The cauchy - Poisson problem for a viscous liquid

ARCHIEF

_{Scheepsbouyikunde}

J. Fluid Mech. (1968), vol. 34, part 2, pp. 359-370

Teihnische

_Hogeschoo159

Delft

The CauchyPoisson problem for

a viscous liquid

By JOHN W. MILE St

Institute of Geophysics and Planetary Physics, University of California, La Jolla

(Received 25 April 1968)

The axisymmetric, free-surface response of a semi-infinite viscous liquid to either

a point impulse or an initial displacement of zero net volume is calculated. The asymptotic disturbance is resolved into three components: (i) a damped gravity wave, which represents a primary balance between gravitational and inertial forces with secondary, but cumulative, modification by viscous forces; (ii) a

diffusive motion, which represents a balance between viscous and inertial forces; (iii) a creep wave, which represents a balance between gravitational and viscous forces. Van Dorn has suggested that the results may be relevant to the concentric

circular ridges that surround the crater Orientale on the Moon.

1. Introduction

Van Dorn (1968) has suggested that the concentric circular ridges that

sur-round the crater Orientale at lat. 200 S. and long. 95°W. on the Moon may have been initiated as gravity waves on a viscous liquid under the impact of a meteorite.

Testing the plausibffity of this conjecture would require not only the solution of the CauchyPoisson problem for a viscous liquid, but, as minimum data,

estimates of the viscosity of the material and of the ultimate shear stress (above

which the material might behave as a liquid); unfortunately, reliable estimates do not appear to be available at this time. Nevertheless, it does appear worth while to obtain a formal solution of the problem and to examine its qualitative

features vis-a-vis the classical problem.

The transient development of one-dimensional surface waves on a viscous fluid

has been considered by Sretenskii (1941), and the basic motions that such a development comprises were discussed by both Basset (1888, §520-2) and Lamb (1932, §349; Lamb concluded with the statement: 'By a proper synthesis of the various normal modes it must be possible to represent the decay of an arbitrary initial disturbance'). These basic motions may be classified into three types: (i) damped gravity waves, which represent a primary balance between gravitational and inertial forces with secondary, but cumulative, modification by viscous forces; (ii) diffusion, which represents a primary balance between viscous and inertial forces; (iii) creep, which represents a primary balance between gravitational and viscous forces. The relative importance of these

motions for a particular initial configuration depends essentially on the viscous length

1 g-ivf, (1.1)

where g is the acceleration of gravity and v is the kinematic viscosity.

We consider a free-surface motion that stems either from an initial

displace-ment of vertical scale d and lateral scale a or from an impulse of magnitude I and lateral scale a. We neglect surface tension, T, on the hypothesis that the capillary

length is negligible: _{(T pg)i < max {l, a}. (1.2)}

The available similarity parameters then are

all = {(ga)ia/v}i 2A-} (1.3)

(we find A and the Reynolds number R convenient in §5 below), dla, and

v/(ga)t (11pa3)1(ga)i, (1.4)

where v is an equivalent impact speed (we do not assume conservation of

momen-tum in the impact, so that v and I may not be simply related to the speed and

momentum of the impacting body).

The classical, Cauchy-Poisson problem for the response of an inviscid liquid

to a concentrated (a -->0) impulse or displacement yields an asymptotic represen-tion of the free-surface displacement in the form

(r, t) A(r,t) cos co + B(r,t) sin a), (1.5a)

where = gt214r c:o, (1.5b)

and A and B are slowly varying functions of r and t. Referring to Lamb's (1932,

§349) discussion of the basic motions enumerated above, we may incorporate

the dominant effect of a sufficiently small viscosity by neglecting motions of types (ii) and (iii) and multiplying the right-hand side of (1.5a) by the damping factor

exp ( - 2pg t) exp ( - vg2t5180), (1.6a)

where ks gt2i4r2 colr (1.6b)

is the wave-number of stationary phase. Having this result, we seek to determine

more precisely those conditions under which motions of types (ii) and (iii) are

either negligible or significant in comparison with the damped gravity wave and

to determine the modification of the damping factor of (1.6a) for an initial

dis-placement of zero net volume.

