Applied Scientific Research

, Volume 30, Issue 3, pp 161–181 | Cite as

The indentation of a Newtonian fluid in a finite cylindrical container by a right circular cylinder

  • M. T. Abdel-Moneim
  • P. G. Kirmser
Article
  • 19 Downloads

Abstract

The indentation of the free surface of a Newtonian fluid in a finite cylindrical container by a right circular cylinder is considered. It is assumed that weight and inertia effects are negligible compared to viscous effects.

A finite difference technique is used to obtain approximate values for initial velocities, pressures, and stresses at any point in the fluid as well as an estimate of the force required to indent the fluid with a given velocity.

The solution obtained forms the basis for a primary indenter viscometer for very viscous fluids which have viscosities in the range of 104–1010 poises.

Keywords

Viscosity Free Surface Finite Difference Initial Velocity Circular Cylinder 

Nomenclature

a

radius of indenter

a0,a1,a2

parameters

b

size ratio defined in the text

c

size ratio defined in the text

d

diameter of indenter

D

diameter of container

f

force of indentation

G

interval size in the axial direction

H

interval size in the radial direction

k

a constant

L

depth of fluid

p(r, s)

fluid pressure

p+(x, y)

dimensionless fluid pressure, =P/ρW 0 2

r

radial coordinate

R

radius of container

RN

Reynolds' number defined in the text

u(x, y)

dimensionless radial velocity component, =vr/W0

vr(r, z)

radial velocity component

vz(r, z)

axial velocity component

V

velocity vector

W0

velocity of indenter

w(x, y)

dimensionless axial velocity component, =vz/W0

x

dimensionless radial coordinate, =r/a

X

body force vector

y

dimensionless axial coordinate, =z/L

z

axial coordinate

A, B, C, ...

Capital letters are matrices defined in the text unless stated otherwise

q1,q2

matrices defined in the text

η, η1,η2

arbitrary constants

μ

fluid coefficient of viscosity

ρ

fluid mass density

σz(r, z)

normal stress in thez-direction

σz/+(x, y)

dimensionless normal stress in the axial direction, =σz/ρW 0 2

τrz(r, z)

shear stress on ther face in thez direction

τrz/+(x, y)

dimensionless shear stress on ther face in thez direction, =τrz/ρW 0 2

ψ(x, y)

stream function

ω

a diagonal weighting matrix defined in the text

$$\nabla e_\gamma \frac{\partial }{{\partial \gamma }} + e_\theta \frac{1}{\gamma }\frac{\partial }{{\partial \theta }} + e_Z \frac{\partial }{{\partial Z}}$$
2
$$\nabla ^2 \frac{{\partial ^2 }}{{\partial \gamma ^2 }} + \frac{1}{\gamma }\frac{\partial }{{\partial \gamma }} + \frac{1}{{\gamma ^2 }}\frac{{\partial ^2 }}{{\partial \theta ^2 }} + \frac{{\partial ^2 }}{{\partial Z^2 }}$$

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References

  1. [1]
    Schlichting, H., Boundary Layer Theory, 6th ed., McGraw-Hill, N.Y., 1968.Google Scholar
  2. [2]
    Goudy, R. S. andP. G. Kirmser, Appl. Sci. Res.,19 (1968) 393.Google Scholar

Copyright information

© Martinus Nijhoff, The Hague/Kluwer Academic Publishers 1975

Authors and Affiliations

  • M. T. Abdel-Moneim
    • 1
  • P. G. Kirmser
    • 1
  1. 1.Dept. of Appl. Mech.Kansas State Univ.ManhattanUSA

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