Summary
This is the second of a series of two papers in which the steady state velocity and temperature distributions in a liquid film are analytically investigated. The liquid is assumed to appear on a flat surface due to transpiration through a porous medium or the melting of a solid material. The liquid is considered to be introduced into the film at the solid-liquid interface, in a direction normal to the interface, such that mass is continually being added to the film along the longitudinal path of flow.
The flow, which is assumed to be laminar, occurs under the influence of gravity or an externally applied shear stress at the film surface. Heat transfer takes place into the film by convection from an atmosphere which is at a higher temperature than the liquid. Evaporation and gaseous boundary layer effects are not considered. Physical properties of the liquid, such as density, viscosity, and thermal conductivity are considered constant.
In the first paper, an investigation is made of the velocity distribution for a liquid film which is subjected to a uniform body force. In this paper, the velocity distribution is determined for a film which is subjected to an externally applied shear stress at the film surface. In addition, temperature distributions are determined for a film subjected to either a gravity body force or a surface shear stress. The externally applied surface shear stress, and the rate at which liquid is introduced into the film are, in general, considered independent of position.
The continuity, momentum, and energy equations are set up in integral form. Then, the problems of determining the film thicknesses, velocity distributions, and temperature distributions are approached in four different ways, referred to as Case I, II, III and IV. These four cases differ from one another by the simplifying assumptions made. These assumptions are made in such a way that each successive case takes different effects into account, and yields a solution based on a somewhat more accurate analysis.
Comparing the solutions resulting from the four cases, it is found that where the liquid film thicknesses are relatively small, for each individual problem the results of all four cases converge to the same expression. This tends to justify the use of certain simplifying assumptions even though one would not initially regard them reasonable.
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Abbreviations
- a, b, c, d :
-
coefficients in velocity and temperature profile polynomials
- A :
-
dimensionless parameter,
$$A = \left( {\frac{{\partial f\upsilon }}{{\partial \eta }}} \right)_{\eta = 0} = \frac{\delta }{U}\left( {\frac{{\partial u}}{{\partial y}}} \right)_0 ;A_T = \frac{{\partial f_T }}{{\partial \eta }} = \frac{\delta }{{T_\delta }}\left( {\frac{{\partial T}}{{\partial y}}} \right)_0 $$ - B :
-
dimensionless parameter,
$$B = \int\limits_0^1 {f\upsilon (\eta ,x)d\eta = \frac{1}{{U\delta }}\int\limits_0^\delta {u{\text{d}}y} } $$ - C :
-
dimensionless parameter,
$$C = \int\limits_0^1 {f_\upsilon ^2 (\eta ,x)d\eta = \frac{1}{{U^2 \delta }}\int\limits_0^\delta {u^2 {\text{d}}y} } $$$$C_T = \int\limits_0^1 {f\upsilon (\eta ,x)f_T (\eta ,x)d\eta = \frac{1}{{U_\delta \delta }}\int\limits_0^\delta {uT{\text{d}}y} } $$ - C p :
-
specific heat of liquid
- E :
-
dimensionless parameter E=C/B 2
- f(x, η):
-
dimensionless velocity of temperature profile function, f, f s , f v , f T ; gravity, surface shear, gravity or surface shear, thermal problems respectively
- g :
-
gravitational acceleration component in x-direction
- h :
-
heat transfer coefficient at free surface of liquid
- H :
-
local Biot number, H=hδ/k
- J :
-
dimensionless parameter, J=C T /B
- k :
-
thermal conductivity of liquid
- M :
-
dimensionless parameter, M=A/B
- P :
-
local Peclet number, P=V 0 δ/α
- R :
-
local Reynolds number, R=V 0 δ/ν
- S :
-
dimensionless parameter, S=1/2(g/V 0 ν)(δ 3/x) (gravity problem) S s =1/2(τ s /V 0 μ)(δ 2/x) (surface shear problem) temperature at any point
- t 0 :
-
liquid temperature at liquid-solid interface, y=0
- T :
-
temperature excess at any point, T=t−t 0
- Tα :
-
ambient temperature excess
- Tσ :
-
temperature excess at film surface, y=δ
- u :
-
liquid velocity in x-direction
- U :
-
surface velocity in x-direction
- V 0 :
-
“feed-in” velocity; y-direction velocity at y=0
- x, y :
-
space coordinates
- α :
-
thermal diffusivity of liquid, α=k/(ρC p )
- α, β, γ :
-
terms defined in Appendix B
- δ :
-
film thickness normal to plate surface
- η :
-
dimensionless space coordinate, η=y/δ
- θ :
-
dimensionless film surface temperature excess fraction, θ=Tσ/Tα
- μ :
-
dynamic viscosity
- ν :
-
kinematic viscosity, ν=μ/ρ
- ρ :
-
liquid density
- σ :
-
variable surface shear stress, see Appendix A
- τ s :
-
externally applied surface shear stress
- ϒ :
-
variable feed-in velocity, see Appendix A
- φ :
-
dimensionless term in Case II solution expressions, φ, defined by (25), paper 1 for gravity problem, φ s , defined by (41), paper 2 for surface shear problem, φ v , denotes either φ or φ s
- Φ T :
-
dimensionless term defined by (48)
- ω :
-
dimensionless term in Case IV solution expressions, ω, defined by (42), paper 1 for gravity problem ω s , defined by (58), paper 2 for surface shear problem
- Ω :
-
volumetric flow parameter,
$$\Omega = \int\limits_0^x {\Upsilon (x){\text{d}}x} $$, see Appendix A
- G, s, v, T :
-
refers to gravity, surface shear, either gravity or surface shear, thermal problems, respectively
- 0:
-
refers to y or η=0, solid-liquid interface
- UL, LL :
-
upper limit, lower limit
- δ :
-
refers to y=δ, film surface
References
Mouradian, E. M. and J. E. Sunderland, Appl. sci. Res. A 14 (1965).
Mouradian, E. M., Velocity and Temperature Distributions in a Liquid Film, Ph. D. Thesis, Northwestern University, Evanston, Ill., September, 1961.
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Mouradian, E.M., Sunderland, J.E. The velocity and temperature distributions in a liquid film. Appl. Sci. Res. 14, 453–470 (1965). https://doi.org/10.1007/BF00382266
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DOI: https://doi.org/10.1007/BF00382266