Under laminar flow, the flow is smooth, steady and ordered such that fluid elements flow along individual layers (laminae) or streamlines that do not cross each other. Flow is dominated by viscosity, it is possible to derive the velocity profile for a fluid under laminar flow conditions with
knowledge of the velocity.
Let us take a long, straight, constant diameter section of a pipe, the force balance on a cylindrical fluid element in this pipe gives,
$$\tau = \frac{r}{2}\frac{\Delta P}{\Delta L}$$
For a power law viscosity fluid, the general equation is given by,
$$\tau = -\mu\left(\frac{\text{d}u}{\text{d}r}\right)^n $$
where if $n=1$ the fluid is a Newtonian fluid.
Integrating the two equations above the velocity profile as in the graph below. Examine how the key values vary the velocity profile.