1) An incompressible, isothermal power law fluid is driven down a pipe of radius R and length L by an applied pressure gradient ∂P/∂z (a known negative value). Assume the flow is axisymmetric and gravitational effects are negligible, such that it is also possibly to assume ∂P/∂θ=0 (it is not necessary to show this, take it as an assumption). Furthermore, assume the flow is steady and fully developed in the downstream z-direction.
a) Solve for the velocity, shear stress distribution and the volumetric flow rate down the pipe. This does not require scaling.
b) Plot the velocity profile(s), varying n over the values 0.1, 0.2, 0.5, 1, 2, 5, 10. This is easiest if you normalize by the average velocity, which equals the volumetric flow rate divided by the cross-sectional area /πR2. Then plot , vz /vz, vave versus r/R
c) Though this solution does not require scaling, now show that the axisymmetric continuity equation specifies the relationship between L and R under which the fully developed assumption is valid.
2) Consider pressure driven flow of an incompressible, isothermal Bingham fluid between parallel plates of length L in the x-direction and gap 2h in the z-direction, where h << L. Assume the 2-D flow (plates are infinite in the y-direction) is steady, gravitational effects are negligible, vy≡0, and ∂P/∂x is a known negative constant. Before you start, think about the best placement of your z-axis.
a) Assume the characteristic velocity in the x-direction is Vx and use scaling to estimate VZC.
b) Now use this result to estimate the components of the rate-of-deformation tensor. Then estimate γ. = √IIγ./2
c) Scale the appropriate component of the momentum equations and solve for vx. What assumptions aboutReL and the geometry were necessary to neglect the inertia terms?
d) Now compute the “exact” value of γ. = √IIγ./2 based on the solution to the flow field. At what value of z is this exact value equal to the estimated value?
e) Plot the velocity profiles for several values of hy /h.
3) Apply the material derivative D / Dt to the mass of a fluid particle m as follows:
Considering that the material derivative measures the rate of change while moving instantaneously with the fluid particle, evaluate Dm / Dt . Now, substitute m = ρ , and use the “alternative” form of compressible continuity in the notes, to relate
(the rate of volume increase of a particle, per unit volume) to the volumetric strain rate εii (the dilatation or normal strain rate).
4) Simplify the general form of continuity for an incompressible, isothermal fluid. What does this tell us about the volumetric strain rate εii (the dilatation or normal strain rate) in such a fluid? As we follow a fluid particle in this flow, what happens to its volume?