Page:Encyclopædia Britannica, Ninth Edition, v. 12.djvu/461

445 HYDRODYNAMICS.] HYDROMECHANICS 445 PAIIT II. HYDRODYNAMICS. In considering the motion of fluids we shall suppose them non-viscous, so that whatever be the state of motion, the stress across any section is normal to the section, and therefore the stress is a pressure and the same in all direc tions about a point, as in Hydrostatics. Two methods are employed in hydrodynamics, called the Eulerian and Lagrangian, although both are due to Euler ; in the Eulerian we fix our attention on particular points of space, and observe the changes of pressure, density, and velocity which take place there, and in the Lagrangian we follow a particle of fluid and observe its changes. The first may be called the statistical and the second the historical method, according to Prof. J. C. Maxwell. The Eulerian method is generally employed except where the fluid has a moving boundary. Tlie Eulerian Form of the Equations of Motion. The first equation to be established is the equation of continuity, which expresses the fact that the increase of matter inside a fixed surface is due to the flow of fluid across the surface into the in terior, supposing there are no hypothetical sources or sinks in the interior of the surface. Lemma, The quantity of fluid, estimated in units of mass, which flows across a plane area in a given time is equal to the product of the area, the den sity, the time, and the resolved part of the velocity per pendicular to the area. For if q be the Fl - 9 - velocity, the quantity of fluid which flows across the area A in the time t will form an oblique cylinder of length qt, with its generat ing lines in the direction of motion (fig. 9). If 9 denote the angle between the normal to the area and the velocity, the mass of the cylinder = ptq cos 9, which is therefore the flux across the area A in the time t. Generally, if S denote any fixed surface, M the mass of the fluid inside it, and the angle which the normal drawn outwards at any point of the surface makes with the velocity q at that point, then - = rate of increase of quantity of fluid inside the surface per unit of time = flux across the surface per unit of time = -j~ pq cos dS ; dt / the integral equation of continuity. In the Eulerian equations of motion u, v, w are taken to denote the components of the velocity q parallel to the coordinate axes at the point xyz at the time t ; u, v, w are therefore functions of x, y, z, t the independent variables, and d is used to denote partial differentiation with respect to these four independent variables. To transfer the integral equation into the differential equation of continuity, we require Green s transformation, namely, d-n d - H dy d; or, individually, M where the integrations extend respectively through the volume and over the surface of a closed space S ; l,m,n denote the direction- osines of the outward drawn normal at the surface element dS, an(1 I. 1?, C are continuous functions of a;, y, z. The integral equation of continuity may now be written ffA -^dxilxdy +ff(lp tt + m p v + which by Green s transformation becomes . dpu dpv d -- = 0, leadin to the differential equation of continuity dpw dp dpu eft dx _ = (2). It is customary to establish the differential equation of continuity immediately by considering the fluid which enters and leaves an infinitesimal parallelepiped, whose edges are dx, dy, dz, in the time dt, but this requires us to suppose in succession each of the elements dx, dy, dz, though infinitesimal, to be infinite compared with the other two, and with the infinitesimal element of time dt ; this viola tion of the principles of the differential calculus is avoided by establishing the equation in its integral form first. We shall establish the equations of motion in a similar&quot; way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the forces acting throughout the space S and by the pressures acting at the surface S. Taking the fixed direction parallel to the axis of x, the rate of increase of momentum in that direction per unit of time, due to the fluid which crosses the surface, is r r f T o Jj (lpu~ + mpuv + npuiv)d$ , which by Green s transformation dpu 2 dpuv dpuw, 7 j -^ h 1- dxdudz ; dx dy dz J and, adding this to the rate of increase of momentum per unit of time of the fluid inside the surface Mr- we obtain, as the total rate of increase of momentum per unit of time of the fluid which fills the space S, rrrfdMdjtf djw dpuw JJJ at dx dy dz J in the direction of the axis of x. The rate of generation of momentum in this direction by tho forces of components X,Y,Z per unit of mass in the interior is ff/pXdxdydz, and by the pressures at the surface is -flpdS , which by Green s transformation is equal to and therefore S/y/dpu dpu 2 dpuv dpuw, , 7 /// -JT + f- + j +- - }dxdyds JjJ dt dx dy dz J leading to the differential equation of motion dpu dpu 2 dpuv dpuw _ ^ dt dx dy dz with two similar equations in y and z. These equations may be slightly simplified dpu dpu 2 dpuv dpuio dt dx dy dz {du du = p ( T7 +u~r + &quot;j~ dt dx dy (3); for du du dy dz J dt dx dy dz which reduces to the first line, the second line vanishing in conse quence of the equation of continuity ; and therefore the equations of motion may be written 1 du du du du + u~- -f v- + w~ = X - dt dx dy with the two similar equations do dv du dt dx dy dv -J- dz Y- dp p dx 1 dp dw dw dw dw -j- + 76-y- + V-T- -f W-j- dt dx dy dz (5), (6). As a rule these equations are established immediately by deter mining the component accelerations of the fluid particle which is at xyz at the instant of time t considered, and saying that these