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Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 14,
- Issue 6,
- pp. 491-504
- (1927)
- •doi: 10.1364/JOSA.14.000491

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- Holborn and Schultze, Ann. d. Phys., 47, p. 1095; 1915.

- Keyes and Brownlee, Thermodynamic Properties of Ammonia, Wiley & Sons, p. 11; 1916.

- Keyes, Joubert, and Smith, Jour. Math. & Phys., M.I.T., p. 195; 1922.

- For the steady motion of a cylindrical shell of incompressible fluid moving upward in the direction of the z axis the usual hydrodynamical equation is:μr∂∂r(r∂w∂r)=∂p∂z=constantwhere μ is the coefficient of viscosity, w the velocity along the direction of the z axis, p the pressure per unit area, and r the radius of the cylinder. The space between the piston and cylinder for the case of exact coincidence of the two axes may be designated (a−b)=σ where a and b are the radii of cylinder and piston respectively. If no “slip” occurs at the solid boundaries ∂w/∂r is zero as usually assumed. Integration of the equation above givesμ∂w∂r=σ2∂p∂za+brSince a is nearly equal to b, (a+b)/r may be taken as very nearly 2 and the total force on the cylindrical surface perpendicular to r becomes:F=πaσ∂p∂zl=πaσ(p1-p0)This shows that the force due to the escaping oil and tending to lift the piston is independent of the viscosity of the oil and remains an invariable fraction of the pressure measured. The equation is in agreement with Michel’s which was obtained by a different method involving the same assumptions.If compressibility of the liquid is allowed for, an equation of state of the liquid must be assumed. The result obtained above, however, would be but slightly modified.

- P. W. Bridgman, Proc. Am. Acad., 47, p. 321; 1911.

[Crossref] - A. Michels, Ann. der Phys., 72, p. 285; 1923.

[Crossref] - A. Michels, Ann. der Phys., 73, p. 577; 1924.

[Crossref] - The Thermodynamic Properties of Ammonia. John Wiley & Sons, p. 11, 1916.

- We take pleasure in expressing our great obligation for the painstaking aid given us and the courtesy shown by Mr. Francis Meredith, Director of Standards of the State of Massachusetts.

A. Michels, Ann. der Phys., 73, p. 577; 1924.

[Crossref]

A. Michels, Ann. der Phys., 72, p. 285; 1923.

[Crossref]

Holborn and Schultze, Ann. d. Phys., 47, p. 1095; 1915.

P. W. Bridgman, Proc. Am. Acad., 47, p. 321; 1911.

[Crossref]

P. W. Bridgman, Proc. Am. Acad., 47, p. 321; 1911.

[Crossref]

Keyes and Brownlee, Thermodynamic Properties of Ammonia, Wiley & Sons, p. 11; 1916.

Holborn and Schultze, Ann. d. Phys., 47, p. 1095; 1915.

Keyes, Joubert, and Smith, Jour. Math. & Phys., M.I.T., p. 195; 1922.

Keyes, Joubert, and Smith, Jour. Math. & Phys., M.I.T., p. 195; 1922.

Keyes and Brownlee, Thermodynamic Properties of Ammonia, Wiley & Sons, p. 11; 1916.

A. Michels, Ann. der Phys., 73, p. 577; 1924.

[Crossref]

A. Michels, Ann. der Phys., 72, p. 285; 1923.

[Crossref]

Holborn and Schultze, Ann. d. Phys., 47, p. 1095; 1915.

Keyes, Joubert, and Smith, Jour. Math. & Phys., M.I.T., p. 195; 1922.

Holborn and Schultze, Ann. d. Phys., 47, p. 1095; 1915.

A. Michels, Ann. der Phys., 72, p. 285; 1923.

[Crossref]

A. Michels, Ann. der Phys., 73, p. 577; 1924.

[Crossref]

P. W. Bridgman, Proc. Am. Acad., 47, p. 321; 1911.

[Crossref]

The Thermodynamic Properties of Ammonia. John Wiley & Sons, p. 11, 1916.

We take pleasure in expressing our great obligation for the painstaking aid given us and the courtesy shown by Mr. Francis Meredith, Director of Standards of the State of Massachusetts.

Keyes and Brownlee, Thermodynamic Properties of Ammonia, Wiley & Sons, p. 11; 1916.

Keyes, Joubert, and Smith, Jour. Math. & Phys., M.I.T., p. 195; 1922.

For the steady motion of a cylindrical shell of incompressible fluid moving upward in the direction of the z axis the usual hydrodynamical equation is:μr∂∂r(r∂w∂r)=∂p∂z=constantwhere μ is the coefficient of viscosity, w the velocity along the direction of the z axis, p the pressure per unit area, and r the radius of the cylinder. The space between the piston and cylinder for the case of exact coincidence of the two axes may be designated (a−b)=σ where a and b are the radii of cylinder and piston respectively. If no “slip” occurs at the solid boundaries ∂w/∂r is zero as usually assumed. Integration of the equation above givesμ∂w∂r=σ2∂p∂za+brSince a is nearly equal to b, (a+b)/r may be taken as very nearly 2 and the total force on the cylindrical surface perpendicular to r becomes:F=πaσ∂p∂zl=πaσ(p1-p0)This shows that the force due to the escaping oil and tending to lift the piston is independent of the viscosity of the oil and remains an invariable fraction of the pressure measured. The equation is in agreement with Michel’s which was obtained by a different method involving the same assumptions.If compressibility of the liquid is allowed for, an equation of state of the liquid must be assumed. The result obtained above, however, would be but slightly modified.

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Pressure gage.

Actual arrangement of gages and stopcocks.

**Table 1** Gage constants at 30° from comparisons made with a mercury column 900 cm in length.

**Table 2** Properties of mercury, steel and oil used in reducing column lengths of mercury to standard conditions.

**Table 3** Mercury column comparisons made with large pistons.

Equations on this page are rendered with MathJax. Learn more.

$$\frac{\mu}{r}\frac{\partial}{\partial r}\left(r\frac{\partial w}{\partial r}\right)=\frac{\partial p}{\partial z}=\text{constant}$$

$$\mu \frac{\partial w}{\partial r}=\frac{\sigma}{2}\frac{\partial p}{\partial z}\frac{a+b}{r}$$

$$F=\pi a\sigma \frac{\partial p}{\partial z}l=\pi a\sigma ({p}_{1}-{p}_{0})$$

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