Our study having been induced by reference to an impact of cataclysmic magnitude, it seems necessary to consider the implications of the hypothesis of small disturbances, without which significant mathematical progress would be impossible. Strictly speaking, this hypothesis is equivalent to the restrictions

d < max (1, a}, v < max {(ga)+, (gl)-}}; (1.7a, b)

however, both intuitive considerations and observational data for underwater

explosions (Cole 1948) suggest that events at sufficient distances from the point of impact may be adequately described by ignoring the details of the impact (or

The Cauchy-Poisson problem for a viscous liquid 361

which must be inferred from both observation and empirical considerations. It seems quite unlikely, on the other hand, that the effects of an impact can be

adequately represented by the assumption of a prescribed impulse unless _{(1.7 b)}

is actually satisfied. The formal solution for a point impulse is, nevertheless, of

interest both as a preliminary example that yields an especially interesting creep wave and for direct comparison with experiments in actual, viscous liquids.

2. Formulation of problem

We consider a liquid that fills the half-space z> 0 (z is positive downwards), is at rest for t < 0, and is subjected to a surface impulse pq50(r) and

free-surface displacement (,(r) (positive downwards) at t= 0. Letting

p(r, z, t) pOi(r, z, t) _(2.1)

be the pressure (we follow Lamb's convention for the sign of the potential 0) and rVr(r, z, t) be a Stokes stream function, such that the radial and downward

components of the particle velocity q are given by

u = -

and invoking the hypothesis of small disturbances, we find that the continuity

and (linearized) Navier-Stokes equations,

V q = 0, q1 = - V(p/p) vV2q, _(2.3)

imply _{Orr+ r-lcbr + Ozz = 0 (2.4a)}

and r-20.

=v'fr1.(2.4b)

The initial conditions are

0(r, 0, 0) = 00(r), 0) = C0(r). (2.5)

The total impulse (positive downwards) delivered to the surface at t = 0 is given

by

I = 2np

f

00(r)r.

00

(2.6)

The potential energy associated with the initial displacement is given by

=

f

E,

where C0(r) satisfies the constraint

co

4.0(r)r dr = 0 (2.8)

fo

if the displaced volume vanishes identically.

Invoking the requirements of kinematic and dynamical equilibrium at the disturbed free surface, z = (r, t), we obtain the linearized boundary conditions

= _(2.9a)

Ot+ gC-2vwz = 0, (2.9b)

362

We may satifify these boundary conditions of z = 0, rather than z = C, without introducing an error greater than that already implicit in the linearization of the

equations of motion.

3. Formal solution

We obtain a formal solution to the problem posed in the preceding section with the aid of the following Laplace and Hankel transformations:

{Z, C!>, W} =fc° e-gids {C, b, w} Jo(kr)r dr, (3.1a)

U} = foc° e-si cis f: ,u} Ji(kr)r dr , (3.1 b)

and {Zo, = c° go, 00) Jo(kr)r dr.. (3.2)

Carrying out the corresponding transformations of (2.2), (2.4) and (2.9), we obtain _{U = le(1) +1F., W = - (Dz - (3.3)}

(Doz- k2(1) = 0, 11roz-K24" = 0, (3.4) sZ - W = Z (z = 0), (3.5a)

80+ gZ - 2vfc = (Do (z = 0), (3.5b)

and U- kW = 0 (z = 0), (3.5c)

where K = {k2 (81v)}i (Ric 0). (3.6)

Invoking the requirement that both (1) and 1r be bounded as z co, we pose

the solution to (3.4) in the form

= A(s,k)e-kz, IF = B(8, k) e-Kz (z 0). (3.7)

Substituting (3.7) into (3.3) and (3.5), we obtain

U = kA e-kz - KB e-Kz, W = k(A e-kz - B e-Kz), (3.8) and the matrix equation

M.{Z, A, B} = {Z0, 00, 0}, (3.9)

