Torsional oscillations of a rod in a layered medium of simple fluids

ht. J. Engns Sci. Vol. 24, No. 4, pp. 631-640, Printed in Circa* Britain.

TORSIONAL LAYERED

1986

0020-7225186 $3.00 + .OO 0 1986 Pcgrmcm Ras Ltd.

OSCILLATIONS OF A ROD IN A MEDIUM OF SIMPLE FLUIDS?

AYDENIZ SIGINER Department of Mechanical Engineering, Auburn University, Auburn, AL 36849, U.S.A. Abstract-Free surface flow of a layered medium of simple fluids driven by a torsionally oscillating, cylindrical rod is investigated. The stress, expressed as a series of multiple integrals of polynomials in the strain histories, is expanded into a Fr&het series in terms of the amplitude of the oscillation of the rod. A domain perturbation method is used to perturb the rest state. The mean shapes and deviations from flatness at the rod of the interfaces between layers are determined together with the flow field up to and including second order in the amplitude of the oscillation. 1. INTRODUCTION IT IS POSSIBLEto

deduce information about the constitutive structure of a non-Newtonian liquid by studying the deformation the free surface of the liquid undergoes in a motion driven from the boundary with steady or unsteady data. In this context Joseph and Fosdick [I] suggested the use of the Weissenberg effect between rotating concentric cylinders and in the limit of the infinite outer radius its use in the rotating rod geometry. Joseph and Beavers [2] extended the study of the Weissenberg effect to the case of the unsteady data when the liquid is confined between two cylinders undergoing torsional oscillations. At the second order in the perturbation series there are two material functions, the shear relaxation modulus G(f) and the quadratic shear relaxation modulus -y(ti, t2). At the lowest order they appear, they are enough to determine the response of a simple fluid of integral type in all motions of small amplitude. It is assumed that G(t) > 0, Vt and that the functional representations of both G(t) and -r(tl, t2) decay rapidly in .time. Joseph and Beavers approximated the shear and the quadratic shear relaxation moduli by generalized Maxwell models and by combining theory with experiments in the limiting case of very large outside radius they determined some of the constants which appear in the representations of G(t) and ~(t,, t2). The use of the oscillating rod geometry versus the concentric cylinder geometry is very much to the point as shown by Siginer [3, 4, 51 and Siginer and Beavers [6]. In the concentric cylinder geometry even the slightest eccentricity gives rise to additional Newtonian effects of large magnitude completely overwhelming the non-Newtonian behaviour of the free surface in the case of liquids which exhibit small normal stress effects. In this paper we study the free surface motion of a layered medium of simple fluids driven by a rod undergoing torsional oscillations of small amplitude. The frequency of the oscillation may be quite large. The flow variables are expanded in a power series in terms of the amplitude of the oscillation which perturb the rest state. The solution requires the computation of the canonical forms of the Frechet stresses which arise at different orders in the perturbation analysis pivoted around the rest state. Frechet stresses are defined as the forms of the extra stress at different orders, obtained by differentiating the extra-stress functional with respect to the perturbation parameter and evaluating the resulting expression on the rest state. The inherent difficulty of the unsteady problem is the monitoring of the history of the deformation. The calculation of the histories in the integral representation of the stress requires the knowledge of the substantial derivatives of the position vector of a given particle and of the appropriate Rivlin-Ericksen tensors evaluated at all past times. It is found that at first order the flow fields are completely decoupled in each layer and that the particles move in circles with no contribution to the pressure field and height ‘I Parts of this paper presented at the 20th Annual Meeting of the Society of Engineering Science, University of Delaware, Newark, Delaware, U.S.A., August 22-24, 1983, and at the XIIth Southeastern Conference on Theoretical and Applied Mechanics, Callaway Gardens, Georgia, U.S.A., May 10-I 1, 1984.

