468 H Y D R M E pressure. Then, if H is the total energy at Q per unit of weight of fluid, HYDROMECHANICS [HYDRAULICS. Differentiating, we get dR = dz + + . for the increment of energy between Q and P. dz = PQ cos (p = ds cos <f> ; 7TJ dp vdv j . . dti = -| 1- ds cos <f> G g But (1) (la), (2). where the last term disappears if the motion is in a horizontal plane. Now imagine a small cylinder of section eo described round PQ as an axis. This will be in equilibrium under the action of its centrifugal force, its weight and the pressure on its ends. But its volume is uds and its weight Goads. Hence, taking the com ponents of the forces parallel to PQ G v 2 wdp -.-- - <ads - Gco cos <p ds . 9 P where p is the radius of curvature of the stream line at Q. Conse quently, introducing these values in (1), gp g g ( p ds Now it is already known that if, through any particle A, lines be drawn through B and C two particles near to A, such that AB and AC are at right angles at the instant considered, then the mean an gular velocity of these lines is the same in whatever direction they are drawn, and is equal to the angular velocity with which a small cylindrical element described round A would rotate if supposed sud denly solidified. This mean angular velocity may be conveniently called the molecular rotation, and will be denoted by (y . In the present case is the angular velocity of the tangent at Q, and p ds is the angular velocity, reckoned in the same direction, of a line per pendicular to the tangent through P and Q. The sum of these is, therefore, twice the molecular rotation, and . (3). 9 Now vds is constant, being the flow in an elementary stream of breadth unity, and thickness ds. Therefore the difference of energy between two consecutive elementary streams is proportional to the molecular rotation at any point of either. CURRENTS. 30. Rectilinear Current. Suppose the motion is in parallel straight stream lines (fig. 38) in a vertical plane. Then p is infinite, and from eq. (2), 29, Comparing this with (1) we see that . . z + -^- = constant (4); or the pressure varies hydrostatically as in a fluid at rest. For two stream lines in a horizontal ft plane, z is constant, and therefore p is constant. Radiating Current. Sup pose water flowing radially between horizontal parallel planes, at a distance apart = 8. Conceive two cylin drical sections of the curr velocities are i and v 2 , and the pressures p t and p%. Since the flow across each cylindrical section of the current is the same, Q = 2ir?- ]l 8r 1 = 2irr. 2 8v y a I i i n t dz i ^^ a u Fig. 38. ent at radii r and / 2 , where th< a~a (5). 2 V l The velocity would be infinite at radius 0, if the current could be conceived to extend to the axis. Now, if the motion is steady, G -2 20 Pa -Pi ~ G Hence the pressure increases from the interior outwards, in a way indicated by the pressure columns in fig. 39, the curve through the free surfaces of the pressure columns being, in a radial section, the Fig. 39. quasi-hyperbola of the form xy^ c 3 . This curve is asymptotic to a horizontal line, H feet above the line from which the pressures are measured, and to the axis of the current. Free Circular Vortex. A free circular vortex is a revolving mass of water, in which the stream lines are concentric circles, and in which the total head for each stream line is the same. Hence, if by any slow radial motion portions of the water strayed from one stream line to another, they would take freely the velocities proper to their new positions under the action of the existing fluid pressures only. For such a current, the motion being horizontal, we have for all the circular elementary streams ! = JL. H = constant ; dR- g Consider two stream lines at radii r and r + dr (fig. 39). (2), 29, p = randfo = ?r, r 2 , vdv A rfr-| =0 , gr g dv dr . (7). Then in (8), precisely as in a radiating current ; and hence the distribution of pressure is the same, and formulae 6, 6 are applicable to this case. Free Spiral Vortex. As in a radiating and circular current the equations of motion are the same, they will also apply to a vortex in which the motion is compounded of these motions in any pro portions, provided the radial component of the motion varies in versely as the radius as in a radial current, and the tangential component varies inversely as the radius as in a free vortex. Then the whole velocity at any point will be inversely propor tional to the radius of the point, and the fluid will describe stream lines having a constant inclination to the radius drawn to the axis of the current. That is, the stream lines will be logarith
mic spirals. When water is delivered from the circumference
