non iso thermal spreading

8/3/2019 Non Iso Thermal Spreading

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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

= >

non iso thermal spreading

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