We study Poiseuille flow of a one-component fluid between adiabatic plates in liquid-gas coexistence with a stationary interface. Using only bulk transport laws and particle and energy conservation, we identify small dimensionless parameters $A^L$ and $A^G$ governed by squared microscopic-to-macroscopic length ratios. In weak driving with macroscopic bulk regions and a regular interfacial temperature response, the pressure drop is concentrated across the interface, strongly suppressing the particle current. For equal-temperature reservoirs, the residual current produces interfacial cooling. Experimental feasibility is also discussed.