Computing the velocity of fluid along the length of a cylinder

I am modelling the flow of fluid through a cylinder of length L.The cylinder is divided into three segments(each segment of length L/3), there are four nodes(i=0,1,2,3). I'm trying to compute the inlet velocity and outlet velocity in each segment.

The known value is the average velocity of the fluid flowing through the cylinder of radius r and length L.

Using the average velocity of the fluid, the pressure drop across the cylinder of length L is computed using the Hagen-Poiseuille equation. $$v=\frac{r^2}{8\mu L}\Delta P$$ $$v=\frac{r^2}{8\mu L}(P_o-P_i)$$

Pressure at the inlet $$P_i$$ and outlet $$P_o$$ of the cylinder of length L.

I'm computing the outlet velocity($$v_1$$) of the first segment assuming equal pressure drop in all three segments. $$v_1=\frac{r^2}{8\mu L/3}\frac{(P_o-P_i)}{3}$$

Is this a right way to proceed? EDIT: I'm confused about the following, If the pressure drop in each segment is not constant, would the variation in velocity along axial direction violate the equation of continuity?

• Your approach is correct if you have assumed fully developed flow at the pipe inlet. Also equation of continuity expresses mass conservation, so it must always hold. – Deep Oct 18 '18 at 5:18
• For cylindrical coordinates, the equation of continuity simplifies to $$\frac{\partial v_z}{\partial z }=0$$ Wouldn't this mean the velocity doesn't change along the axial direction? Please correct me if my understanding is wrong. – Natasha Oct 18 '18 at 7:16
• Yes- that's the definition of "fully developed flow". – Deep Oct 18 '18 at 10:13