  Based on the results published recently [SciPost Phys. 7, 026 (2019)], the
influence of surfaces and boundary fields are calculated for the ferromagnetic
anisotropic square lattice Ising model on finite lattices as well as in the
finite-size scaling limit. Starting with the open cylinder, we independently
apply boundary fields on both sides which can be either homogeneous or
staggered, representing different combinations of boundary conditions. We
confirm several predictions from scaling theory, conformal field theory and
renormalisation group theory: we explicitly show that anisotropic couplings
enter the scaling functions through a generalised aspect ratio, and demonstrate
that open and staggered boundary conditions are asymptotically equal in the
scaling regime. Furthermore, we examine the emergence of the surface tension
due to one antiperiodic boundary in the system in the presence of symmetry
breaking boundary fields, again for finite systems as well as in the scaling
limit. Finally, we extend our results to the antiferromagnetic Ising model.
