Extraction into a flow of a fluid with a velocity gradient

Unsteady extraction from a system of semi-infinite capillaries to the flow of a fluid with a linear velocity gradient is studied. An assumption that the diffusion in the flow is a quasi-steady-state process is taken to obtain a linear equation for the local mass flux from the pore space, which involves fractional derivatives with respect to time and flow coordinate. Limiting solutions for the total diffusion flux at small and large times that cover the entire time interval are found.