8

-k

where

M= g 8+ 2vk2

0 -2k2 k2 K2

Solving (3.9) for Z, we obtain

Z(k =

ZJ

s (1 D(s,k)) D(8,10' (3.11

, s) )

where D(8, k) = (8 ±2vk2)2- 4v1k3(8+ vk2)i + gk. (3.12)

We satisfy the requirement that the real part of the radical K be non-negative by cutting the 8-plane along 8 = [- co, - vk2] and choosing the positive square root along 8 = (-PP, co). We then find that D has two, and only two, zeros in

The Cauchy-Poisson problem for a viscous liquid 363

the cut plane, both of which lie in as < 0 (see §4 below); accordingly, we may

choose the path of integration for the inverse _{Laplace transformation along} = 0 to obtain the formal solution

1cori

C(r,t) = 2-Trifo Jo(kr)kdk Z(k, 8) es 1 ds, _(3.13)

where Z is given by (3.11) and (3.2). We notice that the effects of surface tension could be included simply by multiplying gk by 1_{+ (k1')2 in (3.11) and (3.12).}

4. Point-impulse problem

We assume, in this section, that the radius of the area over which the impulse / is applied is negligible compared with the viscous_{length 1 and that the initial}

displacement vanishes identically. Invoking these hypotheses, together with

(2.6), in (3.2), we obtain

Z0(k) m. 0, 00(k) = 1127rp vlh, (4.1) where 1 is defined by (1.1) and h is an appropriate vertical scale. Substituting

(4.1) into (3.11) and (3.13) and introducing the dimensionless_quantities

= rll, T = (g11)it, a = kl, Cr = (11g)i8, _(4.2)

and A(o-, a) (11g)D(s,k) (0. +2,22)2 4as(0.± ce2).} ± cc, _(4.3)

we obtain (r, t) = h C 0_{J0(av)x(r,a)a2da, (4.4)}

where _{X(T a) =} 1 L ic° ecr

rdo-(4.5)

2772. A(o.,a)

We seek the asymptotic behaviour of C as7--> co.

Deforming the path of integration in the a-plane in the usual_{way, we obtain} contributions to x from the two poles determined by the dispersionequation

A(a, a) = 0, Cr = ±(a), _(4.6)

and from the two sides of the cut along a = [- oo, - a2]. The zeros of A in this cut plane are (cf. Lamb 1932, §349) complex conjugates in ao- < 0 for

0 < a < cc, = 1-20,

negative real in ( - a2, 0) for a> ac, and analytic functions of a in a complex-a plane cut along a = [0, ac]. Their numerical values are plotted in figure 1. Invoking Cauchy's residue theorem to obtain the contributions of the poles to x and introducing the change of variable

o- = - a2(1 + x2) + i0, _{(o- + a2)i = + iax}

on the upper and lower sides of the cut, we obtain

x(T, al 8a4 f exp {_ (1 + x2) ce27}x2dx

364

The dominant contributions to the integral (4.4) as T -÷ CO must come from the neighbourhoods of a = 0 and a = oo since the integrand is exponentially small at

intermediate values of a.t The limiting forms of cr±(a), as determined by (4.3)

and (4.6), are given by

= + jai - 2a2+0(a-V) (a-+0),0), (4.8a)

er+

-

0 04 0-8

FIGURE 1. The roots of the dispersion equation (4.6); o- = or±ioj for 0 < a < 1.20. Substituting these approximations and the corresponding approximations to 06,1.9a, as determined from (4.3), into (4.7) and approximating the x-integral with the aid of Watson's lemma, we obtain

x c4 exp - 2a2T} sin (air) {1 + 0(a1)}- 2(nr3)4 a exp - a2r} {1 + 0(0:IT)}

is the asymptotic approximation to the contributions of the complex-conjugate poles and represents a damped gravity wave;

= 2h(r/r3)-if Jo(an)exp{- a27}a3da (4.12) t The subsequent, asymptotic development makes free use of the methods described by Copson (1965).

