Using Muskhelishvili's method, we present closed-form analytical
solutions for an isolated elliptical inclusion in general shear
far-field flows. The inclusion is either perfectly bonded to the matrix
or, as in the case of a circular inclusion, to a possible intermediate
layer. The solutions are valid for incompressible all-elastic or
all-viscous systems. The actual values of the shear modulus or viscosity
in the inclusion, mantle and matrix can be different and no limits are
imposed on the possible material property contrasts. The solutions
presented are complete 2-D solutions and the parameters that can be
analysed include all kinematic (e. g. stream functions, velocities,
strain rates, strains) and dynamic parameters (e. g. pressure, maximum
shear stress, etc.). We refrain from giving the tedious derivation of
the presented solutions, and instead we focus on how to use the
solutions, how to extract the parameters of interest and how to apply
and verify them. In order to demonstrate the usefulness of
Muskhelishvili's method for slow viscous flow problems, we apply our
results to a clast in a shear zone and obtain important new insights,
even for simple, circular inclusions. Another important application is
the benchmarking of numerical codes for which the presented solutions
are most suitable due to the infinite range of viscosity contrasts and
the strong local gradients of properties and results. To stimulate a
broader use of Muskhelishvili's method, all solutions are implemented in
MATLAB and downloadable from the web.