A great deal of interest was generated recently in the possibility of producing new materials in the reduced gravity environment provided during the forthcoming missions of Spacelab. The range of possibilities extend from producing large crystals of uniform properties to manufacturing materials with unique properties. Most of these processes involve the solidification of materials from the liquid state. Convective motions within the liquid during solidification can influence the local material composite and the shape of the solid-liquid interface which may result in solids with non-uniform properties and crystal defects. The microgravity environment of Spacelab is being viewed as one in which the buoyancy forces are eliminated so that convection driven by thermal gradients does occur, resulting in an improved solidification process. However, convection may occur for other reasons and whether convection is negligible or not during solidification constitutes processing in low-gravity environment. Little information exists presently on convection during solidification under such circumstances. A continuation of an analytical investigation into the nature of convective motion in a binary liquid layer due to surface tension forces during its solidification is reported. The onset of convection will be determined through a stability analysis which is described

Antar, B. N.

Collins, F.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19840003163 2020-03-21T00:40:27+00:00Z
'. .
(ui8-CM- 170916) STODISS Of Co84rMON 18 I	 Wt-11231
Soubl"In -GUUT U1288E AT S289CSD
""Its iiaoBtlly 8epott, 21 Jul. 1982 - 11
IMF IS" eTMae^es ,' AV . Space last. 	 Oacla s
=aLlaboma.t 10 p AC 102/ap A01	 CSCL 07D 23/25 48522
UNIVERSIT:' OF TENNESSEE
SPACE INSTITUTE
TULLAHOMA, TENNESSEE 37388
May 11, 1983
"STUDIES OF CONVECTION IN A SOLIDIFYING BINARY
MIXTURE AT REDUCED GRAVITY"
Reporting Period
	 Z1^
July 21, 1982 - May 11, 1983
0%4 1°^
Bi-Monthly Report No. 7	 aECEty^^^(
Contract NAS8-34268
_a
Drs. B. Antar and F. Collins, Co-Principal Investigators
PREPARED FOR GEORGE C. MARSHALL SPACE FLIGHT CENTER
MARSHALL SPACE FLIGHT CENTER, ALABAMA 3.0
INTRODUCTION
A great deal of Interest has been generated recently In the possibility of producing new materials
In the reduced gravity environment provided during the forthcoming missions of Spacelab. The range
of possibilities extend from producing large crystals of uniform properties to manufacturing materials
with unique properties. Most of these processes involve the solidification of materials from the liquid
state. Convective motions within the liquid during solidification can influence the local material
composite and the shape of the solid-liquid interface which may result in solids with non-uniform
properties and crystal defects. The microgravity environment of Spacelab is being viewed as one in
which the buoyancy forces are climinated so that convection driven by thermal gradients will not
occur. It is hoped that this will lead to an improved solidification process. However, convection may
occur for other reasons and whether convection is negligible or not during solidification constitutes
processing in low-gravity environment. Little information exists presently on convection during
solidification under such circumstances.
The work reported here is a continuation of an analytical investigation into the nature of
convective motion In a binary liquid layer due to surface tension forces during its solidification. The
onset of convection will be determined through a s. ability analysis which is described below.
STATEMENT OF THE PROBLEM
The occurrence of convective motions in a til I has been studied both theoretically and ex-
perimentally for approximately a century. The problem is very well documented in several looks
and numerous articles with all of its variations. It is obvious that In order to study analytically
the detailed convective motions in a fluid in any configuration requires nothing less than the total
solution of the Navier-Stokes equations and Energy Conservation equations. This is well known to
be a formidable, although not an impossible, task requiring considerable amounts of time as well as
financial resources.
In certain applications, it is sufficient to be able to know whether or not under certain conditions
