HYDRODYNAMICS A point in, or a "particle" of, a fluid in motion generally de scribes a curve, in the course of time, which is called the path of the particle. The rate of change of position of the particle may be represented by a directed quantity, or vector, q called the velocity of the particle. The magnitude of this velocity will be denoted by the symbol q, and the components in the directions of a selected set of rectangular axes by u, v, w. If the position of a point in space is specified by its rectangular co-ordinates x, y, z relative to these axes, and the time by a variable quantity t, then at each place occupied by fluid there will be a vector at each in stant t, and its components u, v, w can be regarded as functions of x, y, z and t.
At a given time t the distribution of velocities in the fluid may be conveniently specified by lines of motion and surfaces of constant velocity. A line of motion is a curve such that the tangent at each point is in the direction of the velocity of the fluid particle which happens to be at the point at time t. The lines of motion are not generally the same as the paths of particles and do not indicate the magnitude of the velocity at each point. This is why it is advantageous to use also a set of surfaces over each of which the velocity has some constant magnitude.
When at every point occupied by fluid the velocity is constant in magnitude and direction (u, v, w independent of t), the motion is said to be steady. The lines of motion are in this case the same as the paths of the particles and are called stream-lines. The stream-lines drawn through an infinitesimal contour form what is called a stream-tube. Such a tube pos sesses the property that fluid in the tube remains in the tube, and important infer ences may be drawn by using the principles of the conservatism of mass and energy.
If a quantity of fluid is represented by its mass, the rate of discharge of fluid across the area A may be represented by the quantity Q= pAq, and the equation pAq = p'A'q' may be regarded as a form of the equation of continuity. A more general form of the equation, which is due to Leonhard Euler (1707-83), may be obtained by considering the amount of fluid which in a small interval of time dt crosses the faces of a small parallelopiped with sides parallel to the axes of co-ordinates. Denoting the lengths of the sides by dx, dy, dz respectively, the change in the mass of the fluid enclosed in time dt is approximately p dxdydz. Equat ing this to the change due to the flow across the faces, an equa tion is obtained, which may be finally written in the form represents the rate of change of p for a particle of fluid. This quantity is zero when the fluid is incompressible, and the equa tion of continuity takes the simple form aua v aw which means that the volume of an element of fluid remains un changed during motion. These equations were given by Euler in 1755 in the great memoir in which he laid the foundations of the science of hydrodynamics. The principle that the volume of an element of an incompressible fluid remains unchanged had, how ever, been laid down by Jean le Rond d'Alembert in 1752. He also indicated the type of modification needed for the case of a compressible fluid.
The Equations of Motion, Energy and Pressure.—The equations of motion were obtained by Euler by a method, sug gested by d'Alembert, which amounts to an application of what is now called d'Alembert's principle ; they are of type du p dt _ - pX ax where X, Y, Z are the components of the total external force (per unit mass) including gravitation.
Bernoulli's Equation.—Referring again to the stream-tube, the fluid within the slice AB will at some later time occupy the slice A'B' and will have the same total energy as before. If V denotes the volume of the slice AB, the total energy is made up of three parts when the fluid is incompressible. These are the kinetic energy the pressure-energy pV, and the potential energy of external forces (such as gravitation) which may be represented by pVSl. Assuming that there are no other external forces, and using primes to denote corresponding quantities in the second position of the small mass of fluid, the principle of the conserva tion of energy, when no heat is transferred, gives This equation which has many important applications was first given by Daniel Bernoulli 0700-82).
Pressure of Impact.—If a stream of fluid moving with velocity is deflected by a plate, or other obstacle, and the velocity is zero at a point I (where the fluid strikes the plate normally), the pressure p at this point is given by the equation (where p, q and 12 refer to a point far in front of the body). Furthermore, if SZ is constant, then pi = p+ z p and is greater than at any other point in the fluid except at a point of confluence or stagnation, where the velocity again is zero.
Barotropic Fluids.—When the fluid is compressible there is an equation, analogous to Bernoulli's equation, in the case when p is a function of p alone. This condition is not always satisfied in a real fluid, and a study of the general type of compressible fluid has been made by Vilhelm Bjerknes, who calls a fluid in which p is a func tion of p a barotropic fluid, and uses the name baroclinic f or a fluid in which the surfaces of constant density are not the same as the surfaces of constant pressure.
