Previous proposed modifications in the respective constitutive forms of the Newtonian deviatoric stress tensor and the no-slip boundary condition imposed upon viscous fluids at solid surfaces, wherein the fluid's mass velocity is replaced by its volume velocity, furnishes a complete continuum-hydrodynamic description of thermal transpiration phenomena occurring in a closed capillary tube filled with a single-component gas or liquid, the former at negligibly small Knudsen numbers. The resulting expression for the steady-state thermomolecular pressure difference $\Delta p$ existing between the two ends of the capillary, the latter maintained at different temperatures, is free of empirical parameters, such as Maxwell's thermal-slip coefficient, upon which current non-continuum theories of the phenomenon are based. The predicted $\Delta p$ (with the pressure highest at the hotter end) is shown to agree well with experimental data for gases in the near-continuum limit of vanishingly small Knudsen number. Also discussed is the experimentally observed lack of dependence of $\Delta p$ upon the physicochemical properties of the capillary walls, an observation which accords with the predictions of our theory. Our proposed volume velocity-based rationalization of the phenomenon of thermal transpiration offers a strictly continuum no-slip alternative to Maxwell's widely-accepted thermal creep explanation thereof, involving slip of the fluid's mass velocity at a non-isothermal surface. The agreement of our theoretical predictions of the thermomolecular pressure difference with experimental data, which is essentially indistinguishable in accuracy from that provided by Maxwell's thermal creep theory, provides further support for the viability of the generic volume velocity-based framework underlying our theory, the latter having recently been used to also rationalize related thermophoretic and diffusiophoretic phenomena in gases, as well as thermal diffusion in liquids.