a fluid could sustain convective motions. The answer to such a question requires far less work;
but, again, the information is essentially just of a binary form. Such information can be obtained
through hydrodynamic stability analysis. Essentially one introduces perturbation into a well-known
basic state and studies the evolutions of these perturbations in time. If the perturbations decay with
time, then the basic state Is said to be stable and no convective motion will ensure. If, on the other
hand, the perturbations are found to grow with time, then the basic state is said to be unstable
and convective motion will rake place. Fundamental to such a stability analysis is the existence of a
stationary basic state. Unfortunately, the problem under consideration does not have such a basic.
However, it has been shown that it is possible to carry out a meaningfull stability analysis of such
a basice state through several techniques. We have chosen the energy stability method as the best
suited for the problem under consideration. This technique is elucidated in detail in the monographs
of Joseph 111.
In this work, we consider the stability of a binary fluid layer which is being solidified from below
and has an upper free surface. The fluid layer Is assumed to be of infinite extent in the horizontal
direction. Since It is assumed that the process is being carried out in the low 'gravity'environment of
Spacelab, It Is anticipated that the driving force in the surface tension force at the free surface. The
solution to the basic state has already been obtained In a previous report (Antar (21). As is expected,
the basic state is a function of both time and space. Thus, the stability analysis used must account
ORit NAL PAN It
OF POOR QUALITY
for the variations of the basic state with time.
The stability analysis starts with the perturbation equations which may be written in the
following form:
au
et +
u•VU + U.VU +u •VU=— .Vp +VV 2V + gL8ie^-027Ik	 (1)
F+ u•Vry+U•V ey+u•VC =DV2ry 	(2)
of
9t +U•V$+u•V8+u•VT= KVa8 	 (3)
where u (u, v, w) is the perturbation velocity vector, U (U, V, W) is the basic state velocity vector.
and C are the perturbation and basic state concentration and B and T are the perturbation and
basic state temperature, respectively. v, and D and x are the kinematic viscosity, thermal diffusivity
and the solutal diffusion coefficient, respectively. These equations are for a Boussinesque fluid and
subject to the following boundary condition
U=0=7=0
on the lower surface.
While at the upper surface, we must have the following boundary conditions:
k'9z = —q9
	 (4a)
8ry = 0
	 (4b)
8x
µ( 8z + 8z ) 8x	 (4c)
µ( 8y + 8z ) ey	 (4d)
where o is the surface tension. On all the sidewalls we must have the perturbation function vanishing.
The governing equations are first non-dimensionalized using the fluid depth d as a length scale,
d2/x for a time scale, k/d for a velocity scale, T, — T. and Cl — C. for temperature and concentration
2
In
ORIGINAL PAGE
'`I
OF POOR t^JA
scales respectively. Upon introducing these scales into equations (1) - (4) we get the following
equations for the perturbations functions:
8t+u•Ott+U•Vu+u•VU=—Vp +PV2u+P(R6+LRe'I)
	 (5)
0i +U•VB+u•VB+u•VT =V 20	 (6)
8t +U• V'7+u•V'ry+U. VC =LV2ry	 (7)
with the following boundary conditions:
0=u=ry=0	 (8)
on all side except the upper surface and
of8z + B9 = 0	 (9a)
6z 
= 0	 (9b)8a
8u	 888'r _
8z + MeZ + M`L8x 0	 (9c)
8z + Mey + M°L8y 
= 0	 (9d)
at the upper surface. In these equations we have the following non-dimensional numbers.
P = Prandtl No. = K
R — Rayleigh No. = #Ig6T Vic
Re = 3olutal Rayleigh No. = gg26C^6*
L = Lewis No. = d	 -
B = Blot modulus = 4d
M = Marangoni No. (=HqF-)STdµic
3
Ir
no
ORHANAL PAGE 19
OF POOR QUALITY
Me — Solutai Mo	 ni No.Ta go 
when ff and if are the constants from the surface tension variations with both temperature and
concentrations.
Now to obtain the energy equations we first take the dot product equation (5) with u and
integrate about the volume under consideration to get
dt J 
P— I q2/2dv = — I V u : V udv + R J w8dv + LRe I wrydv
—
M L, B 8s dady — MeL Ll  ry 8s 
dxdy	 (10)
Similiarly It Is possible to obtain equations for 82/2 and ry2/2 in the following integral form
dt f 0/2dv = — f wB 8^ dv —Bj ^ l Badxdy —fV V 8 • V$dv	 (11)
d f ry?/2dv = — 1 w.y'C — L f V ry 
V ydv
	
