Abstract
A theoretical study is made of the flow behavior of thin Newtonian liquid films being “squeezed” between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations.
Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.
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Abbreviations
- a :
-
half of the distance, or gap, between the two plates
- a 0 :
-
the value of a at time t=0
- adot :
-
da/dt
- ä :
-
d2 a/dt 2
- \(\dddot a\) :
-
d3 a/dt 3
- a i :
-
components of a contravariant acceleration vector
- f :
-
unknown function of z 0 and t defined in (6)
- f i :
-
function defined in (9) f 1=r 0 g(z 0, t) f 2=θ 0 f 3=f(z 0, t)
- F :
-
force applied to the plates
- g :
-
unknown function of z 0 and t defined in (6)
- g′ :
-
∂g/∂z 0
- h :
-
grid dimension in the z 0 direction (see Fig. 5)
- \(\left\{ \begin{gathered} i \hfill \\ jk \hfill \\ \end{gathered} \right\}\) :
-
Christoffel symbol
- i, j, k, l :
-
indices
- k :
-
grid dimension in the t direction (see Fig. 5)
- r :
-
radial coordinate direction defined in Fig. 1
- r 0 :
-
radial convected coordinate
- R :
-
radius of the circular plates
- t :
-
time
- v r :
-
fluid velocity in the r direction
- v z :
-
fluid velocity in the z direction
- v θ :
-
fluid velocity in the θ direction
- x i :
-
cylindrical coordinate x 1=r x2=θ x3=z
- z :
-
vertical coordinate direction defined in Fig. 1
- z 0 :
-
vertical convected coordinate
- θ :
-
tangential coordinate direction
- θ 0 :
-
tangential convected coordinate
- μ :
-
viscosity
- ν :
-
kinematic viscosity, μ/ρ
- ξ i :
-
convected coordinate ξ 1=r0 ξ2=θ0 ξ3=z0
- ρ :
-
density
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Grimm, R.J. Squeezing flows of Newtonian liquid films an analysis including fluid inertia. Appl. Sci. Res. 32, 149–166 (1976). https://doi.org/10.1007/BF00383711
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DOI: https://doi.org/10.1007/BF00383711