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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

    index-100_1.png

    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.

    index-101_1.png

    index-101_2.png

    92