non iso thermal spreading
TRANSCRIPT
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The Spreading of a Non-Isothermal
Liquid Droplet
Steven W. Benintendi
Marc K. Smith
The George W. Woodruff School of Mechanical Engineering
Georgia Institute of Technology
Atlanta, Georgia 30332-0405
Abstract
The effect of the slip coefficient and the mobility capillary number on the spreading of a thin
axisymmetric liquid droplet with uniform heating/cooling of the solid surface is examined. The results
show that increasing the slip coefficient reduces the spreading/shrinking behavior of the droplet and that
the final equilibrium states are slip dependent. These results are explained by the development of a return
flow inside the droplet. We show how a speed-dependent slip coefficient can be used to remove the
dependence of the final state on the slip coefficient. It is also shown that increasing the mobility capillary
number decreases the spreading/shrinking rate of the droplet. For thermocapillary-driven droplets, there
is a capillary-number-dependent time delay for the onset of motion. The entire effect of the mobility
capillary number on the spreading process is explained in terms of the deformability of the free surface.
PACS: 47.55.Dz, 68.15.+e, 68.10.Cr, 68.10.Gw
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Introduction
The spreading of a liquid over a smooth solid surface is a complicated free-boundary problem
characterized by the presence of a moving contact line. A contact line is formed when three immiscible
material phases are in mutual contact, such as when a water droplet spreads on a glass substrate in an air
environment. This fundamental spreading process is of vital importance in a host of technological
processes such as coating, materials processing, and film-cooling applications. The motion of the liquid
during the spreading process is controlled by the fluid dynamics occurring in the immediate neighborhood
of the contact line. The importance of this contact-line region and the difficulties in modeling it have
been reviewed by Dussan1
and de Gennes2
. The main modeling difficulties stem from the fact that the
exact physics governing the behavior of the fluid in this microscopic region is unknown. Nevertheless,
several ad hoc assumptions can be made about the contact-line region and its effect on the overall motion
of the droplet in order to make the problem tractable in terms of continuum theory.
In classical fluid mechanics, the accepted boundary condition between a fluid and a solid surface is
the no-slip condition. However, in moving contact-line problems, the application of no-slip gives rise to a
non-integrable shear-stress singularity at the contact line3 4,
. Several alternative methods have been
suggested to overcome this singularity, such as using a non-Newtonian description of the fluid near the
moving contact line or relaxing the no-slip condition by allowing the fluid to slip along the solid surface.
Imposing a slip law in the vicinity of the contact line is the most common method used to remove the
singularity and has been used extensively in spreading film and droplet geometries5 10
.
Under static conditions, the solid-liquid-gas interactions in the vicinity of the contact line are
accounted for by prescribing the value of the contact angle, thereby providing a boundary condition for the
interface shape. As a natural extension to the static case, it is appealing to use a dynamic contact angle to
describe the interactions in the contact-line region under dynamic conditions. To this end, one approach
is to assume that the microscopic contact angle is always equal to its static value even under dynamic
conditions. In this approach, a local analysis near the contact line is used and matched to an outer
solution using asymptotic techniques. This approach has been taken by Hocking6 7,
, Lowndes11
, Cox12
,
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Dussan13
, and others. Another approach is to assume a constitutive relation between the apparent
contact angle and the contact-line speed. Here the constitutive relation connects the mathematical
description of the outer region of the flow field to the contact-line speed. The precise micro-physics in the
contact-line region is contained in this relation and so a detailed analysis of this region is not required in
order to compute the flow for the bulk droplet. This approach is supported by the experiments of
Schwartz and Tejeda14
, Hoffman15
, Tanner16
, and Chen17
that demonstrate some dependence of the
dynamic apparent contact angle on the speed of the contact line. Greenspan5
, Ehrhard and Davis8
,
Haley and Miksis9
, and Smith10
have all used this approach to successfully compute the bulk droplet
behavior.
The spreading of an axisymmetric liquid droplet on a solid surface is a classic example of a geometry
that incorporates a moving contact line. Greenspan5
was the first to examine the spreading of an
isothermal axisymmetric droplet due to a non-equilibrium initial shape. He adopted an analytical model
using a Maxwell slip law and a linear relationship between the apparent contact angle and the contact-line
speed. Using lubrication theory, a single evolution equation was derived in terms of the droplet shape and
was coupled to the dynamic contact-line boundary condition to describe the bulk motion of the droplet.
