a nozzle in air
3. flow of a thin sheet of liquid over a solid surface.

Here the significant parameter for dynamic similarity is the magnitude ratio of the surface tension force to the inertia force.

 This can be written as  


The term    is usually known as Weber number, Wb (after the German naval architect Moritz Weber who first suggested the use of this term as a relevant
parameter. )

Thus for dynamically similar flows (Wb)_m =(Wb)_p



Dynamic Similarity of Flows with Elastic Force

When the compressibility of fluid in the course of its flow becomes significant, the elastic force along with the pressure and inertia forces has to be considered.

Therefore, the magnitude ratio of inertia to elastic force becomes a relevant parameter for dynamic similarity under this situation.

Thus we can write,
    (18.2h)

The parameter   is known as Cauchy number ,( after the French mathematician A.L. Cauchy)

If we consider the flow to be isentropic , then it can be written
    (18.2i)

(where E_s is the isentropic bulk modulus of elasticity)

 Thus for dynamically similar flows (cauchy)_m=(cauchy)_p

 

• The velocity with which a sound wave propagates through a fluid medium equals to   .

• Hence, the term    can be written as V² / a² where a is the acoustic velocity in the fluid medium.

The ratio V/a is known as Mach number, Ma ( after an Austrian physicist Earnst Mach)

It has been shown in Chapter 1  that the effects of compressibility become important when the Mach number exceeds 0.33.

The situation arises in the flow of air past high-speed aircraft, missiles, propellers and rotory compressors. In these cases equality of Mach number is a condition for dynamic similarity.
Therefore,

  (Ma)_p=(Ma)_m

i.e.

   (18.2j)

Ratios of Forces for Different Situations of Flow