The thermomechanical effect in superfluid helium is used to create an initial chemical potential difference ∆u_0 across a solid 4_He sample. This ∆u_0 causes a flow of helium atoms from one reservoir filled with superfluid helium, through a sample cell filled with solid helium, to another superfluid-filled reservoir until potential equilibrium between the reservoirs is restored. The solid helium sample is separated from each of the reservoirs by Vycor rods that allow only the superfluid component to flow. With an improved technique, measurements of the flow F at several fixed solid helium temperatures T have been made as a function of ∆u_0 in the pressure range 25.5-26.1 bar, and measurements of F have been made as a function of temperature in the range 180 < T < 545 mK for several fixed values of ∆ u. The temperature dependence of the flow above 100 mK shows a reduction of the flux with an increase in temperature that is well described by F = F_0^*[1- a exp(-E/T)]. The nonlinear functional dependence F_( ~) (∆〖u)〗^b, with b < 0.5 independent of temperature but dependent on pressure, documents in some detail the dissipative nature of the flow and suggests that this system demonstrates Luttinger liquid-like one-dimensional behavior. The mechanism that causes this flow behavior is not certain, but is consistent with superflow on the cores of edge dislocations.