632

A. SIGINER

rise. At second order the motion is decomposed in each layer into a mean term and a time-periodic term oscillating with frequency 20, where o is the oscillation frequency of the rod. At second order there is no flow in the mean in each layer and a nontrivial pressure field exists in every layer. Like the velocity fields in the layers, there is no interaction between the mean shapes of the interfaces between superposed layers. We determine the shapes in the mean of the interfaces and the mean up or down climb of each interface at the oscillating rod. The oscillatory part of the motion and of the deformation of the interfaces is presently under study and will be reported in a later paper. 2. MATHEMATICAL

FORMULATION

We consider the flow configuration made up of the three layers of air and two simple fluids set in motion by a slender rod oscillating with amplitude 6 and angular frequency w, Fig. 1. The physical problem is mathematically defined by the field equations P,$f=-V4j+V’S’

&j =

J,

Pj + PjfX

V-u=0

j = 1, 2, 3;

(2.1)

inu,,

vg = {I; 8, zlrO S r < co, 0 I8

< 2a, -co < z < co},

the boundary and asymptotic conditions u(rO, z, t) = eehro sin wt n(r, z, t),

4j
u(r, z, 0 - 0,

NT,

t),

(2.2) Z-++CQ,

4jtr9 t),

(2.3b)

r-co,

4jCr9 z~ t) --) 4(Z),

(2.3a)

and interface conditions W =

hj,, + uhj,r

j=

(&r@)j = <&@>j+l3

1,2;

u = u-e,,

h=u.e,; j=

Csm>j = (snt)j+l

1,2,

(2.4a) (2.4b)

with

St, = &I - h,Sre,

(2Sa)

S,, = h,,(Sz, - S,,) + (1 - hf,)&.

(2.5b)

INTERFACE

INTERFACE

Fig. 1. Oscillating rod in a layered medium.

1

2

Torsional oscillations of a rod in a layered medium of simple fluids

633

We note that the interfaces are flat away from the rod h*(co, 0, t) = h*(co, -5

t) = h,,,(m, 0, 0 = hZ,r(% -z

0 = 0,

(2.6)

and that at each interface the jump in the normal stress is balanced by surface tension nj -

[-Jll

Sj] .

+

tlj

=

j=

gjJi

1,2;

j no sum,

(2.7)

with aj and Jj representing the interfacial value of the surface tension and the mean curvature at each interface. The indices in (2.1) represent the layer of air and the two superposed layers of simple fluids as indicated in Fig. 1. The extra stress Sj in each layer is given by S1 = pA, = 2pD, Sj

=

,,m

~(s)G(s,

A)ds

+

p

r

{@jCSl,

Q)G(~I,

(2.8)

A)G(s2,

A)

s

sss m

+ aji(Sl, S2)[Tr WI, NlG(s2, N@b2

+

0

al

0

cc

0

. ’ *3

(2.9)

j = 2, 3,

II, A, and D represent the dynamic viscosity, the 1st Rivlin-Ericksen tensor and the stretching tensor, respectively. Upon taking N terms in the representation (2.9), where the Nth term is an N-tuple integral of polynomials in the history G(s, A) of the motion, we obtain the representation of the extra stress for a fluid of order N, Coleman and No11 [7]. sj, @j, aj are the material functions specified for each simple fluid. It can be shown that if the strain history is expanded into a series G(s, A) = AGi(s) + A2Gz(s) + A3G&) + o(A4),

(2.10)

the extra-stress functional (2.9) may be reduced to Sj = A %

h(S)cij(s)dr al

+ A2

ss0

j = 2, 3.

ss0

c(SK&j(S)ds

m 0

cc + A3

+ A2 % 1

Pj(si 3 s2)Gljtsl )GI j(s2Mlb

00

0

aj(Si 3 sd[Tr Gj(SI JIGIj(S2)dsldrsz+ o(A3), (2.11)

j no sum.

The coefficient of the material function aj(Si, 32) is o(A3) because for small strains, G,(s) in (2.10) may be shown to be given by AGl(s) = 2[D(t - s) - D(t)], where D is the stretching tensor to be evaluated in a fixed reference configuration, instance the rest state ~0. Then for incompressible materials Tr G,(s) = 0,

for

Tr G(s, A) = o(A2).

