jats:pWe consider the steady flow of a granular current over a uniformly sloped surface that is smooth upstream (allowing slip for jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline1" />jats:tex-math$x<0$</jats:tex-math></jats:alternatives></jats:inline-formula>) but rough downstream (imposing a no-slip condition on jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline2" />jats:tex-math$x>0$</jats:tex-math></jats:alternatives></jats:inline-formula>), with a sharp transition at jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline3" />jats:tex-math$x=0$</jats:tex-math></jats:alternatives></jats:inline-formula>. This problem is similar to the classical Blasius problem, which considers the growth of a boundary layer over a flat plate in a Newtonian fluid that is subject to a similar step change in boundary conditions. Our discrete particle model simulations show that a comparable boundary-layer phenomenon occurs for the granular problem: the effects of basal roughness are initially localised at the base but gradually spread throughout the depth of the current. A rheological model can be used to investigate the changing internal velocity profile. The boundary layer is a region of high shear rate and therefore high inertial number jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline4" />jats:tex-math$I$</jats:tex-math></jats:alternatives></jats:inline-formula>; its dynamics is governed by the asymptotic behaviour of the granular rheology for high values of the inertial number. The jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline5" />jats:tex-math$𝜇(I)$</jats:tex-math></jats:alternatives></jats:inline-formula> rheology (Jop jats:italicet al.</jats:italic>, jats:italicNature</jats:italic>, vol. 441 (7094), 2006, pp. 727–730) asserts that jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline6" />jats:tex-math$d𝜇/dI=O(1/I2)$</jats:tex-math></jats:alternatives></jats:inline-formula> as jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline7" />jats:tex-math$I\to \infty$</jats:tex-math></jats:alternatives></jats:inline-formula>, but current experimental evidence is insufficient to confirm this. We show that this rheology does not admit a self-similar boundary layer, but that there exist generalisations of the jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline8" />jats:tex-math$𝜇(I)$</jats:tex-math></jats:alternatives></jats:inline-formula> rheology, with different dependencies of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline9" />jats:tex-math$𝜇(I)$</jats:tex-math></jats:alternatives></jats:inline-formula> on jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112019003574_inline10" />jats:tex-math$I$</jats:tex-math></jats:alternatives></jats:inline-formula>, for which such self-similar solutions do exist. These solutions show good quantitative agreement with the results of our discrete particle model simulations.</jats:p>