# hydrodynamics

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## hydrodynamics:

see mechanics**mechanics,**

branch of physics concerned with motion and the forces that tend to cause it; it includes study of the mechanical properties of matter, such as density, elasticity, and viscosity.

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## Hydrodynamics

The study of fluids in motion. The study is based upon the physical conservation laws of mass, momentum, and energy. The mathematical statements of these laws may be written in either integral or differential form. The integral form is useful for large-scale analyses and provides answers that are sometimes very good and sometimes not, but that are always useful, particularly for engineering applications. The differential form of the equations is used for small-scale analyses. In principle, the differential forms may be used for any problem, but exact solutions can be found only for a small number of specialized flows. Solutions for most problems must be obtained by using numerical techniques, and these are limited by the computer's inability to model small-scale processes. *See* Conservation of energy, Conservation of mass, Conservation of momentum, Fluid flow, Fluid mechanics

Applications of hydrodynamics include the study of closed-conduit and open-channel flow, and the calculation of forces on submerged bodies.

Flow in closed conduits, or pipes, has been extensively studied both experimentally and theoretically. If the pipe Reynolds number,

given by the equation below, where*V*is the average velocity and

*D*is the pipe diameter, is less than about 2000, the flow in the pipe is laminar. In this case, the solution to the continuity, momentum, and energy equations is readily obtained, particularly in the case of steady flows. If Re

_{D}is greater than about 4000, the flow in the pipe is turbulent, and the solution to the continuity, momentum, and energy equations can be obtained only by employing empirical correlations and other approximate modeling tools. The Re

_{D}region between 2000 and 4000 is the transition region in which the flow is intermittently laminar and turbulent.

*See*Laminar flow, Reynolds number, Turbulent flow

Confined flows that have a liquid surface exposed to the atmosphere (a free surface) are called open-channel flows. Flows in rivers, canals, partially full pipes, and irrigation ditches are examples. The difficulty with these flows is that the shape of the free surface is one of the unknowns to be calculated.

In most open-channel flows the bottom slope and the water depth change with position, and the free surface is not parallel to the channel bottom. If the slopes are small and the changes are not too sudden, the flow is called a gradually varied flow. An energy balance between two sections of the channel yields a differential equation for the rate of change of the water depth with respect to the distance along the channel. The solution of this equation, which must be accomplished by using one of many available numerical techniques, gives the shape of the water surface.

Flow over spillways and weirs and flow through a hydraulic jump are examples of rapidly varying flows. In these cases, changes of water depth with distance along the channel are large. Here, because of large accelerations, the pressure distribution with depth may not be hydrostatic as it is in the cases of gradually varied and uniform flows. Solutions for rapidly varying flows are accomplished by using approximation techniques. *See* Hydraulic jump, Open channel

The force exerted by a fluid flowing past a submerged body is calculated by integrating the pressure distribution over the surface of the body. The pressure distribution is determined from the simultaneous solution of the continuity and momentum equations along with the appropriate boundary conditions. In almost all cases, this solution must be accomplished by using an appropriate approximation. *See* Boundary-layer flow

Usually the force exerted on the body is resolved into two components, the lift and the drag. The drag force is the component parallel to the velocity of the undisturbed stream (flow far away from the body), and the lift force is the component perpendicular to the undisturbed stream.

## Hydrodynamics

a branch of hydromechanics studying the motion of incompressible fluids and their interaction with solids. The methods of hydrodynamics can also be used to study the motion of gases, if the velocity of this motion is significantly lower than the velocity of sound in the gas being studied. If the gas moves with a velocity approaching or exceeding the velocity of sound, the compressibility of the gas becomes significant. In this case the methods of hydrodynamics are no longer applicable; this type of gas motion is studied in gas dynamics.

