Tutorial examples on dimensional analysis and rational expressions

1 (a) A sphere of diameter  and surface roughness  moves at a uniform velocity  in a fluid of density  and viscosity . If the resistance to its motion is taken to be some function of  show by the method of dimensions that :-

                                        

 (b)(i) A smooth sphere of material of density  falls at its terminal velocity  through a liquid of density  and of viscosity . If the drag on the sphere is given by the rational expression :-

                                               where  is the projected area of the sphere,  is its velocity,  is the density of the fluid and  is the coefficient of drag show that  :-

                                              

    (ii) A smooth sphere of 30 mm diameter and made of steel falls through water at 10°C and reaches its terminal velocity. Given that, in the range of Reynolds number, defined as , from 1,000 to 100,000,  = 0.5 show that the terminal velocity will be about 2.3 m/s.

 

For steel take  = 7,800 kg/m³ and for the properties of water use the data in the steam tables. 

 

2(a)(i) A sphere of radius  moves in a straight line through a fluid of density  and viscosity . If the drag  exerted on the sphere by the fluid is taken to depend on the velocity  and the acceleration  of the sphere as well as on  show that :-

                                       

       (ii) Measurements are made of the velocity of small steel balls falling vertically in oil and these show that,  at the terminal velocity, the drag is proportional to the product of  and . Show from this observation and the outcome of dimensional analysis above that the drag is proportional to .

 

 (b) A steel ball has a diameter of 1.59 mm (1/16²) and, in oil having density 850 kg/m³ and viscosity 0.188 kg/ms, has a terminal velocity of 51 mm/s. If the drag on the ball is given by the rational expression :-

                                                     Drag =  find the value of .

For steel take  = 7,800 kg/m³.

 

3(i)  When a fluid flows at a steady rate through a long horizontal pipe the force exerted on the water to sustain the flow  must be resisted by a tangential force acting on the inner surface of the pipe. Some would argue that there is a shearing stress  exerted on the wall of the pipe. Show that :-           where  is the pressure drop over a length  of pipe of diameter .

  (ii)  The rational expression due to Darcy for friction loss in pipe is :-

Use this and the expression for  to show that :-      

  (iii) Using dimensional analysis show that for the pipe in (i) :-

           and that  where  are the density and viscosity of the fluid,  is the mean velocity and  is the surface roughness.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solutions to examples on dimensional analysis and rational expressions