even though the wave-pattern is fixed in space and constant in time.
So, how do we mathematically express ''differentiation following the motion''? To follow particles in a continuum, a special type of differentiation is required. Arbitrarily small variations of C(x, y, z, t), a function of position and time, are given to the first order by
SC = — St +--Sx +--Sy +--Sz, dt dx dy dz where the partial derivatives d/dt etc. are understood to imply that the other variables are kept fixed during the differentiation. The fluid velocity is the rate of change of position of the fluid element, following that element along. The variation of a property Cfollowing an element of fluid is thus derived by setting Sx = uSt, Sy = vSt, Sz = wSt, where u is the speed in the x-direction, v is the speed in the y-direction, and w is the speed in the z-direction, thus
fixed particle dC dC dC 3C\ --H u--H v--H w— St, dt dx dy dz where (u, v, w) is the velocity of the material element, which by definition is the fluid velocity. Dividing by St and in the limit of small variations we see that fixed particle dC dC dC dC --H u--H v--H w —
dt dx dy dz
in which we use the symbol D to identify the rate of change following the motion
Dt dt dx dy dz dt
Here u = (u, v, w) is the velocity vector, and V=( dx, jdy, d) is the gradient operator. D/Dt is called the Lagrangian derivative (after Lagrange; 1736-1813) (it is also called variously the substantial, the total, or the material derivative). Its physical meaning is time rate of change of some characteristic of a particular element of fluid (which in general is changing its position). By contrast, as introduced above, the Eulerian derivative d/dt expresses the rate of change of some characteristic at a fixed point in space (but with constantly changing fluid element because the fluid is moving).
Leonhard Euler (1707—1783). Euler made vast contributions to mathematics in the areas of analytic geometry, trigonometry, calculus and number theory. He also studied continuum mechanics, lunar theory, elasticity, acoustics, the wave theory of light, and hydraulics, and laid the foundation of analytical mechanics. In the 1750s Euler published a number of major works setting up the main formulas of fluid mechanics, the continuity equation, and the Euler equations for the motion of an inviscid, incompressible fluid.
Some writers use the symbol d/dt for the Lagrangian derivative, but this is better reserved for the ordinary derivative of a function of one variable, the sense it is usually used in mathematics. Thus for example the rate of change of the radius of a rain drop would be written dr/dt, with the identity of the drop understood to be fixed. In the same context D/Dt could refer to the motion of individual particles of water circulating within the drop itself. Another example is the vertical velocity, defined as w = Dz/Dt; if one sits in an air parcel and follows it around, w is the rate at which one's height changes.2
The term u.V in Eq. 6-1 represents advec-tion and is the mathematical representation of the ability of a fluid to carry its properties with it as it moves. For example, the effects of advection are evident to us every day. In the northern hemisphere, southerly winds (from the south) tend to be warm and moist because the air carries with it properties typical of tropical latitudes northerly winds tend to be cold and dry because they advect properties typical of polar latitudes.
We will now use the Lagrangian derivative to help us apply the laws of mechanics and thermodynamics to a fluid.
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