Page:EB1911 - Volume 14.djvu/87

dQθ or GdQθ pounds falls through a vertical height z + y₁ − y₀, and the work done by gravity is

GdQθ(z + y₁ − y₀).

Putting p_a for atmospheric pressure, the whole pressure per unit of area at a₀ is Gy₀ + p_a, and that at a₁ is −(Gy₁ + p_a). The work of these pressures is

G(y₀ + p_a/G − y₁ − p_a/G) dQθ = G (y₀ − y₁) dQθ.

Adding this to the work of gravity, the whole work is GzdQθ; or, for the whole cross section,

GzQθ.

(2)

Work expended in Overcoming the Friction of the Stream Bed.—Let A′B′, A″B″ be two cross sections at distances s and s + ds from A₀B₀. Between these sections the velocity may be treated as uniform, because by hypothesis the changes of velocity from section to section are gradual. Hence, to this short length of stream the equation for uniform motion is applicable. But in that case the work in overcoming the friction of the stream bed between A′B′ and A″B″ is

GQθζ(u² / 2g) (χ / Ω) ds,

where u, χ, Ω are the mean velocity, wetted perimeter, and section at A′B′. Hence the whole work lost in friction from A₀B₀ to A₁B₁ will be

GQθ $\int _{0}^{l}$ζ(u2 / 2g) (χ/Ω)ds.

(3)

Equating the work given in (2) and (3) to the change of kinetic energy given in (1),

α(GQθ / 2g) (u₁² − u₀²) = GQzθ − GQθ $\int _{0}^{l}$ ζ(u² / 2g) (χ / Ω) ds;

∴ z = α(u₁² − u₀²) / 2g + $\int _{0}^{l}$ ζ(u² / 2g) (χ / Ω) ds.

Fig. 120.

§ 116. Fundamental Differential Equation of Steady Varied Motion.—Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, a₁b₁, taken for simplicity normal to the stream bed (fig. 120). For that short length of stream the fall of surface level, or difference of level of a and a₁, may be written dz. Also, if we write u for u₀, and u + du for u₁, the term (u₀² − u₁²)/2g becomes udu/g. Hence the equation applicable to an indefinitely short length of the stream is

dz = udu/g + (χ/Ω) ζ(u²/2g) ds.

(1)

From this equation some general conclusions may be arrived at as to the form of the longitudinal section of the stream, but, as the investigation is somewhat complicated, it is convenient to simplify it by restricting the conditions of the problem.

Modification of the Formula for the Restricted Case of a Stream flowing in a Prismatic Stream Bed of Constant Slope.—Let i be the constant slope of the bed. Draw ad parallel to the bed, and ac horizontal. Then dz is sensibly equal to a′c. The depths of the stream, h and h + dh, are sensibly equal to ab and a′b′, and therefore dh = a′d. Also cd is the fall of the bed in the distance ds, and is equal to ids. Hence

dz = a′c = cd − a′d = i ds − dh.

(2)

Since the motion is steady—

Q = Ωu = constant.

Differentiating,

Ω du + u dΩ = 0;

∴ du = −u dΩ/Ω.

Let x be the width of the stream, then dΩ = xdh very nearly. Inserting this value,

du = −(ux / Ω) dh.

(3)

Putting the values of du and dz found in (2) and (3) in equation (1),

ids − dh = −(u²x / gΩ) dh + (χ / Ω) ζ(u² / 2g) ds.

dh/ds = {i − (χ/Ω) ζ(u2/2g)} / {1 − (u2/g) (x/Ω)}.

(4)

Further Restriction to the Case of a Stream of Rectangular Section and of Indefinite Width.—The equation might be discussed in the form just given, but it becomes a little simpler if restricted in the way just stated. For, if the stream is rectangular, χh = Ω, and if χ is large compared with h, Ω/χ = xh/x = h nearly. Then equation (4) becomes

dh/ds = i (1 − ζu² / 2gih) / (1 − u²/gh).

(5)

§ 117. General Indications as to the Form of Water Surface furnished by Equation (5).—Let A₀A₁ (fig. 121) be the water surface, B₀B₁ the bed in a longitudinal section of the stream, and ab any section at a distance s from B₀, the depth ab being h. Suppose B₀B₁, B₀A₀ taken as rectangular coordinate axes, then dh/ds is the trigonometric tangent of the angle which the surface of the stream at a makes with the axis B₀B₁. This tangent dh/ds will be positive, if the stream is increasing in depth in the direction B₀B₁; negative, if the stream is diminishing in depth from B₀ towards B₁. If dh/ds = 0, the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4)

dh/ds = 0, if i − (χ/Ω) ζ(u²/2g) = 0;

∴ ζ(u²/2g) = (Ω/χ) i = mi,

which is the well-known general equation for uniform motion, based on the same assumptions as the equation for varied steady motion now being considered. The case of uniform motion is therefore a limiting case between two different kinds of varied motion.

Fig. 121.

Consider the possible changes of value of the fraction

(1 − ζu² / 2gih) / (1 − u² / gh).

As h tends towards the limit 0, and consequently u is large, the numerator tends to the limit −∞. On the other hand if h = ∞, in which case u is small, the numerator becomes equal to 1. For a value H of h given by the equation

1 − ζu² / 2giH = 0,
H = ζu²/2gi,

we fall upon the case of uniform motion. The results just stated may be tabulated thus:—

For h = 0, H, > H, ∞,

the numerator has the value −∞, 0, > 0, 1.

Next consider the denominator. If h becomes very small, in which case u must be very large, the denominator tends to the limit −∞. As h becomes very large and u consequently very small, the denominator tends to the limit 1. For h = u²/g, or u = √ (gh), the denominator becomes zero. Hence, tabulating these results as before:—

For  h = 0, u²/g, > u²/g, ∞,

the denominator becomes −∞, 0, > 0, 1.

Fig. 122.

§ 118. Case 1.—Suppose h>u²/g, and also h>H, or the depth greater than that corresponding to uniform motion. In this case dh/ds is positive, and the stream increases in depth in the direction of flow. In fig. 122 let B₀B₁ be the bed, C₀C₁ a line parallel to the bed and at a height above it equal to H. By hypothesis, the surface A₀A₁ of the stream is above C₀C₁, and it has just been shown that the depth of the stream increases from B₀ towards B₁. But going up stream h approaches more and more nearly the value H, and therefore dh/ds approaches the limit 0, or the surface of the stream is asymptotic to C₀C₁. Going down stream h increases and u diminishes, the numerator and denominator of the fraction (1 − ζu²/2gih) / (1 − u²/gh) both tend towards the limit 1, and dh/ds to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line D₀D₁.

The form of water surface here discussed is produced when the flow of a stream originally uniform is altered by the construction of a weir. The raising of the water surface above the level C₀C₁ is termed the backwater due to the weir.

Fig. 123.

§ 119. Case 2.—Suppose h > u²/g, and also h < H. Then dh/ds is