  The chiral phase structure, mesonic properties and transport properties in a
momentum anisotropic quark matter induced by the preferential expansion of
medium are phenomenologically investigated based on the two flavor
Nambu-Jona-Lasinio model. The calculations of various transport coefficients
have been performed using the kinetic theory in the relaxation time
approximation, where the momentum anisotropy is embeded in the estimation of
both distribution function and the relaxation time. Our results indicate that
an increase in anisotropy parameter $\xi$ can considerably lead to a catalysis
of chiral symmetry breaking. The critical endpoint (CEP) is shifted to smaller
temperatures and larger quark chemical potentials as $\xi $ increases, and the
influence of momentum anisotropy on temperature of CEP is almost the same as
that on the quark chemical potential of CEP. The meson masses and the
associated decay widths also significantly are affect by the variation of
$\xi$. The temperature behavior of scaled shear viscosity $\eta/T^3$ and scaled
electrical conductivity $\sigma_{el}/T$ exhibit a similar dip struture,
however, their qualitative behaviors with $\xi$ are different. Nevertheless,
the minima of both $\eta/T^3$ and $\sigma_{el}/T$ shift toward higher
temperatures with an increase of $\xi$. Furthermore, in this work the Seebeck
coefficient $S$ exhibits a decreasing function of temperature and its sign is
positive, indicating the dominant carriers of converting the temperature
gradient to the electric field are $up$ quarks. With a rise in $\xi$ the value
of $S$ has a significant enhancement for the temperature below the critical
temperature.
