Question

Only need help with Question 3, thanks ;)

The actual velocity profile is given by the Poiseuille equation as follows: ()APR where AP is the pressure driving force, R is the tube radius, u is the fluid viscosity, Lis the tube length, and r is the radial position of interest. A couple of notes: . Assume that the fluid has a viscosity similar to water (rester 103 Pa s) Winter 2019 Il. The radial position r is defined to be zero at the centerline, and the vessel radius IIWatch your units! Use metric units for all variables (Pascal for pressure, meter for IV. This assignment will require that you calculate a very simple integral and is R (i.e.r R at the wall). length). Do feel free to use unit prefixes though when appropriate (i.e. micrometers um) derivative. If you have had calculus previously, please briefly review your notes from that class. If not, see one of the TAs for help. For a pressure driving force (AP) of 2000 Pa, vessel radius (R) of 100 μm, and vessel length of 10 cm, what is the velocity I. A) at the vessel center? B) at the vessel wall? 2. How do these results, for both cases, change for A) a more viscous fluid (3 X Pwater)? Note: all else is the same B) a narrower vessel (R 50 um)?

Answer 1

Write the expression for the actual velocity profile 1) At vessel center Substitute the given values in equation (1) APR 2000 x(100x10) 4x10-1x0.1 0.05 m/s Thus, the velocity at the vessel center is 0.05 m/s B) At vessel wall r=R Substitute the given values in equation (1)

V,(r)=0 Thus, the velocity at the vessel wall is 0 2) At vessel center Substitute the given values in equation (1) vz (r) 2000x 100x10 ½ (r) 4×3×10" × 0.1 0.0167 mjs Thus, the velocity at the vessel center is 0.0167 m/s At vessel wall r=R Substitute the given values in equation (1)

v,(r) 0 Thus, the velocity at the vessel wall is 0 B) At vessel center Substitute the given values in equation (1) 2000x(50x10-5) ½ (r) 4x10" 0.1 = 0.0125 m/s Thus, the velocity at the vessel center is 0.0125 m/s At vessel wall Substitute the given values in equation (1).

Thus, the velocity at the vessel wall is 0 3) Write the expression for the average velocity profile (v)-歜计 At vessel center Substitute the given values in equation (2) ΔΡ/02 04 Thus, the velocity at the vessel center is Om/s

At vessel wall Substitute R for r in equation (2) AP R2 R2 Substitute the given values in equation (3) 2000x(100x109 8x10-3 x0.1 0.025 m/s Thus, the velocity at the vessel wall is0.025 m/s B) At vessel center Substitute the given values in equation (2) (v)-0 m/s

Thus, the velocity at the vessel center is 0m/s At vessel wall Substitute the given values in equation (3) 2 2000x(100x 10-5) 8 x3x10-3 x0.1 (1) = 8.33x10-3 m/s Thus, the velocity at the vessel wall is 8.33x103 m/s C) At vessel center Substitute the given values in equation (2) (v)-0 m/s Thus, the velocity at the vessel center is 0m/s At vessel wall Substitute the given values in equation (3)

2000x(50x10) (v) = 8x10-3 x 0.1 -6.25x103 m/s Thus, the velocity at the vessel wall is 6.25x10 m,/s