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Homework answers / question archive / 1)Flow of a Casson fluid in a pipe, The flow of blood is often described by the Casson fluid model described briefly in Section 5

1)Flow of a Casson fluid in a pipe, The flow of blood is often described by the Casson fluid model described briefly in Section 5

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1)Flow of a Casson fluid in a pipe, The flow of blood is often described by the Casson fluid model described briefly in Section 5.7. The model equation takes the following form for pipe flow: √−τ = √τ0 + s −(dvz/dr) (6.93) since both the shear stress, τ , and the strain rate (dvz/dr) are negative here. Develop an equation for the velocity profile for flow of this type of fluid in a circular pipe and also an expression relating the volumetric flow rate to the pressure drop for this case.

2)Power-law fluid: flow in parallel pipes. A 1-m long pipe delivers a fluid with a power-law index of 0.5 at the rate of 0.02 m3/s. The pressure drop of the pipe across the system is 3.5 × 104 Pa. Now, if an additional parallel line of the same size is laid in parallel, what will the new flow rate be? How much is the increase in capacity? Assume laminar flow holds in both the cases.

3)A power-law fluid example. A solution of 13.5% by weight of polyisoprene has the following power-law parameters: = 5000 Pa ·sn and n = 0.2. Consider the flow of such a solution in a pipe of internal diameter 1 cm and length 100 cm. Calculate and plot the volumetric flow rate as a function of the imposed pressure difference across the pipe. If the pipe diameter is doubled, how does the volumetric flow change?

4)Film flow over a cone and over a solid sphere. Consider the flow over a cone as shown in Fig. 6.21. Find the film thickness as a function of distance along the surface of the cone. Use the lubrication approximation. Repeat for the solid sphere in Fig. 6.21.

5)Sqeezing flow. Two parallel disks of radius R are separated by a distance H. The space between them is filled with an incompressible fluid. The top plate is moved towards the bottom at a constant velocity, causing the fluid to be squeezed out. Use the lubrication model and find an expression for the force needed to keep the squeezing motion.

6)Consider the flow of water in a pipe of length 2 m with an imposed pressure difference of 1000 Pa. Find the flow rate if (a) the pipe has a uniform cross-section of diameter 2 cm and (b) the pipe is tapered with an inlet diameter of 2 cm and an exit diameter of 3 cm. Assume the flow is laminar in both cases.

7)For the slider–block problem (Section 6.8.1) find the point where the pressure is a maximum and find the value of this pressure. Calculate the tangential and normal stresses on the top plate. Calculate the tangential stress on the bottom plate. Explain why the tangential stress on the top plate is not the same as that on the bottom plate.

8)Show that the superposition of a radial velocity field and a torsional field results in a spiral flow.Sketch typical streamlines. Spiral flows are good prototypes for tornadoes. Explain why the pressure field cannot be superimposed. How would you compute the pressure field.

9)For flow in a square channel the shear stress is not a constant along the perimeter, unlike in a cylindrical channel. Obtain the stress distribution in a rectangular channel. At what point is the stress maximum? Derive an expression for the volumetric flow rate for a square channel. Compare it with that for pipe flow for the same area of cross-section. Which geometry gives the higher flow rate? Why?

10)Show that the vorticity transport equation can also be written as Dω Dt = ν 2ω + [ω · ]v How does it simplify for 2D flows? How does it simplify for slow flows? Show all the steps leading to the pressure Poisson equation.

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