Oral) 
V.—On the Resistance to Flow of Water through Pipes or Passages having 
Divergent Boundaries. By Professor A. H. Gibson, D.Sc., University College, 
Dundee. Communicated by Professor W. Prppiz, D.Sc. 
(MS. received March 20,1911. Read July 3, 1911. Issued separately August 30, 1911.) 
CONTENTS. 
PAGE PAGE 
1. Introduction. > ae: : : oS 4, Rectangular Pipes with Uniformly Diverging 
2. Circular Pipes with Uniformly Diverging Boundaries : : é : : . 103 
Boundaries : : é 3 ; 98 5. Pipes of Square Section with Uniformly 
3. Effect of Projecting Smaller Pipe into Space Diverging Boundaries : : é . 105 
bounded by Diverging Walls. : . 102 6. Form of Pipe for Minimum Loss of Head . 106 
7. Summary and Conclusions ; : 2 . 113 
§ 1. [yrRopUCTION. 
Some time ago the author published* the results of a series of experiments on the 
flow of water through tubes having uniformly diverging boundaries, in which the loss 
of energy corresponding to given angles of conicity of the tubes was determined. These 
pipes, some circular, others square or rectangular in cross section, had the same initial 
and the same final areas, these being respectively the same as those of circles of 1°5 
inch and 3°0 inches diameter, the ratio of initial to final mean velocity of flow being 
in each case 4 to 1. 
The present experiments were planned with a view of extending the investigation 
to cover a series of-values of the ratio of enlargement, and also of ascertaining whether 
the loss in similar pipes having identical mean velocities of flow varies with or is in- 
dependent of their dimensions. 
In some cases where such divergent passages form an essential feature of a hydraulic 
machine it is the practice to project the boundaries of the smaller pipe for some distance 
into the space bounded by the divergent walls, and the work has been extended to 
investigate the effect of this method of construction on the efficiency of energy trans- 
formation. 
The question as to the shape of pipe giving the least loss of energy with given 
initial and final areas and with a given length—-a problem of much practical import- 
ance—was also touched upon in the former paper, and further investigation has been 
directed to this point. 
Altogether upwards of ninety pipes have been examined. The experiments have 
been carried out in the engineering laboratories of University College, Dundee, the 
apparatus used and the methods of making measurements and of carrying out the 
work being substantially illustrated and described in the former paper.t Of the pipes 
examined those of rectangular section were of wood carefully made to template and 
* Proc. Roy. Soc., A, vol. 1xxxiii., 1910, p. 366. + Ibid., p. 368. 
TRANS. ROY. SOC. EDIN., VOL. XLVIII. PART I, (NO. 5), 16 
