You are a research scientist working at a company to develop hydrophilic surface of the pipe for better flow of some non-Newtonian liquid with polarity. For this liquid, your colleague in your team found that the fluid follows laminar flow profile and the shear stress of the fluid inside a horizontal cylindrical pipe, T , can be described as

T=-deltaT(eta)(du/dr)^2

where eta denotes viscosity, delta t denotes some characteristic response time of the fluid, u denotes the horizontal velocity of the fluid, and r is for the radial direction variable. Given the pressure gradient over the longitudinal direction of the pipe such that dp/ dz = -delta p/ L, please address following questions

1. Please find the velocity profile of the fluid u(r).

2. Please calculate the total volumetric flow rate Q.

3. One of your colleague suggested that the characteristic response time of the non- Newtonian fluid can be described by comparing the total volumetric flow rate your found in 3.2 with the original flow rate of the Newtonian fluid (Hagen-Poiseuille flow). Please calculate delta t.

4. Please calculate the total momentum, M , delivered by the fluid during delta t . For delta t , you can use your result in 3.3.

5. Using the results 3.3 and 3.4, please calculate the average velocity, u momentum , corresponding to total mass (W ) and total momentum.