  The momentum distribution of particles accelerated at strong non-relativistic
shocks may be influenced by the spatial distribution of the flow speed around
the shock. This phenomenon becomes evident in the cosmic-ray modified shock,
where the particle spectrum itself determines the flow velocity profile
upstream. However, what if the flow speed is not uniform downstream as well?
Hydrodynamics indicates that its spatial variation over the length scales
involved in the acceleration of particles in supernova remnants (SNRs) could be
noticeable.} {In the present paper, we address this issue, initially following
Bell's approach to particle acceleration and then by solving the kinetic
equation. We obtained an analytical solution for the momentum distribution of
particles accelerated at the cosmic-ray modified shock with spatially variable
flow speed downstream.} {We parameterized the downstream speed profile to
illustrate its effect on two model cases, the test particle and non-linear
acceleration at the shock.The resulting particle spectrum is generally softer
in Sedov SNRs because the flow speed distribution reduces the overall shock
compression accessible to particles with higher momenta. On the other hand, the
flow structure in young SNRs could lead to harder spectra. The diffusive
properties of particles play a crucial role as they determine the distance from
the shock, and, as a consequence, the flow speed that particles encounter
downstream. We discuss the effect of the plasma velocity gradient to be
(partially) responsible for the evolution of the radio index and for the
high-energy break visible in gamma rays from some SNRs. We expect that the
effect from the gradient of the flow velocity downstream could be prominent in
regions of SNRs with higher diffusion coefficient and lower magnetic field,
i.e. where acceleration of particles is not very efficient.
