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# Hydrodynamics

## pressure, forces, fluid, surface, dp, ap, plate, bd and bodily

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HYDRODYNAMICS (vdwp. water; &moms, force), or HYDROMECHANICS, is that part of Dynamics which treats of the motion or rest of fluids under the action of forces. A perfect fluid is defined as a body whose parts are perfectly free to move under the action of the smallest forces, or otherwise, as a body such that the reactions between any two portions of it are normal to the surface separating them. If there is any tangential drag tending to prevent the one portion of the substance from slipping past the other, the fluid is said to be viscous. A perfect fluid is an abstraction, like the material particle or the rigid body, but many of the ordinary fluids, like water, alcohol, air and other gases, are so slightly viscous that for many purposes they may be consid ered as perfect. The normal reaction, which alone we suppose to be present, is called the pressure, and is measured by the limit of the ratio of the force exerted on an element of surface to the area of the element, when both diminish without limit. The usual gravitational unit of pressure is the pound-weight per square inch, the usual scientific, absolute unit (see MECHANICS), is the dyne per square centimeter. The pressure of the atmosphere may be con sidered as equal to 1,000,000 dynes per square centimeter.

The fundamental theorem of hydrodynamics is that the pressure on an element of surface is independent of the direction of the normal to the surface. This may be proved by con sidering the equilibrium of a small tetrahedron, ABCD, Fig. 1, and resolving the forces on its faces in the direction BD. The pressure being normal, the forces on ABD, BCD have no component along BD, while if the pressures on ABC, ACD, and the areas of those sides are respectively pi, p,, S,, S., their normals es,, n,, we have for equilibrium piS, cos (ni,BD) NS, cos (n,, BD). But S, cos (ni, BD), S. cos (n,, BD) are the projections of the areas on a plane perpendicular to BD, which are equal, therefore pa pi. If there are other forces applied to the fluid besides the pressure, such, for instance, as its weight, these will be proportional to the volume of the tetrahedron, and when its size is diminished indefinitely, the volume vanishes to the third order of small quantities, and may thus he neglected in comparison with the area, which is of the second order, so that the result is not affected.

We will first consider hydrostatics, or that part of our subject which deals with fluids at rest. Suppose that the fluid is subject to the action of forces whose Components along the co-ordinate axes are equal to X, Y, Z per unit of mass. These we call bodily forces. Now consider the equilib rium of an infinitesimal rectangular parallelo piped, Fig. 2, whose edges, parallel to the co ordinate axes, have lengths dx, dy, dz. Sup pose that the mean value of the pressure on the side ABCD, which is at a distance x from the origin, is p, then the X-component of the force on this side is pdydz. On the side EFGH,

which lies at a distance dx farther from the origin, the mean pressure will be p + and the component on the face EFGH, acting on the parallelopiped, being in the opposite a direction will be  (p + dydz. Now if ax the density of the fluid is p, the amount of matter in the parallelopiped, being the product of the density and volume, will be pdxdydz, and the force exerted upon it in the X-direction will be Xpdxdydz. We must therefore have for equilibrium pdydz ap  dx dydz pXdxdydz 0, ax and passing to the limit by decreasing the dimensions, p will be the pressure at any point, and dividing by dxdydz, we have In a similar manner we have  ap p =- ' (1) ap = of Thus the fluid can be in equilibrium only under the influence of bodily forces such that the components of the bodily forces, multi plied by the density, are the derivatives of the same functions of the co-ordinates. Now there is in general a physical relation between the pressure of a fluid and the density at any point. If we put dP 1 p ' dp p' we have aP dPap aP dP op 1 aP dp p ax! dp ay p aP dP ap dp as P and our equations (I) become a P  8 P , 8P (2) ' az Now this is the condition that the bodily forces are conservative. (See MECHANICS). In that case the potential energy for unit mass is called the potential of the forces, and will be denoted by V. Thus we shall have dp + const., and dV If twofluids of different densities are in contact, we have at their common surface  dii=pitiV=poriV, so that (,p,pi)d V=0, and since th  p, is not zero we must have di 1= 0, Consequently the surface of separation is a surface of constant potential and constant pressure. In the case of gravity we have, if the Z-axis is measured vertically upward V= gs, so that the surfaces of con stant V are horizontal planes, and a surface where water is in contact with the atmosphere must be a horizontal plane, or level surface, the pressure being the constant atmospheric pres sure. If we suppose the fluid to be incom pressible, we have p constant, P= pip, (3) P  --V const. =  gs coast., (4) p so that, if we neglect the atmospheric pressure, and count the depth from the plane z=0, we have the fundamental theorem for heavy liquids, namely, that the pressure is propor tional to the depth. This may be proved ex perimentally by placing a well-fitting plate under a tube, Fig. 3, communicating with a vessel of any shape, and holding the plate up by a wire hung from a balance, while water is poured in above. The weight required to hold the plate up is found to be independent of the shape of the vessel, and to depend only on the depth and the area of the plate. The vase and plate may also be immersed in liquid, while, instead of liquid within, weights are placed, on .the plate.; the weight supported will then be proportional to the depth.

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