The irrotational flow round an obstacle is derived by finding a solution of o which satisfies the boundary condition that there is no normal component of relative velocity between the fluid and the obstacle. When the motion is steady the variation of pressure over the surface of the stationary obstacle may be obtained by means of Bernoulli's theorem, and it is found that the forces exerted by the fluid on the body either balance or reduce to a couple. This is quite contrary to experience, for it is well known that fluid flowing past a body exerts a force on the body. This contradiction between the irrotational theory and experiment is called the paradox of d'Alembert.
The theory of vortex motion was developed greatly by Hermann von Helmholtz (1821-94) and by William Thomson (Lord Kelvin 1824-1907). Helmholtz showed that the circula tion is the same for all circuits embracing a vortex tube and drawn on it, and is equal to wa, where a is the area enclosed by the circuit, and w = (2+,72.+2)t is the resultant vorticity of the fluid at the point. He also showed that, in the case of a barotropic fluid subject to forces with a single valued potential, vortex lines move with the fluid and so remain vortex lines. Kelvin amplified this theorem, and proved it anew by showing that the circulation in any circuit moving with the fluid remains constant. These results led to the conclusion that in a baro tropic fluid circulation can neither be created nor destroyed so long as all external forces have a single-valued potential.
Principia (1687) commenced the study of tangential forces in a moving fluid by con sidering the case of flow in one direction (that of Ox), when the velocity u varies in a perpendicular direction Oy (fig. 5). A layer I., moving faster than the lower layer II., exerts on the latter a viscous drag depending on the difference of velocities. Newton's hypothesis means, mathematically, that the tangential force, acting across a small area A separating I. from II., is µA where it is a physical quantity called the coefficient of viscosity of the fluid. µ has the dimensions of a momentum, divided by an area. Some times the kinetic viscosity v= µ/p is used in place of µ ; this has the dimensions of a velocity multiplied by a length.
In the kinetic theory the viscous drag is attributed to fast moving molecules crossing from I. to II., and slow-moving mole cules crossing from II. to I. In the type of fluid motion which is called turbulent, there is a similar transfer of momentum from one layer to another, in which aggregates of fluid particles endowed with vorticity are the wander ing elements instead of single molecules.
In the case of viscous flow un der pressure along a straight tube, the condition for the steady motion of a cylindrical element (fig. 6) is that the difference of pressure at the two ends should be equal to the surface integral of the viscous drag over the curved surface. This leads to the equation when A is small. The velocity u is subject also to the boundary condition that there should be no slipping at the surface of the tube, i.e., u= o at the boundary. In the case of a circular tube of radius a, the conditions are satisfied by where r is the distance of a point from the axis of the tube. The amount of fluid passing through the tube in unit time is thus This law, which is due to Jean Louis Marie Poiseuille, has been confirmed by experiments in the case of flow through capillary tubes and very slow motion through wide tubes. The applicability of the law seems to depend upon the value of the ratio which is called the Reynolds number, of ter Osborne Reynolds (1842-1912) who showed that the flow ceases to be viscous when this number exceeds a certain critical value. For a very viscous liquid like oil, Poiseuille's law is usually applicable because R is small.
In the case of a tube of arbitrary section, the equation for u may be compared with an equation which occurs in Saint Ve nant's theory of the torsion of a straight bar, or with the equation which may be deduced from equation (I), and gives the mate form of a liquid film covering a hole in a diaphragm, and subjected to a greater pres sure on one side than on the other (fig. 7) . A soap film method of solving problems of viscous flow and of the tor sion of prisms was proposed by Ludwig Prandtl, and has been developed by English investigators. The inclination of each element of the film, the contour lines and the volume under the film are determined by simple measurements. Two Bel gian investigators have recently devised an improved arrange ment depending upon the use of the boundary surface of two liquids of equal density which do not mix.
In the case of flow through a long tube, the boundary layer (fig. 9) will eventually fill the tube, and the flow at the entrance is materially different from the turbulent flow in the main part of the tube, for in this latter part the boundary layer theory is inapplicable. The turbulent flow leads to a much more uniform distribution of velocity over the cross section of the tube than the viscous flow, and to an entirely different law of re sistance. (See HYDRAULICS.) The flow in the boundary layer at the surface of a solid im mersed in the fluid is regular when it is in the direction of de creasing pressure, but, when the flow is from low pressure to high pressure, the momentum of the fluid in the boundary layer may not be sufficient to overcome the resistances, and the stream lines break away from the boundary. Isolated vortex filaments, or eddies, form behind the body more or less close to the place where the stream-lines break away, and to the place where there is a confluence of stream-lines which separated at the front of the body.