INTRODUCTION - Hydrology: An Introduction - page 17

The viscosity of a liquid at standard pressure depends on temperature. For most practical
purposes, in the range 0 ≤ T ≤ 40 ◦ C the kinematic viscosity of water (in units of m 2 s − 1 )
as a function of temperature (in ◦ C) can be calculated with sufficient accuracy by means
of ν = 10 − 6 (1 . 785 − 0 . 057 89 T + 0 . 001 128 T 2 − 0 . 9671 × 10 − 5 T 3 ); this yields roughly

1.00 × 10 − 6 m 2 s − 1 at 20 ◦ C . This expression is based on measurements at the National
Bureau of Standards, made available by J. F. Swindells. Similarly, for most applications
in hydrology the density of water (in units of kg m − 3 ) at 1 atmosphere can be calculated
in the same temperature range by means of ρ = (999 . 8505 + 0 . 060 01 T − 0 . 007 917 T 2

+

4 . 1256 × 10 − 5 T 3 ).

Finite control volume
Like Equations (1.8) and (1.9), also here Equations (1.11), (1.12) and (1.13) describe the flow
phenomenon at a point. Again, they can be extended to a larger control volume by integrating
out the spatial dependence of the terms. This can be accomplished by multiplication of each
term in Equation (1.13) by the differential volume d s · d A (in which d s and d A represent
the differential flow path and cross-sectional area vectors, the latter pointing in the direction
of flow) and by subsequent integration along all flow paths inside the control volume and
across all areas of entry and exit of the control volume. For example, in the case of a conduit
fixed in space occupied by a fluid volume S of constant density ρ , this yields for, say, the
x -direction, approximately,

d ( S V x )
dt

ρ + ρ ( Q e V x e − Q i V x i ) = F x (1.14)

where F is the sum of all forces acting on the fluid in the control volume, Q i and Q e are
the inflow and outflow rates of the control volume, V is the average fluid velocity inside
the control volume, V i and V e the average fluid velocity over the entry and exit section,
respectively, of the control volume, and the subscript x denotes the component direction of
the momentum and of the forces.

1.5.4 The kinematic approach

In principle, the description of fluid flow phenomena should involve conservation of mass,
conservation of momentum, and conservation of energy. However, whenever, the relevant
phenomena are isothermal, most of the energy is mechanical, and the energy conservation
equation becomes redundant, so that it is often not included in the formulation. In this topic,
the energy conservation equation will be used only in relation to atmospheric phenomena,
where it will be discussed further. In hydrologic applications, whenever both mass and
momentum conservation principles are made use of, the mathematical description of the
flow phenomena is called a dynamic formulation. However, in some situations, momentum
changes, both temporal and spatial, are so small that they can be neglected. In such cases,
the terms on the left-hand side of Equation (1.12) can be omitted and this greatly simplifies
the formulation. In practice, the right-hand side of Equation (1.12) can then often be param-
eterized by an explicit functional relationship between the flow velocities in the system and
some other variables such as pressure, water depth or water level height. Whenever only the
continuity equation is required, and the momentum equation can be replaced by this type of
relationship, the mathematical description is referred to as a kinematic formulation. The
same idea can also be applied to larger control volumes. In this case, the combination of the