# 3.27: Untitled Page 53

- Page ID
- 18186

## Chapter 3

is 0.15 cm, at the particular instant of time, what is the volumetric flow rate at the exit of the artery?

3‐16. A variety of devices, such as ram pumps, hydraulic jacks, and shock absorbers, make use of moving solid cylinders to generate a desired fluid motion.

In Figure 3.16 we have illustrated a cylindrical rod entering a cylindrical cavity in order to force the fluid out of that cavity. In order to determine the *force* acting on the cylindrical rod, we must know the velocity of the fluid in the annular region. If the density of the fluid can be treated as a constant, the velocity can be determine by application of the macroscopic mass balance and in this problem you are asked to develop a general representation for the fluid velocity.

*Figure 3.16*. Flow in an hydraulic ram

3‐17. In Figure 3.17 we have illustrated a capillary tube that has just been immersed in a pool of water. The water is rising in the capillary so that the height of liquid in the tube is a function of time. Later, in a course on fluid mechanics, you will learn that the average velocity of the liquid, v , can be *z*

represented by the equation

*Single component systems *

91

8 v

2

o

r

*z h*

*gh*

(1)

2

o

*r*

**capillary**

**gravitational**

** force**

** force**

**viscous**

** force**

in which v is the average velocity in the capillary tube. The surface tension *z*

, capillary radius , and fluid viscosity

o

*r*

can all be treated as constants in

addition to the fluid density and the gravitational constant g. From Eq. 1 it is easy to deduce that the final height (when v

*z*

0 ) of the liquid is given by

*h* 2 *g * o

*r *

(2)

In this problem you are asked to determine the height *h* as a function of time (Levich, 1962) for the initial condition given by

I.C.

*h * 0 *, t * 0

(3)

Part (a). Derive a governing differential equation for the height, (

*h t*) , that is to be

solved subject to the initial condition given by Eq. 3. Solve the initial value problem to obtain an implicit equation for (

*h t*) .

*Figure 3.17* Transient capillary rise

Part (b). By arranging the implicit equation for *h*( *t*) in dimensionless form, demonstrate that this mathematical problem is identical in form to the problem described in Example 3.5 and Problem 3‐13.

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