On thin evaporating drops: When is the -law valid?

2016 ◽  
Vol 792 ◽  
pp. 134-167 ◽  
Author(s):  
M. A. Saxton ◽  
J. P. Whiteley ◽  
D. Vella ◽  
J. M. Oliver
Keyword(s):  
Mass Transfer ◽  
Asymptotic Analysis ◽  
Numerical Simulations ◽  
Viscous Dissipation ◽  
Square Root ◽  
Vapour Transport ◽  
Asymptotic Results ◽  
Rate Of Evaporation ◽  
And Diffusion ◽  
Matched Asymptotic Analysis

We study the evolution of a thin, axisymmetric, partially wetting drop as it evaporates. The effects of viscous dissipation, capillarity, slip and diffusion-dominated vapour transport are taken into account. A matched asymptotic analysis in the limit of small slip is used to derive a generalization of Tanner’s law that takes account of the effect of mass transfer. We find a criterion for when the contact-set radius close to extinction evolves as the square root of the time remaining until extinction – the famous $d^{2}$-law. However, for a sufficiently large rate of evaporation, our analysis predicts that a (slightly different) ‘$d^{13/7}$-law’ is more appropriate. Our asymptotic results are validated by comparison with numerical simulations.

scholarly journals On contact-line dynamics with mass transfer

2015 ◽  
Vol 26 (5) ◽  
pp. 671-719 ◽  
Author(s):  
J. M. OLIVER ◽  
J. P. WHITELEY ◽  
M. A. SAXTON ◽  
D. VELLA ◽  
V. S. ZUBKOV ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Surface Tension ◽  
Asymptotic Analysis ◽  
Contact Line ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Leading Order ◽  
Vapour Transport ◽  
Gravity Surface ◽  
Matched Asymptotic Analysis

We investigate the effect of mass transfer on the evolution of a thin, two-dimensional, partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, other effects, such as gravity, surface tension gradients, vapour transport and heat transport, are neglected in favour of mathematical tractability. Our focus is on a matched-asymptotic analysis in the small-slip limit, which reveals that the leading-order outer formulation and contact-line law depend delicately on both the sign and the size of the mass transfer flux. This leads, in particular, to novel generalisations of Tanner's law. We analyse the resulting evolution of the drop on the timescale of mass transfer and validate the leading-order predictions by comparison with preliminary numerical simulations. Finally, we outline the generalisation of the leading-order formulations to prescribed non-uniform rates of mass transfer and to three dimensions.

Initial motion of a viscous vortex ring

1994 ◽  
Vol 446 (1928) ◽  
pp. 589-599 ◽  
Keyword(s):  
Reynolds Number ◽  
Asymptotic Analysis ◽  
Vortex Ring ◽  
Kinematic Viscosity ◽  
Reynolds Numbers ◽  
Initial Radius ◽  
Dimensionless Form ◽  
Ring Radius ◽  
Initial Motion ◽  
Matched Asymptotic Analysis

The behaviour of a viscous vortex ring is examined by a matched asymptotic analysis up to three orders. This study aims at investigating how much the location of maximum vorticity deviates from the centroid of the vortex ring, defined by P. G. Saffman (1970). All the results are presented in dimensionless form, as indicated in the following context. Let Γ be the initial circulation of the vortex ring, and R denote the ring radius normalized by its initial radius R i . For the asymptotic analysis, a small parameter ∊ = ( t / Re ) ½ is introduced, where t denotes time normalized by R 2 i / Γ , and Re = Γ/v is the Reynolds number defined with Γ and the kinematic viscosity v . Our analysis shows that the trajectory of maximum vorticity moves with the velocity (normalized by Γ/R i ) U m = – 1/4π R {ln 4 R /∊ + H m } + O (∊ ln ∊), where H m = H m ( Re, t ) depends on the Reynolds number Re and may change slightly with time t for the initial motion. For the centroid of the vortex ring, we obtain the velocity U c by merely replacing H m by H c , which is a constant –0.558 for all values of the Reynolds number. Only in the limit of Re → ∞, the values of H m and H c are found to coincide with each other, while the deviation of H m from the constant H c is getting significant with decreasing the Reynolds number. Also of interest is that the radial motion is shown to exist for the trajectory of maximum vorticity at finite Reynolds numbers. Furthermore, the present analysis clarifies the earlier discrepancy between Saffman’s result and that obtained by C. Tung and L. Ting (1967).

scholarly journals On the Influence of Soret and Dufour Effects on MHD Free Convective Heat and Mass Transfer Flow over a Vertical Channel with Constant Suction and Viscous Dissipation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Ime Jimmy Uwanta ◽  
Halima Usman
Keyword(s):  
Mass Transfer ◽  
Heat And Mass Transfer ◽  
Viscous Dissipation ◽  
Convective Heat ◽  
Vertical Channel ◽  
Skin Friction Coefficient ◽  
Physical Parameters ◽  
Soret And Dufour Effects ◽  
Constant Suction ◽  
Dufour Number

The present paper investigates the combined effects of Soret and Dufour on free convective heat and mass transfer on the unsteady one-dimensional boundary layer flow over a vertical channel in the presence of viscous dissipation and constant suction. The governing partial differential equations are solved numerically using the implicit Crank-Nicolson method. The velocity, temperature, and concentration distributions are discussed numerically and presented through graphs. Numerical values of the skin-friction coefficient, Nusselt number, and Sherwood number at the plate are discussed numerically for various values of physical parameters and are presented through tables. It has been observed that the velocity and temperature increase with the increase in the viscous dissipation parameter and Dufour number, while an increase in Soret number causes a reduction in temperature and a rise in the velocity and concentration.

scholarly journals The rate of evaporation of droplets. Evaporation and diffusion coefficients, and vapour pressures of dibutyl phthalate and butyl stearate

1946 ◽  
Vol 186 (1006) ◽  
pp. 368-390 ◽  
Keyword(s):  
Diffusion Coefficients ◽  
Atmospheric Pressure ◽  
Dibutyl Phthalate ◽  
Droplet Evaporation ◽  
Medium Size ◽  
Good Accordance ◽  
Butyl Stearate ◽  
Rate Of Evaporation ◽  
Low Pressures ◽  
And Diffusion

The rate of evaporation of drops of dibutyl phthalate and butyl stearate of radius approx. 0.5 mm. has been studied by means of a microbalance over a range of atmospheric pressures down to approx. 0*1 mm. of mercury. Wide departures from Langmuir’s evaporation formula were found to occur at these low pressures, but results are in good accordance with the theory of droplet evaporation advanced by Fuchs which hitherto has not been tested experimentally. This experimental verification of Fuch’s theory for droplets of medium size evaporating at low pressures shows that the theory can be applied to the evaporation of very small drops at atmospheric pressure. The vapour pressures of the above liquids have been measured by Knudsen’s method and the evaporation and diffusion coefficients calculated fro n the experimental data.

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