These results, originally due to Coleman and No11[7] and Pipkin [8], respectively, combine to show that the coefficient of the material function aj in (2.1 I) is o(A’). 3. SOLUTION

FOR

THE

FLOW

FIELD

The algorithm developed by Joseph [9] for the study of the unsteady motions of order fluids will be used. The algorithm is based on an expansion of the extra stress in a Frkhet

634

A. SIGINER

series in terms of the amplitude A of the oscillatory motion. Used in conjunction with the theory of domain perturbations which allows the problem to be solved in the undisturbed rest state v. with flat interfaces and then continued analytically to the physical domain vA, the algorithm yields a succession of linear problems. In the context of the theory of domain perturbations field variables are formally expanded into Taylor series in powers of A where the coefficient of each term is the partial derivative with respect to A evaluated in the rest state v. and indicated by ( )@, i = 1,...,a). l

3.1 Solution at 1st order At 1st order we obtain in the successive layers

,o,us’)= p,A&),

V-u(‘) = 0

c vo,

pju$‘) = V.S(‘) J )

V.“W

f vo

= 0

(3.1) j = 2, 3,

uQ)(r, 2, t) = ego sin cot, uQ)(r, 2, t) -

uQ)(r, t),

u(‘)(r, z, t) -

0,

w(‘)(r, 0, t) = w(‘)(r, -2,

(S@),” = C&e)%

(3.3)

2-+CXl,

(3.4a)

00,

(3.4b)

r-

t) = hi”) 5 0,

(&)j’)

(3.2)

= (&I$~,

j=

1,2,

(3.5)

j=

1,2,

(3.6)

where vois the rest domain with flat interfaces between layers. Introducing the shear relaxation modulus S;(S) = [dGj(r)]/& in (2.11) and integrating by parts it is easy to obtain $1) = s0

m Gj(S)A$“(s)ds ,

j = 2, 3,

(3.7)

where A, is the first Rivlin-Ericksen tensor. In cylindrical coordinates (3.1) becomes

(

p,u$‘)= pl erd, +

T f3, +

1

e& (u(‘)e, + u(‘)ee + w(‘)e,).

(3.8)

We note that the radial and vertical components of the velocity are even functions of A. Consequently u(l), w(I) are zero at first order. We seek solutions of the following form: u(l)(r, t) = e&‘)(r,

t),

(3.9a)

v(l)(r, t) = R(r)qt)

= rW(r)e’“’ + rW(r)eP’.

(3.9b)

From (3.8), (3.9), (3.3) and (3.4b) we obtain

4, +

1r R, -

&(rW R(r) = r. KhA)

R = 0,

’

R(m)

R(ro) = ro,

W(r) = F

,

Ag!!?.

= 0,

(3.10a)

(3. lob) ccl

Asymptotic conditions (3.4a, b) are identically satisfied. We turn now to the layers of simple fluids. With the representation (3.9) for the velocity field, (3.7) becomes

Torsional oscillations of a rod in a layered medium of simple fluids

and the momentum

635

balance (3.2) is transformed into

$[r(

pjU$‘)=

l

Gj(S)U")(r, t - s)~s, 5 OD

),l],r -

j = 2, 3.

(3.11)

The velocity field (3.9) and (3.11) yield a modified Bessel equation of order one, similar to (3.10a) with Ai’ =

‘piw Vjli(W) ’

b(W)

s

=

j = 2, 3,

Gj(S)e-‘““ds,

*

0

(3.12)

where q(w) is the complex viscosity, the solution of which is given by (3.10b). Now we consider the coupling conditions (3.6) between layers. These represent the continuity of the shear tractions at the interfaces. We recognize that because of the particular form of the velocity field (3.9) both conditions (3.6) are identically satisfied at both interfaces. There is no coupling of the velocity fields at the 1st order. Turning now to the stress jump condition (2.7) at either interface we get J’(I)=f[(l

+r%,),,Z]l”_O,

I--PZ+ @M”) 172

- @2M”) + pi’)

1,2,

- [-PI + ~r,(Dd,,l(‘)= 0,

(I) = Pi’) Pl

-

j=

at

z = 0,

- t-L+ + @dn”P z=-2.

at

= pi’)

= 0,

3.2 Solution at 2nd order At second order we obtain the following problems in successive layers: P,u$2) - &u(z)

&42>

-

v.

