  Thermal conduction has been suggested as a possible mechanism by which
sufficient energy is supplied to the central regions of galaxy clusters to
balance the effect of radiative cooling. Here we present the results of a
simulated, high-resolution, 3-d Virgo cluster for different values of thermal
conductivity (1, 1/10, 1/100, 0 times the full Spitzer value). Starting from an
initially isothermal cluster atmosphere we allow the cluster to evolve freely
over timescales of roughly $ 1.3-4.7 \times 10^{9} $ yrs. Our results show that
thermal conductivity at the Spitzer value can increase the central ICM
radiative cooling time by a factor of roughly 3.6. In addition, for larger
values of thermal conductvity the simulated temperature and density profiles
match the observations significantly better than for the lower values. However,
no physically meaningful value of thermal conductivity was able to postpone the
cooling catastrophe (characterised by a rapid increase in the central density)
for longer than a fraction of the Hubble time nor explain the absence of a
strong cooling flow in the Virgo cluster today. We also calculate the effective
adiabatic index of the cluster gas for both simulation and observational data
and compare the values with theoretical expectations. Using this method it
appears that the Virgo cluster is being heated in the cluster centre by a
mechanism other than thermal conductivity. Based on this and our simulations it
is also likely that the thermal conductvity is suppressed by a factor of at
least 10 and probably more. Thus, we suggest that thermal conductvity, if
present at all, has the effect of slowing down the evolution of the ICM, by
radiative cooling, but only by a factor of a few.
