Calculations based on the standard equations for the flow He II in tub
es predict large pressure and moderate temperature maxima when the tub
e is long and narrow. This effect has not, to our knowledge, been prev
iously observed. Experimental data are presented that confirm the exis
tence of the pressure maxima in capillaries of diameter 0.3 mum. The m
easured data are in good agreement with predictions. The data validate
the model equations for flow in capillaries and porous media using th
e bulk value of the Gorter-Mellink parameter. This effect may impact i
nterpretations of experimental searches for size effects.