696 



HYDRODYNAMICS. 



Fig. 16. 



VLATE 

 cccxvin. 



Fig. 5. 

 No. -1. 



Fig. S. 

 No 5. 



large in the apartments ; but that it will be sufficient 

 if they be enlarged at their upper terminations, accord- 

 ing to the form CD, Fig. 16. This divergency of the 

 upper part will carry off the smoke very well, even 

 when it is not practicable to afford chimnies of suffi- 

 cient length to the upper apartments. The same ob- 

 servation is applicable to chemical furnaces for strong 

 fire." 



PROF. VI. 



The eddies of the water in currents and rivers are 

 produced by motion, communicated from the more ra- 

 pid parts of the stream to the lateral parts, which are 

 more at rest. 



"The water which moves in the channel MNH, Fig. 5. 

 No. 4. meets the obstacle BA, which impedes its course, 

 and causes it to rise and discharge itself in the direction 

 AC, with an increased velocity. Suppose the water in 

 BDCA to be dormant, the current AC communicates its 

 motion to the lateral particles E, (Prop. 1.) and conveys 

 them forward ; the surface of the dormant water be- 

 comes depressed at E, and the most remote particles 

 towards D are urged, according to the laws of the equi- 

 librium of fluids, to fill the depression. The current 

 AC continues to carry them off, and the space BDCA 

 continues to be exhausted. The water of the current 

 AC, by virtue of the same laws, is acted upon by a 

 constant force which urges it towards the cavity E, 

 while its natural course or projection carries it towards 

 AC. Under the agency of these two forces, the water 

 AC acquires a curve-lined motion in CD, and descends 

 as it were through an inclined plane, becoming retro- 

 grade in DE, whence it would proceed to strike the ob- 

 stacle BA, and the current AC, after which it would 

 undergo several oscillations previous to acquiring a state 

 of equilibrium and repose. ' ut the current AC con- 

 tinues its lateral action ; a second time it draws away 

 the water through CD into E, and forces it to renew 

 its motion through the curve CDE ; in which manner 

 the eddy continues without ceasing. 



If the river should pass through a contraction of its 

 bed at N, it will produce eddies at both sides, at P and 

 at Q, similar to those we have considered at DC. 



Suppose the stream of water, after having struck the 

 bank GH, to be reflected into a new direction HS, the 

 lateral communication of motion will excite eddies in 

 the angle of reflection R. 



When two currents of unequal velocity meet ob- 

 liquely in the middle of the river, the most rapid cur- 

 rent will produce eddies in that which is the least ra- 

 pid. 



Suppose a stream of water to flow over a bed of un- 

 equal depth. If the longitudinal section of the inequa- 

 lities of the bottom exhibit a gentle slope, as at ABC, 

 Fig. 5. No. 5. the superior water will impress its mo- 

 tion by lateral communication upon the inferior water 

 which is near the bottom, beneath the line AC, and a 

 current will take place through the whole depth of the 

 section MB. The current, which is formed near the 

 bottom at B, is turned out of its course by the slope 

 BC, and proceeds to rise above the surface at Q, some- 

 times in the form of a curling wave, or vertical whirl- 

 pool. If the extremities of the hollow place form an 

 abrupt angle, as DEFG, eddies will be produced even 

 at the bottom, in the vertical direction at D, and some- 

 times also at G. These phenomena may be observed 

 in an artificial channel with glass sides. 

 Every eddy destroys a part of the moving force of the 



Lateral 

 



. 



Motion in 



Fluid*, 



current of the river. For the water which descends by 



a retrograde motion in the inclined plane CDE, Fig. 5. 



No. 4. cannot be restored in the direction of the current 



of the river but by a new impulse. It is as it were a 



ball, which is forced to rise on an inclined plane, '^^y^^' 



whence it continually falls back again to receive new PLATE 



impulsions. cccxvni. 



Hence we deduce, as a primary consequence, that in ^'8- * 

 a river, of which the course in permanent, and the sections 

 of its bed unequal, the mater continues more elevated than 

 it mould have done, if the whole river had teen equally 

 contracted to the dimensions of its smallest section. The 

 cause of this phenomenon is the same as that which re- 

 tards the expenditure through the tube with enlarged 

 parts. (Prop. 7- No. 4.) The water which descends 

 from the elevation above the contracted part N into the 

 bason PQ, Fig. 5. No. 4. loses nearly the whole of the 

 velocity it acquired by descending from it ; because the 

 narrow part has a curved slope towards the lower part of 

 the river, which directs the velocity of the stream in an 

 horizontal direction. Guglielmini has well remarked, 

 that a fall does not influence the velocity of the lower 

 stream, because the eddies of the water in the bason 

 PQ destroy the velocity produced by the fall. This 

 velocity increases the depth, and enlarges the width of 

 the channel at PQ. Eddies are formed on each side, at 

 the bottom, and at the surface, both in the horizontal 

 and vertical directions. It would be to no purpose to 

 attempt to prevent this hollowing out and enlargement 

 of the channel by such a fall by adopting the means of 

 close walls, for the bason would then obtain its enlarge- 

 ment where these constructions might end. 



If the channel have a number of successive contrac- 

 tions and dilatations MN, without cascade or dam, there 

 will still be formed, at each dilatation, eddies which 

 will diminish the velocity more than if the channel had 

 an uniform section equal to that in M or N. It will 

 therefore follow, that the surface of the water, after each 

 dilatation, must rise, in order to recover the velocity it 

 lost by the eddies. If we call the height to which the 

 water must rise, above the elevation necessary to have 

 overcome the retardations of a bed of uniform section, 

 = a, and the number of equal and successive alter- 

 nate dilatations and contractions be = m, the height 

 of the rise in the stream thus alternately dilated beyond 

 that of the same river uniformly contracted, will be 

 = am. We here suppose the bottom of the river to be 

 uniform. If this bottom be of such a nature to be at- 

 tacked by the current, the contracted parts will be hol- 

 lowed out, and the matter will be deposited in the en- 

 larged parts. 



The second consequence which we draw from the 

 principle here established, respecting the loss of force 

 caused by the eddies, is of considerable importance in 

 the theory of rivers, and appears to have been neglect- 

 ed by those who have treated on this subject. The 

 friction of the water along the wet banks, and over the 

 bottom of rivers, is very far from being the only cause 

 of the retardation of their course, which consequently 

 requires a continued descent to maintain its velocity. 

 One of the principal and most frequent causes of re- 

 tardation in a river, is also produced by the eddies, 

 which are incessantly formed in the dilatations of the 

 bed, the cavities of the bottom, the inequalities of the 

 banks, the flexures or windings of its course, the cur- 

 rents which cross each other, and the streams which 

 strike each other with different velocities. A consider- 

 able part of the force of the current is thus employed 

 to restore an equilibrium of motion, which that current 

 itself does continually derange." 