The principal laws and methods of mechanics are used in solving various problems of hydrodynamics. If necessary allowances are made for the general properties of fluids, solutions are obtained making it possible to determine the velocity, pressure, and the shearing stress at any given point of the space occupied by the fluid. This also makes it possible to calculate the forces of interaction between a fluid and a solid. From the point of view of hydrodynamics the main properties of a fluid are its high mobility, or fluidity, as evidenced by its low resistance to shear strain and its continuity (in hydrodynamics a fluid is considered to be a continuous, homogeneous medium). In hydrodynamics it is also assumed that a fluid has no tensile strength.

The primary equations of hydrodynamics are obtained by applying the general laws of physics to an element of mass, isolated in the fluid, with the subsequent transition to a limit as the volume occupied by this mass approaches zero. One of the equations, the so-called equation of continuity, is obtained by applying the law of the conservation of mass to the element. Another equation (or three equations, if projected on the coordinates axis) is obtained by applying the law of momentum to an element of the fluid. According to this law, a change in the momentum of an element of fluid must coincide in magnitude and direction with the momentum of the force applied to this element. In hydrodynamics, the solution of general equations can be exceedingly complex. Complete solutions are not always possible; they can be obtained only for a limited number of special cases. Therefore, many problems must be simplified; this is done by neglecting in the equations those members that are nonessential in determining the flow characteristics for a given set of conditions. For instance, in many cases it is possible to describe the actually observed flow with sufficient accuracy if the viscosity of the fluid is neglected. In this manner the theory for an ideal liquid is obtained; this theory can be used in solving numerous problems of hydrodynamics. In cases where the moving fluid is highly viscous (for example, thick oils), acceleration can be neglected because the change in flow velocity is insignificant. This approach yields another approximate solution for several problems of hydrodynamics.

The so-called Bernoulli equation is of particular importance in the hydrodynamics of an ideal fluid. According to this equation, throughout the length of a small stream of fluid there exists the following relationship between the pressure *p*, the flow velocity *v* (for a fluid of a density ρ), and the height *z* above the reference plane: *p + ½ρv ^{2} + ρgz* =constant. Here

*g*is the acceleration due to gravity. This is the principal equation in hydraulics.

An analysis of the equations for the motion of a viscous fluid shows that, for geometrically and mechanically similar flows, the quantity *ρvl/μ = Re* must be constant. Here, l is the linear dimension appropriate for the problem (for instance, the radius of a streamlined body, the cross-sectional radius of a pipe), ρ is the density, *v* is the velocity, and *μ* is viscosity coefficient. The quantity *Re* itself is the Reynolds number; it determines the nature of motion associated with a viscous fluid. A laminar flow occurs at low values of *Re*. For instance, in pipelines laminar flow occurs if Re = *v*_{cp}*d/v* ≤ 2,300 where *d* is the diameter of the pipe and *v* (nu) = μ/ρ. If *Re* is large, the striation in the fluid disappears and the individual masses are displaced in a random fashion; this is so-called turbulent flow.

The principal equations of the hydrodynamics of viscous fluids turn out to be solvable only for extreme cases—that is, either for very small *Re*, which (for usual dimensions) corresponds to high viscosity, or for very large *Re*, which corresponds to flow conditions for low-viscosity fluids. Problems concerning the flow of low-viscosity fluids (such as water or air) are especially important in many technological applications. For this special case, the hydrodynamic equations can be simplified significantly by isolating a layer of fluid that is immediately adjacent to the surface of the body in contact with which the flow occurs (the so-called boundary layer) and for which the viscosity cannot be neglected. Outside the boundary layer the fluid can be treated as an ideal fluid. In order to characterize fluid motions where gravity is of primary importance (such as waves on the surface of water caused by wind or by a passing ship), another dimension-less quantity is introduced: the Froude number *v*^{2}/ *gl* = *Fr*.

The practical applications of hydrodynamics are extremely diverse. Hydrodynamics is used in designing ships, aircraft, pipelines, pumps, hydraulic turbines, and spillway dams and in studying sea currents, river drifts, and the filtration of groundwater and of underground oil deposits. For the history of hydrodynamics, see HYDROAEROMECHANICS.

### REFERENCES

Prandtl, L.*Gidroaeromekhanika*. Moscow, 1949. (Translated from German.)