(T --oo, a-.-0) (4.9a)

ia-2exp { - la)}{1 + 0(a-3)} (7. -> CO, a oo). (4.9b)

Invoking these approximations in (4.4), we place the result in the form

gg-qd+Cc (r-*00),

where: _g

-h

f

(4.10)

The Cauchy-Poisson problem, for a vi8cous liquid ₃₆₅ is the asymptotic approximation to the contribution of the continuous spectrum of the cut, a- = [-00, - a2], and represents an essentially diffusive disturbance; and

=

hf:

and represents a creep wave. We observe that our definition of_{a damped gravity} wave is precise by virtue of the requirement that o-± be complex. The distinction between diffusive and creeping disturbances is sharp only for T co; in particular,

the disturbance associated with o-_ in a > a, is distinctly diffusiveas a -+00 (in which limit it is also negligible), but is essentially identical with that

associ-ated with cr, as a a0+ .

Carrying out the stationary-phase approximation to (4.11), which is equivalent

to a saddle-point approximation to the oscillatory component, 0 < a < a0, of

(4.4), we obtain

24h-r3r4exp ( -1-T5v-4) sin (17-277-1) _{(1 < T < < 7-2) (4.14a)}

= 2i1/40 exp { - 2(r/2) co2} sin co, _{(4.14 b)}

where co is the similarity variable of (1.5 b).

Invoking the Hankel-transform pairs §8.3(5) and §8.2 (53) in the tables of Erdelyi, Magnus, Oberhettinger & Tricomi (1953), we evaluate the integrals in

(4.12) and (4.13) as follows:

= 2h(7Tr3)4 (alar) fo J0(co7) exP { - a27-} da (4.15a)

= h(77.74)-4 {1 - (712/47-)}ex-p ( -772/47-) _(4.15b)

and 2h(aiaT)f Jo(av) exp ( -7-/4a) sinh (r/4a) da _(4.16a)

= 4-1(a/ar)tai(Ernli)K1([77/])1} (4.16b)

= hv-Ly{Jo(y)Ki(r) Ji(y)Ko(y)} (4.16c)

eV cos y{1 ± 0(y-1)}, _(4.16d)

where y = = (grthi)i. _(4.17)

The disturbance described by (4.14) tends to the classical, Cauchy-Poisson solution of the point-impulse problem (Lamb 1932, §255) as v2Ir-> oo with CO

fixed, in which limit it appears as a true similarity solution in the variable CO and

represents a balance between gravitational and inertial forces. It dominates both

and in 1 47-49747-2 but ultimately decays quite rapidlyas 7-400 oo with

fixed. It is plotted in figure 2 for 7/ in the neighbourhood of the maximum of the

envelope (the coefficient of sin Cu), namely

241/7-37r4exp ( - i-T577-4) = 0.518h/7-2 at 71 =1m = (1-75)1. (4.18) The accuracy of the asymptotic approximation may be poor for < v [in consequence of the approximation (4.8 a)], but we expect it to be qualitatively.

The disturbance described by (4.15) is a similarity solution in the variable

n2Irand represents the diffusion of vorticity under the action of viscous and

inertial forces. It is negligible for T 1, even though it dominates both Cu and

Ccif T-1 < < 711.

Fmcrsz 2. The asymptotic form of the gravity wave generated by a point impulse, as given by (4.14) for 72 = 4, 12 and 20; is positive downwards.

The disturbance described by (4.16) is a similarity solution in the variable y

and represents a balance between gravitational and viscous forces. It dominates

bothCgand if 7/ = 0(T-1) and is especially interesting for a very viscous fluid.

It is oscillatory in character, but the resulting wave pattern is damped at the asymptotic rate of lie per cycle. It is plotted as a function of the similarity

variable y in figure 3a and as a function of Tv in figure 3 b. The latter plot gives a

linearly scaled representation of the subsiding cavity, although it must be re-called that the asymptotic approximation (4.13) is not uniformly valid as

O.

