1913-14.] Viscous Flow through a Circular Tube. 
61 
is measured, and the coefficient //. is then determined from Poiseuille’s 
equation, 
h = ^ (3) 
Where h is measured by the difference of pressures at piezometer 
orifices at two points in the wall of the tube, this equation is rigorously 
true. Where, however, as is more often the case, the upper end of the 
tube projects into a reservoir of the fluid, while its lower end discharges 
freely, the head being measured from the free surface in the reservoir 
to the centre of the discharging end of the tube, the true equation 
becomes 
h — 32/^ _j_ k.E. of discharge + head loss at entrance to tube. . (4) 
The two last terms become decreasingly important as the length of the 
tube increases, but, with a fairly short tube, account for a very appreciable 
portion of the whole head. In many cases in which the details of viscosity 
experiments have been published, the kinetic energy of discharge has been 
calculated erroneously from the mean velocity as in formula (2), while 
various allowances, varying from zero to have been made for the head 
9 
loss at entrance to the tube. In a thin- walled tube projecting into a 
reservoir, the loss at entrance, with an inviscid fluid, may readily be shown 
to be equal to while the effect of viscosity is to reduce this loss some- 
what. As the walls of the tube become thicker the conditions approximate 
more nearly to that of a tube whose end opens flush with the sides of the 
reservoir, in which case, in large tubes conveying water, the loss is 
approximately ‘5^-. The true value of this loss for any actual projecting 
Zg 
tube may therefore be expected to be given by c^- , where c increases with 
^9 
the relative thickness of the walls, and, for such a fluid as water, has a 
value somewhere between - 5 and kO. 
The following experiments have been carried out with a view of 
checking the accuracy of the deductions leading to formula (1), and 
of determining the value of c for tubes of small bore. In each case 
the tube used projected for some distance into the upper reservoir 
and discharged freely at its lower end. The head from the free 
surface in the reservoir to the centre of the outlet was measured, and 
the discharge was collected and measured. The fluid was water, 
