  Influence of fluid dispersion on the Saffman-Taylor instability in miscible
fluids has been investigated both in the linear and nonlinear regimes. The
convective characteristic scales are used for the dimensionless formulation
that incorporates P\'{e}clet number (Pe) into the governing equations as a
measure for the fluid dispersion. A linear stability analysis (LSA) has been
performed in a similarity transformation domain using the quasi-steady-state
approximation. LSA results show that systems with large Pe become more unstable
and the onset of instability occurs earlier compared to the case when Pe is
smaller. Variations of the most unstable wave number and the cut-off wave
number with Pe have been analyzed. Fourier spectral method has been used for
the numerical simulations of the fully nonlinear system. The results indicate
that the wave numbers of the unstable modes increase with Pe. Influence of the
anisotropic dispersion on the onset in both the linear and nonlinear regimes
has been analyzed. Large transverse diffusivity increases the length scale of
the fingers quickly and merges the fingers to generate coarser fingers. Finally
the combined effect of the Korteweg stress and Pe in the linear regime has been
perused. In the presence of the Korteweg stresses and depending upon various
flow parameters, a fluid system with larger Pe exhibits smaller instantaneous
growth rate than with smaller Pe, which is a counter-intuitive result.