V.“(2)

= -V@$2) - &‘).

V.“(2)

= 0,

SW J

= -v(pp

= 0,

uQ)(r,

z, uQ)(

t),

4 (2) (I,

r, z,

t) -

z,

t) -

0,

pj”o>.

-

z,

uqr,

f#P)(r,

z,

(3.13)

V”W,

(3.14)

f vo

j =

6 vo,

t&2+0,

Vu(‘),

t) =

0,

t),

@(r,

t) -

2, 3, (3.15) Z-fC0,

t),

r-a3

p(z),

(3.16b)

w(‘)(r, 0, t) = #1>, w(‘)(r, -Z, (S”di’2)

(3.17)

t) = hi;‘,

(S,*),(2)

= (sdj~i,

(3.16a)

(3.18)

= (S Kl f2) J+I,

j=

1,2.

(3.19)

In the Newtonian air layer the motion is driven at 2nd order by 2 p,"w.vuw

=

_p,e,l)o

r

(3.20)

To solve (3.13, 3.15, 3.16a, b, 3.17, 3.19) we note that at 2nd order the component of velocity in the azimuthal direction is zero because it is an odd function of A. uc2)(r, z, t) is an axially symmetrical field and we introduce a stream function +c2) p

u(2)= V/leg -

r

.

(3.21)

636

A. SIGINER

Using (3.20) and (3.21) we transform (3.13) into + e Ip,r[21W,]Z + W$z*‘~*+ ~~eVziw’].

= -V@

(3.22)

The right hand side of (3.22) suggests that the motion is driven by a time-periodic term oscillating with frequency 2w and a time independent term. Following Joseph [2] we adopt the representations #)

= *jm + +je*ior + qje-*iwr,

j=

4(*) = 4j)im+ 4$iwt + &e-*ior I J 3 J

1,2,3.

(3.23a)

j = 1,2, 3.

(3.23b)

for the field variables. The indice (m) represents time-independent average quantities, averaged over a cycle of period 27r/w or mean quantities and the overbar indicates complex conjugates. We note that the curl of (3.22) eliminates completely the right hand member of the equation and substitution of (3.23a, b) results into the following equation, tier some manipulation, for the mean stream function at 2nd order for the layer of Newtonian air (3.24)

The conditions (3.15) and (3.16a, b) yield rtdr0,z)

= h,Ar0,4

=

hdao, 4 = ~L,,(ab 4 = 0, 00) = 0.

hno-, ao) = h,rk

(3.25)

To obtain the conditions at the interface between air and the top layer of simple fluid we consider (3.19) with j = 1 and (2.5a, b). The first of (3.19) is identically satisfied. The second gives (S,)?

1

= (S&f)

(3.26)

. I z-o

2-O

Using the algorithm developed in Chap. XIII of [9] we obtain from (2.11) and (3.26)

- [(+),r

- ($),I

c

G2(S)e2i~t-s)& + Comp. Conj.} = 0

which yields for the mean motion P1[e)_r

- (*),j

= PZ[(+),~

at

- (?),,1

z = 0.

(3.27)

It is clear from (2.7) that interface elevations will exhibit the same behavior as the pressure field. Consequently we assume hi”) = hj,,, + hje*iot + 7;ie-*iut

j=

1,2

(3.28)

Torsional oscillations of a rod in a layered medium of simple fluids

637

and obtain from (3.17) and (3.23a) (3.29)

1,2.

i=

rCjm,Ar, 0) = 0

The solution of (3.24, 3.25, 3.27, 3.29) is $1, = 0. We obtain the mean pressure distribution from (3.22)

#Jim

=

9

(3.30)

lK,(r0A,)1-2 s’ t-11K,(tA,)12dt = Fr,(r, w).

