  We consider the influence of quenched noise upon interface dynamics in 2D and
3D capillary rise with rough walls by using phase-field approach, where the
local conservation of mass in the bulk is explicitly included. In the 2D case
the disorder is assumed to be in the effective mobility coefficient, while in
the 3D case we explicitly consider the influence of locally fluctuating
geometry along a solid wall using a generalized curvilinear coordinate
transformation. To obtain the equations of motion for meniscus and contact
lines, we develop a systematic projection formalism which allows inclusion of
disorder. Using this formalism, we derive linearized equations of motion for
the meniscus and contact line variables, which become local in the Fourier
space representation. These dispersion relations contain effective noise that
is linearly proportional to the velocity. The deterministic parts of our
dispersion relations agree with results obtained from other similar studies in
the proper limits. However, the forms of the noise terms derived here are
quantitatively different from the other studies.