Under the limit of a small mobility capillary number, an asymptotic solution was found that described the
quasi-steady behavior of a spreading/retracting liquid droplet. Haley and Miksis 9 considered this same
spreading problem but with a goal of investigating various contact-line models. They did not invoke the
small mobility capillary number limit, but instead solved the full transient problem numerically. Their
results showed the dependence of the droplet spreading rates on the mobility capillary number, the
mobility exponent, and various formulations of the slip coefficient.
Ehrhard and Davis 8 considered non-isothermal effects by investigating an axisymmetric liquid
droplet spreading on a uniformly heated/cooled solid surface. As with Greenspan5
these researchers
considered the quasi-steady limit of a small mobility capillary number and rightly set the slip coefficient to
zero. They showed that this type of thermal forcing induces a thermocapillary-driven flow that alters the
final equilibrium radius of the droplet. Since thermocapillarity causes fluid particles on an interface to
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move from a hot region to a cold region, a heated solid surface establishes an interfacial flow from the
contact line toward the droplet center that decreases the final equilibrium radius of the droplet with
respect to the isothermal case. On the other hand, when the solid surface is cooled, an interfacial flow is
established from the droplet center toward the contact line that increases the final equilibrium radius of
the droplet with respect to the isothermal case.
In the current work, we shall extend the work of Ehrhard and Davis 8 and Haley and Miksis 9 by
examining the effect of the mobility capillary number and the slip coefficient on the transient spreading
behavior of a thermocapillary-driven axisymmetric droplet. Whereas Ehrhard and Davis8
performed a
quasi-steady analysis emphasizing the final spreading radius as a function of the thermal forcing
parameter, we perform a fully transient analysis to determine the effect of the physical parameters on the
spreading rates under non-isothermal conditions. This essentially extends the work of Haley and
Miksis9
to non-isothermal spreading. These thermal effects produce some interesting and counter-
intuitive results that we shall explain.
Problem Formulation
We follow the formulation of Ehrhard and Davis8
by considering the motion of an axisymmetric
liquid droplet on a uniformly heated/cooled solid surface as shown in Fig. 1. A cylindrical coordinate
system with the radial direction embedded in the solid surface and the z-axis normal to the solid surface is
used. The droplet is composed of an incompressible Newtonian liquid with the density , thermal
conductivity k, dynamic viscosity , specific heat cp , and the unit surface thermal conductance hg all
considered constant. A passive gas at a temperature T surrounds the droplet from above while a solid
surface held at a constant temperature T0 bounds from below. The velocityrv u w= ( , ) , pressure p, and
temperature T in the droplet are governed by the continuity equation, the Navier-Stokes equations, and the
energy equation,
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u u wz r= + ( ) , w = 0 , T T= 0 , (5)
where is the slip coefficient. Whereas Haley and Miksis 9 considered slip coefficients that were
constant, inversely proportional to the droplet height, and inversely proportional to the square of the
droplet height, this analysis will only consider constant slip coefficients.
The contact line is located at the point r c t= ( ) with the contact condition and the value of the
apparent contact angle given by
h c t( , ) = 0 , h c t tr ( , ) tan ( )= . (6)
We shall use a power-law form for the dynamic contact-line boundary condition that relates the apparent
contact angle to the contact-line speed Ucl ,
( )
( )U
K
Kcl
A
m
A
R
m
R
= >
0 , the solid surface is uniformly heated, and if $M < 0 ,
the solid surface is uniformly cooled. In this analysis we have assumed that $M is an order-one
parameter.
The boundary conditions and other constraints applied to the evolution equation (17) are zero film
thickness at the contact line, zero slope at the center, and constant volume:
( )h c t, = 0 , (19)
( )h tr 0 0, = , (20)
( )h r t r drc
,
0
1
2 = , (21)
The scaled constitutive relation governing the contact-line speed is
( )
( )ct
A
m
A
R
m
R
= >