96 



HYDRAULICS AND ITS APPLICATIONS 



a vessel. Here a vortex is usually formed naturally, some initial disturb- 

 ance of the water determining the direction of rotation, although in the 

 northern hemisphere the earth's rotation 

 would itself tend to cause a rotation in 

 an anti-clockwise direction as viewed 

 from above. This is termed a Free 

 Spiral Vortex. The water moves spirally 

 towards the centre with stream line 

 motion, so that, neglecting viscosity, its 

 energy per unit mass is everywhere the 

 same. If, while the mass is rotating, 

 the orifice be plugged, the motion 

 becomes one of simple rotation in 

 horizontal planes, and forms a Free 

 Cylindrical Vortex. 



Forced Vortex Motion. Since the angular velocity w is constant, we 

 have at any radius r, v = w r. (1) 



The increase in pressure radially is given by 

 djp_W w* r 2 _ W 2 

 dr 



FIG. 49. Forced Vortex. 



9*9 



Integrating between the limits i\ and r 2 we have 



W 



(2) 



If p = p Q where r = o, 



W 



= , 



or, putting - = h (Fig. 49), 





h = 7i + 



which is the equation to a parabola. 



Since the pressure at any point is that equivalent to the column of 

 water supported at the point, it follows that all surfaces of equal pressure, 

 including the free surface of the vortex, form paraboloids of revolution 

 having the axis of rotation as their common axis. Near the sides of the 

 vessel the liquid lags owing to viscosity, and here the surface level will 

 fall below that of the paraboloid. 



Free Cylindrical Vortex Motion. Here, since we have stream line 



motion, the equation-^; + ^ -f z = constant, holds. 



