d$.$ Before we get to the new model, let's think about what factors would increase or decrease the pressure gradient needed to move viscous fluid in a pipe $($think about drinking a viscous fluid through a straw and what would make it harder or easier to drink$).$ A larger required pressure gradient would mean you have to suck harder $()$

i$.$ Increase viscosity of fluid will increase$/$decrease$/$not change the pressure gradient needed to maintain flow

ii$.$ Increase length of pipe wil increase$/$decrease$/$not change the pressure gradient needed to maintain flow

iii. Increase flow rate $($volume$/$time$)$ willincreaseydecrease$/$not change the pressure gradient needed to maintain flow

iv$.$ Increase radius of straw will increase decrease$/$not change the pressure gradient needed to maintain flow