Abstract
A simplified Navier-Stokes equation is applied to the solution of the velocity profile in the liquid meniscus adhering to long flat supports withdrawn continuously from baths of quiescent liquids. The inertial term is included using an Oseen approximation, the inhomogeneous boundary condition is transformed, and the resulting differential equation is solved by the method of eigenfunction expansions. The series describing the velocity profile and volume flowrate are both found to be rapidly convergent.
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Abbreviations
- a s :
-
coefficients defined by (19)
- A s :
-
constant for each s, (36)
- b :
-
width of plate, cm
- B s :
-
constant for each s, (37)
- C s :
-
coefficients defined by (22)
- f :
-
function of y, (10)
- g :
-
gravitation acceleration, cm/sec2
- h :
-
film thickness in the meniscus, cm
- h 0 :
-
film thickness in constant thickness region, cm
- I :
-
integral, (21A)
- k :
-
constant, μ/ρU w
- M s :
-
function of x and s, (20A)
- N :
-
function of h and constants, (11)
- P :
-
pressure (Appendix A), dyne/cm2
- Q s :
-
parameter defined by (23)
- r :
-
function of y, (14)
- R :
-
nondimensional inertial parameter, hU w ρ/μ
- s :
-
term in series, such as (19)
- S :
-
stress tensor (Appendix A)
- T :
-
function of x, (17)
- T s :
-
exp(−λ s kx)
- u :
-
x component of velocity, cm/sec
- u′ :
-
perturbation velocity for u, cm/sec
- u x :
-
∂u/∂x
- u yy :
-
∂ 2 u/∂y 2
- U w :
-
constant velocity of plate withdrawal, cm/sec
- v :
-
y component of velocity, cm/sec
- v′ :
-
perturbation velocity for v, cm/sec
- V :
-
volume flow rate, cm3/sec
- w :
-
function of x and y, (19)
- W :
-
velocity vector (Appendix A)
- x :
-
vertical coordinate (parallel to plate); Fig. 1
- y :
-
horizontal coordinate (perpendicular to plate), Fig. 1
- z :
-
function of x and y, (21)
- β :
-
function of R, (29)
- δ s :
-
π(2s+1)/2
- λ s :
-
eigenvalues, (18)
- μ :
-
viscosity, poise
- φ s :
-
eigenfunctions, (18A)
- ρ :
-
density, gm/cm3
- σ :
-
surface tension, gas-liquid interface, dyne/cm
- τ :
-
integration variable, (21)
References
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White, D. A. and J. A. Tallmadge, Chem. Engr. Sci. 20 (1965) 33.
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Tallmadge, J. A. and A. J. Soroka, Chem. Engr. Sci. 24 (1969) 377.
Levich, V. G., Physicochemical Hydrodynamics, Chapter 12, Prentice-Hall, New York (1962).
Schlichting, H., Boundary Layer Theory, 6 ed., McGraw-Hill Book Co., New York (1968).
Berg, P. W. and J. L. McGregor, Elementary Partial Differential Equations, Holden-Day Inc., San Francisco (1966).
Drake, R. L., Private Communication, December 1968.
Soroka, A. J. and J. A. Tallmadge, “The Inertial Theory for Plate Withdrawal,” Fluid Mechanics Symposium, November, Amer. Inst. of Chem. Engr. meeting, Washington, D.C. (1969).
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Soroka, A.J., Tallmadge, J.A. Velocity profiles for plate withdrawal at high speeds. Appl. Sci. Res. 25, 413–430 (1972). https://doi.org/10.1007/BF00382314
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DOI: https://doi.org/10.1007/BF00382314