Tutorial examples on dimensional analysis

The following examples have the main objective of giving practice in the manipulative phase of dimensional analysis.  None of the examples involves a decision on the relevant physical quantities in any of the given physical system and the combinations chosen may not be complete. The style of the questions shows to some extent the way in which dimensional analysis is applied.

1.  This question is about the orifice tank that is described in the laboratory sheets at the end of this whole text. It is evident that the principal physical quantities involved are, the orifice diameter, , the head above the orifice , the density of the fluid , its viscosity , and the acceleration due to gravity g.  If we take account of the diameter  of the tank we can say that the flow from the tank

                                                    

Show that :-                    .

Compare this result with the rational expression for the steady flow through an orifice.

2.  When a ship moves in water there is a resistance to motion.  This resistance may be attributed to two main effects, the viscous force between the underwater part of the ship and the water, and the disturbance of the free surface that causes waves which carry energy away from the vicinity of the ship.  We consider this system by the method of dimensions by supposing that the resistance to motion  depends on :-

(i)     the shape and size of the ship which we represent by one dimension, its length ,

(ii)      the speed of the ship ;

(iii)     the viscosity of the water :

(iv)   the density :

(v)    the acceleration due to gravity , because the energy of the wave is

stored in the gravitational field.

Show that :-                              .

3.  Suppose that a sphere of radius  was allowed to fall through a fluid at rest which has density  and viscosity .  There would be a phase during which the sphere would be accelerating to its terminal velocity. We could say that, at some instant during this phase, when the velocity is  and the acceleration is , the drag is given by :-

                                            

4   An aeroplane wing has a special shape to its cross-section called an aerofoil section. When this shape is made to move through the air at a steady speed , a force is produced on the wing. This force is clearly dependent on the size of the wing, that is, on the length of the aerofoil  and on the span of the wing . It also depends on the angle  that the aerofoil makes with the direction of motion. Then the force might be considered to be a function of  where  are the density and viscosity of air.

Show that:-                        

 

5  A dashpot is a device that is used to provide a resistance to motion. It is often used as a damper, that is a device to damp out unwanted oscillations as in a vehicle suspension system. The essential features of a dashpot are, a cylinder that is filled with oil and a loose-fitting piston that moves coaxially with the cylinder in the oil. When an axial force is applied to the piston, oil is made to flow through the annular gap between the piston and the cylinder and a pressure difference develops between the two sides of the piston and a net force on the piston.

 

It may be argued that the force  on the piston is a function of the diameter  of the piston, the length  of the piston, the annular gap , the speed of the piston  and, of course, the viscosity  and the density  of the oil. Show that :-

                                       .

 

6   Figure 6 represents a journal bearing. Such bearings are normally used in pairs to support a loaded shaft that is required to rotate. The shaft is plain, that is, it is cylindrical and it runs in two bearing brasses that together form a cylindrical sleeve round the shaft. The brasses are supported and retained by the bearing block. These bearings are fed continuously with oil that fills the space between the shaft and the brasses. This space is nominally specified as a radial clearance  but in fact the shaft does not run coaxially with the bearing. It runs with some eccentricity  that leads to a wedge action in the oil thay separates the shaft and the brasses to produce only viscous friction and no solid friction.

 

It is clear that there must be some relationship between the load  that can be supported by the bearing and the shaft diameter , the length of the bearing , the viscosity , the eccentricity , the speed of rotation  and the radial clearance .

Show that :-                  

 

 

 

 

 

 

Solutions to examples on dimensional analysis