  The correlation between the mean transverse momentum and the harmonic flow
coefficients is an observable which is of great interest; it is sensitive to
shape fluctuations in the initial state of a relativistic nuclear collision.
The measurement of that correlation coefficient in central collisions allows
one to infer about the intrinsic deformation of the colliding nuclei. We
propose to study the momentum dependent covariance and correlation coefficient
between the mean transverse momentum and the harmonic flow in a given
transverse momentum bin. Two possible constructions of such observables are
provided and predictions are obtained from a viscous hydrodynamic model. We
find that such momentum dependent correlation coefficients between the mean
transverse momentum and the harmonic flow show a strong and nontrivial momentum
dependence. We also explore the effects of granularity (nucleon width) in the
initial state, the nuclear deformation, and the shear viscosity on this
momentum dependent correlation coefficient. The shape of the momentum
dependence of the correlation coefficient for the triangular flow is found to
be sensitive to the size of small scale fluctuations in the initial state. On
the other hand, the shape of the momentum dependence of the covariance between
the mean transverse momentum and the harmonic flow coefficients is found to be
sensitive to the value of the shear viscosity and to the granularity of the
initial state.