The integral in (3.30) is to be evaluated numerically for a given geometry. Turning now to the layers of simple fluids and introducing the quadratic relaxation modulus ~(tr, t2) derived from the material function /3(tr, t2) f2) =

/3j(tl)

“:i:;12) 3

1 2

we obtain from (2.11)

ss co

00

+

0

Y.(SI I J 3 s~)A~‘~(s,)A~‘~(~,)ds,dsz,

0

(3.31)

as previously shown by Joseph [9]. Here 4 = 6(X, T, A) is the position vector of the particle at time 7 < t which particle was at X in v. when A = 0. t represents the present time. The symbol [n] indicates the substantial derivative of order n with respect to the perturbation parameter A, evaluated at the rest state vo. The gradient operator is with respect to rest state coordinates and [[p]]

=

pyx, 7) -

p(x, t) =

1 uqx, sjds.

(3.32)

Using (3.31), (3.32), (3.9a) and (3.21) we obtain from (3.14)

+ 2eJV2j(U)(r'I

Wj,J2),r

+

2pjeJl Wj12

+

el[jj(r)e2iwf

+

J(r)e-2i~,

i = 2, 3, j no sum

(3.33)

where Nzj(w) is the second normal stress function for time-periodic flows of order fluids

s co

N2j(W) = -2

0

q(S)

ss co

03

Y++

0

0

‘Yjts19

S2) cos W(SI -

S2Wldr2,

(3-34)

and A(r) and its conjugate A(r) are complicated expressions pertinent to the oscillatory motion of frequency 2~. The use of (3.23a) together with the curl of (3.33) allows us to construct for the mean stream function the following problems in the layers of simple fluids, in uo,

638

A. SIGINER +jmtr09 Z) = rcljm,r(rO, Z) = 0, $jmjm(m9 z, = \Cj*,r(~, 4 hn(~,

-co)

Pj+jm,rz = Pj+ I+;+

= 09

= 1C/3m,r(r, -co) j =

l+( j+ I)m,zz

ti2m,r(b 0) = $2m,&,

-Z)

= 0,

1,2; j no sum,

= J/3m,rk -Z)

= 0,

which solutions are +jrn = 09

j =

2, 3.

The mean pressure fields in the layers of simple fluids are obtained from (3.33)

4) = 2(r21M&12)N2j(w) - 4Fzj(r,

G,(s) %

W) [ +

ds

F~jtr, ~1,

j no sum

(3.35)

where

Fljk

WI =

$

]Kr(roAj)l-2 J

t-‘IKr(tAj)12dt,

j=

1,2,3,

(3.36)

r tlWj,J2dt

F2jtrv ~1 =

= z IK,(roAj)(‘[

Jr tw31Kl(tAj)12dt+ J

t-‘IKl,~(tAj)12dt

s

-

s

, -2 3

Re[Kl(tAj)Kl.~(tAj)Idt

1 j= ,

2, 3.

(3.37)

The integrals involving modified cylinder functions with complex arguments are to be numerically evaluated for a given geometry. 4. MEAN

INTERFACE

SHAPES

At first order particles move on circular paths and there is no change in the hydrostatic pressure distribution. Consequently the contribution to height rise is zero. At second order the following equation obtained from (2.7) holds at both interfaces after the mapping from vA- vo; from here after it will be that the flow variables in the equations are mean quantities and the use of the indice (m) will be discontinued

Where [[ ]b and $22 represent the jump across thejth interface in the indicated quantity and the extra stress component at second order in the direction of the normal to the interface in the rest domain vo, respectively. We note that because of the lack of motion in the mean at 2nd order we have co

[S

s$‘,>, - s$‘,>, =

1

Gz(s)ds - ~1 wf) = 0,

0

s’,‘,>, - sg>, =[s,”

[G3($ - G2W+~~’

= 0.

