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

## 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 10^{4}–10^{10} poises.

## Keywords

Viscosity Free Surface Finite Difference Initial Velocity Circular Cylinder## Nomenclature

*a*radius of indenter

*a*_{0},*a*_{1},*a*_{2}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

*R*_{N}Reynolds' number defined in the text

*u*(*x, y*)dimensionless radial velocity component, =

*v*_{r}/*W*_{0}*v*_{r}(*r, z*)radial velocity component

*v*_{z}(*r, z*)axial velocity component

**V**velocity vector

*W*_{0}velocity of indenter

*w*(*x, y*)dimensionless axial velocity component, =

*v*_{z}/*W*_{0}*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

*q*_{1},*q*_{2}matrices defined in the text

*η, η*_{1},*η*_{2}arbitrary constants

*μ*fluid coefficient of viscosity

*ρ*fluid mass density

*σ*_{z}(*r, z*)normal stress in the

*z*-direction*σ*_{z}^{/+}(*x, y*)dimensionless normal stress in the axial direction, =

*σ*_{z}/*ρW*_{0}^{2}*τ*_{rz}(*r, z*)shear stress on the

*r*face in the*z*direction*τ*_{rz}^{/+}(*x, y*)dimensionless shear stress on the

*r*face in the*z*direction, =*τ*_{rz}/*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]Schlichting, H., Boundary Layer Theory, 6th ed., McGraw-Hill, N.Y., 1968.Google Scholar
- [2]