By M. L. Ge, C. N. Yang
Yang C.N., Ge M.L. (eds.) Braid staff, knot conception, and statistical mechanics (WS, 1989)(ISBN 9971508281)(331s)
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Extra resources for Braid group, knot theory, and statistical mechanics
If ~ is expressed as energy per unit mass of water, with SI units Jkg- 1 , then the permeability takes units of (m 2 ). The form of k(6) in systems that change their volume with 6 is well-known. Total Potential of the Water In a one dimensional (vertical) swelling material, the total potential of the water is given by ~ - ~(6) + 0 + gz (12) In (12), g is the acceleration due to gravity, so gz is the gravitational potential of water at z, defined positive upwards relative to a convenient datum; 0 is the overburden potential, the work associated with vertical displacement of the system following addition of unit mass of water at position z; and ~(6) is the water content dependent potential that arises as a result of interaction of the water with the solid surfaces and their geometry.
E. ~ the differential equation for ~ minimizing - G defined by (18». The first term on the right of (18) is associated with an anisotropic microscopic (Maxwell) stress tensor due to the electric field; and the second is connected with the osmotic pressure due to the ionic excess in the double layer. 2 Microscopic stress tensor. The local microscopic stress tensor g has as its principal components ~~ [~~]2 (T, D 8~ (T3 + kT [~~]2 [N(~) - N(O)] ; + kT [N(~) (19) - N(O)] . is the component of g in the direction of the electric field with in directions normal to it.
There is, nevertheless, a substantial literature, in particular in soil physics and hydrology, devoted to analytical, quasi-analytical and numerical methods of solving this equation subject to relevant conditions. In practice, however, this full solution may be unnecessary for many situations because the gravitational component of potential acts in an opposite sense and tends to cancel 0 (Smiles 1974). The importance of gravi ty also diminishes as the imposed pressure Pi is increased (cf . (14».