(12)
dt v	 v	 8z	 v
Now multiplying equation (11) by a, and equation (12) by X j and adding equations (10) - (12) we
Pt
dtlf	 f.
=—D +[(R — X t 8z )w9 + (ReL — X r 8z )wy)dv
-- 
1 
=1[M8 + Mc" 8 dxdy	 (13)
where
IC — 2 J (^192 -t' > r0
2
 + )"'ya)dv
D= J [V u: V u + X rV $• V 8+ a,yLV ry• V ryJdv + B j l X,82dxdy
Equations ( 13) can be cast in a symmetric form with the following change of variables
	
"'	 ^=^^i^y=Try
	
_^	 4
1ORIGINAL €'A IM 1s
OF POOR QUA -
R— MN; Re=MeNe
which after dropping the primes takes the following forms:
dK =—D-f-^II
wtste
K = z 1(P-19s + e2 + ?)
d
 v
D= f(VU: u + Vd•vB + Vry •ory)dv+B L
I
dzdy02
II — a0µ le/2(f C8wfdv —
 fx I 8 8z dxdy)
+a,,p.1 /2(f Crywrydv — f 
.1 ry 8z dxdy),
and
as =
0,1 = M^
C#=N—p#OT
C7=LN`•—k78z
(14)
X —AS+MC
5
ORIGMAL PACE tR
Now let us focus our Mention on Eq. (14). Since then, the energy of the perturbation will
Inerease or decrease with time depending on whether v/X—II/DjO. Of course the direct and only
wW of gaining this Information Is by solving the Integro-diRerential system explicitly. However Eq.
(14) can be written as an inequality In the following form:
1 dt —1+^II ^	 —I+^	 (15)
D dt	 D —	 p
where
	
