Dr. Vernon and his post-docs, Dr. Celeste Riley and Dr. Hanin Bearat, are making an in situ gelling material (gels after itis inserted into the patient) for treating aneurisms (protrusions in the side of a blood vessel – the material fills theprotrusion). The chemicals required to create the gel are mixed in the operating room, the surgeon waits until the materialhas become viscous (but not solid), then the surgeon injects the material through a catheter into the aneurism site where itfinishes gelling into a solid material to plug the aneurism site. The material must be injected after it is viscous enough thatit will stay in the aneurism, but the surgeon can’t wait too long or the material will be too viscous to be pushed through thecatheter. If the material’s viscosity as a function of time can be described by the equation:
where μ0 is the final viscosity of the material (after gelling), t is the time after the chemicals are initially mixed, and β is aconstant describing how fast the viscosity changes. Assume the material behaves as a Bingham Plastic with critical shearstress τ0. [Note: The fact that viscosity changes with time does not affect the relationship between shear stress and velocitygradient – viscosity still acts as a constant in that relationship]. Assume the pressure at the aneurism site is essentiallyatmospheric. The catheter has an inner radius of R.
(a) Derive the equation(s) for velocity of material as a function of position within the catheter. [Note: Your answer shouldbe a function of time – i.e., you should substitute in the equation above for viscosity.]
(i) Where is the velocity at a maximum?
(ii) Write an equation for the velocity at that position / those positions.
(iii) Write an equation for the velocity at the other positions as a function of that spatial variable. State the range of thespatial variable for which this equation is valid.
(b) Derive the equation for the pressure the must be applied to the syringe (feeding the catheter) to achieve a volumetricflow rate of fluid (Q) if the catheter is of length L.