639

Torsional oscillations of a rod in a layered medium of simple fluids

Rearranging (4.1) we obtain

(4.2) j= (Tj

1,2,

’

to be solved subject to Z&co)

= Zz’2)(co) = 09 J

ZZ$‘(ro) =

(4.3a) (4.3b)

t@j.

In (4.3b) we assumed that the contact angle of the interface at the rod oscillates in time around a mean contact angle 0. The solution to (4.2) and (4.3) is

s m

+

2t2[&?jN2j(W)I

wj,ll’

-

-

FZ(j+l)

~2~j+l~(~)l~~j+*~,~121 - 4 a2jF2jtt9

sm

ds dt,

Gj+l(S) y

j=

@)

1,2.

0

sin ws

Gj(S) -ds w

j no sum

(4.4)

II

0

The integral in (4.4) is to be evaluated numerically for a given geometry. lo, & are modified Bessel functions and Frj, F2j, N2j are defined in (3.36), (3.37) and (3.34), respectively. TO evaluate (4.4) assumptions have to be made about the functional representations of the shear relaxation modulus and the quadratic shear relaxation modulus. The simplest approximation could be G(s) = _ 5 &/Pl~,

(4.5)

y(sr ) s2) = 4ce-k(s’+~),

(4.6)

chosen for compatibility with the Coleman-Markovitz

P=

sco s ss

formulas [lo]

wh

0

(4.7)

cc

a1

=

-

0

sG(w,

m

a2

=

m

Yh

0

(4.8)

9 s2wl~2.

0

(4.9)

With this choice the second normal stress function becomes

N2(w) =

(4.10)

The material constant k is the only parameter to be determined from a combination of the analytical and experimental results with the approximations (4.5) and (4.6). The Rivlin-

640

A. SIGINER

Ericksen constants (Y~and cz2may be obtained from cone and plate and Weissenberg effect experiments, respectively. It is possible to build higher order approximations similar in structure to (4.5) and (4.6) as demonstrated by Joseph and Beavers [2]. 5. CONCLUDING

REMARKS

The results obtained in this paper demonstrate that it is possible to determine the constitutive constants which are enough to describe any motion of a simple fluid forced from an oscillating boundary from a geometry of rotating rod and several layers of superposed simple fluids. One of the simplest approximations for the shear and quadratic shear relaxation moduli introduce only one more material constant at second order of the perturbation analysis in addition to the two Rivlin-Ericksen constants which are enough to determine any smooth, slow, steady motion of the same fluid. It is feasible to determine this additional constant for several fluids successively from a single experiment in a layered medium of simple fluids driven by an oscillating rod. The accuracy of the determination will increase as the interfaces show appreciable deformations at low rates of shear. This is equivalent to requiring the ratio of the second normal stress function for time-periodic flows of order fluids for two successive layers not to be in the vicinity of one. Finally we note that there is no coupling between layers and interfaces at the first order and second order in the mean although correct coupling conditions between layers have been used in the analysis. REFERENCES [1] D. D. JOSEPH and R. L. FOSDICK, Arch. Rat. Mech. Anal. 49(5), 321-401 (1973). [2] D. D. JOSEPH and G. S. BEAVERS, Arch. Rat. Mech. Anal. 62(4), 323-352 (1976). [3] A. SIGINER, J. Non-Newtonian F. Mech. 15(l), 93-108 (1984). [4] A. SIGINER, Z. angew. Math. Phys. 35(4), 545-558 (1984). [5] A. SIGINER, Z. angew. Math. Phys. 35(5), 618-633 (1984). [6] A. SIGINER and G. S. BEAVERS, J. Non-Newtonian F. Mech. 15(l), 109-126 (1984). [7] B. D. COLEMAN and W. NOLL, Rev. Mod. Phys. 33,239-249 (1961). [8] A. C. PIPIUN, Rev. Mod. Phys. 36, 1034-1041 (1964). [9] D. D. JOSEPH, Stability of Fluid Motions, Springer Tracts in Natural Philosophy (1976). 101 B. D. COLEMAN and H. MARKOVITZ, J. Appl. Phys. 35(l), l-9 (1964). (Received 28 January 1984)