P-1 : FM
	
(16a)
with
D =1	 (16b)
where now the problem is cast as a variational problem defined by (15) and (16).
Let us introduce the Iagrange multipliers 11^,,(t) and p(z, y, z, t) Into the variational problem (15)
• (16) with the following optimization, procedure resuting
M (V • u)dv — M D) = 0	 (17)O	 M
Rom variational calculus we know that the solution to (17) is the same as the solution of the
Euler-Lagrange equation resulting from (17). For the present case, the Euler-Lagrange system of
equations are the following
O— V2u = 0	 (18)
8p —Vw=O	 (19)y
i MNCsB+ ^7 , Cryr F. + O2W.0	 (20)
M" as Csw + V38 = 0	 (21)
z f0
'6
au
FS `^ a j.asM 0	 (23)
with the following boundary conditions
8u +1 M^ ae 88 + 1 M, a7 t = 0	 (24a)K 2 ^e 8x 2 ^,r ax
8v + 1 M^, ae 8, + 1 M^ a7 8ry = 0	 (24b)Sx 2 )—p# By 2 f7 ay
BB + as + 2 MJ, at as = 0	 (24c)
^e
8ry + 1M 87 aw s 0	 (24d)
8x 2 r vp7 as
and z = 1 and
u=v=waB=q=O	 (24e)
atz=0.
Now as discussed earlier since the basic state itself a function of time then the term H in the
energy equation (14) is a function of time. Also, the lagrange multiplers and the term in (16a) are
also function of time. Thus, the variational problems (15) - (16) must be solved at each instant of
time which requires the solution of the Euler-Lagrange equations at each point in time. Thus, the
procedure is to solve the system ( 18) - (24) at each instant of time and obtaining the optimization
parameter M1, (t) as a function of time.
REFERENCES
1. Joseph, D. D., Stabl!!j of Fluid Motions I and H. Sprinser 'Facts in Natural Philosophy,
Vol. 27 and 28, Springer Verlag, Berlin Heidelberg ew ork, 1978.
2. Autar, B. N., "Solidi>lcation of a Binwy Mixture" in
Rammh Reports - 1981 NASA/ASEESumr ,.*r Faculty Fellowship Program. Edited by A. Karr,
^^ 'r "' 161W5. 1982.
lbd
i
r ~
M
1
W
OO N
V
IA N
W W
9 M
x c
4c ^
0. w
v v
W x u!r	 Ir_
a K LL
h 0. W
► .
z
.J
nN
LW0N
U
WhU.
O
h
n
Y
ui
z
uJ
L
WM
N O
^p
^ eVVply
aN GCX
CK
W
W X
1
z ►-
r z
l_ l	 A
ai
u
rn
W
QC
r
Q
z
W
Q.
}C
W
2
V
W
N
W
W
a
O
N
Z
U
a
N
• •	 s	 i	 i t "	 _ /
F,
$ aRits^NAL PAGE t..
OF POOR QUALITY
• V4 A w Nco fD p .N-7 Nl "den
H
•
Q^
S
{!pg p
w
!^
•
N ^ P >,D ^ @ 10
a
V 4i ID ^
G r N N
w .yfh N mF) Na^ppt^) M A •	 •M r wt^pp1rf N
^•
•1t Zn
.
pO. 8 tOV N N N $N' p0 2 u O821 p$N
OQd N
0-
Z 4 SO 80 OS 00 88 OS SS OSWx p 40 00 00 00 00 00 00 00a
W G 8° p80 0OO 8.0. So O Q S° 8°
.j O ON Ob O^N^pp 0N^ pN ON ON
a N N N N N t^DJ O^^NppN N
N
8 Sc 8 gr8 8^
QN
th2.1
MM ON AN ON pqj ON ON OCJ
r 8 88 88 8c°^ 08 08 o c°J. 8`0' So
.h. W p 00 00 00 00 00 00 00 00J $
M N
v D
U.
O 1•')p OQ rd't fdi 8 OdA1'f Cq NOd f^01w &n C3 p.vO•N
tJ. M
O
0 003 b N W .a-• n P^'f •4.1IJ_C f"f ^'f m K1 co
co
of Cc •n rl a 1. SAM
h w In .r) Ih 1"! Ch In ^. .o
m
S V .na$ N a KOI Y'! U1 atW t"9 d enfT M
!O i'1 M .A ^ 4•V h t!N
t\ d rfppt^ 1n .+!n O v wO^ J"f Is dO v M M•n La •'• !p•"• O OI .ON Kf NM A
CY W N
(y
1 W W .D Y'! CT U1 Q! q N i0 N O.
0
=0 men
K O M N ••• Pi w U .-. b nr A
in }
^_C) O cf O -;t NO W N EO d Ki N MO %V d !
'•: N b i^.ly1 N fl  d en W a Mc
r
d N CM•9 M OM• .p Uf
u^ NN .V 4T NN bO• M/'1 AV Wf^ WA
^: t c M i ^. u'i w
.^•,
w H .•+ '-^ N '.+ N •-r N M N
lU Na
ggg N •.
f:
lei
N W
A :C
N t.J
-
.^-W
^,C
W
t
^- W
NI- i •'! 1-+M• •f i"1 r•to /"1 h-X'^1-4!
>! i9 .Ll 1 • LJ G7
O © OJ •1: ft C3
yp.(
i1
. l.1
.JtTJ'
Lm] L)
•S 1-
1
Y+ , W
F-
- L 1 C1U N
---
C) L,i- 1-- t7C.l !?
^1	 C: -
• /
all•J Q4. i-1r
iYy A}.. l+
a
Ni -P


Studies of convection in a solidifying binary mixture at reduced gravity

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server