We conclude that the free-surface response of a viscous liquid to an impulse is

asymptotically separated into three, distinct zones:

The CauchyPoisson problem fora viscousliquid 367

a null zone, 7-1 < < T, in which each of Cg, Cd and Ce is small in

conse-quence of the joint action of diffusion and dispersion;

a creep-wave zone, = 0(7--1), in which C

= 0(h/).

The scales of C, and Cg as functions of for fixed T 1 are so disparate that it is

not feasible to compare them in a common plot.

Ti'

FIGURE 3a. The asymptotic form of the creep wave generated by a point impulse, as given by the similarity solution (4.16) ; C is positive downwards.

FIGURE 3b. The asymptotic form of the creep wave generated by a point impulse, as given by the similarity solution (4.16);

C is positive downwards.

5. Initial-cavity problem

We now consider the disturbance produced by the initial displacement (see figure4)

Co(7) = d exp _{(rla)2) {1 (rla)2}, (5.1)}

which represents a cavity 'with a lip such that the volumetric constraint of (2.8)

is satisfied and _{Eo = krpgd2a2.}

(5.2) Substituting (5.1) into (3.2) and invoking §8.3(5) in Erdelyi etal. (1953),we

obtain _{Z0(k) = -ida2(ka)2exp { i(ka)2} (00 =}

0). _(5.3)

Substituting (5.3) into (3.11) and (3.13) and introducing the dimensionless

quantities of (4.2), (4.3) and (1.3), we obtain C(r,t) = 2A2d f Jo(av) exp

1 lice

where Xi(T,ct)

27n,

Comparing (5.5) with (4.5) and invoking

26(7,a) =cxj X(27,a)d71

exp 2a27}cos (car)

2(7773)4 exp { a2r} exp { la)} {1+0(a3)}

ya2}xi(r, a) a3da,

a

le, da

368

Substituting (5.6 b, c) into (5.4), letting T ---> 00 , and decomposing the asymptotic

representation as in (4.10), we obtain

= 2A2df Jo(cev)exp { (A + 2-r) a2} cos (ccir)cc3doc, (5.7)

OD

= 2A2d(nT3)if Jo(cei) exp (A + T) ceda, (5.8)

2A2d

I

Jo and

FIGURE 4. The cavity and lip described by (5.1). The volumetric displacement in r > a (the lip) is equal and opposite to that in r < a (the cavity).

00

FIGURE 5. The asymptotic response of an inviscid liquid to the initial displacement of figure 4; g is positive downwards.

The CauchyPoisson problem for a viscous liquid 369 Evaluating the integrals of (5.7) and (5.8) as in §4 and approximating that in

(5.9) by Laplace's method, we obtain

- 2-tA2dr6/-7exp { - (A + 2r) (r/2v)4} cos (fr2/-1)

(1 <r<v<re),

and i(p)idAirJ0{1(r/4A)-1} exp {- i(Ar2)*}. (5.12)

ggld

050

rla

Fictraz 6a. The asymptotic response of a viscous liquid to the initial displacement of

figure 4 for gt2la = 8 and R = 10, 100 and co; C is positive downwards.

R==

0-5 1-5 25

rla

FIGURE 6b. The asymptotic response of an inviscid liquid to the initial displacement of figure 4 for gt2la = 16 and R = 10, 100, and co; C is positive downwards.

24 _{Fluid Mech. 34} 0 10 0-1 0-2 0-3 0-4 100

370

The approximations (5.11) and (5.12) are uniformly valid with respect to for fixed A, but (5.12) is not uniformly valid as A--->0. O. We again may identify Cg, Cd and Cc as a damped gravity wave, an essentially diffusive disturbance, and a

creep wave, respectively, but only (5.11) as a similarity solution. Both Cd and C, are asymptotically negligible for all 7/ as T --> 00 with A fixed, and there is no counterpart of the creep-wave zone of the point-impulse problem unless A < 1. The results for A 4 1 are similar to those for the point-impulse problem.

Restoring the original variables in (5.10), we place the result in the form

-

where coc = gt2/4a, 6 = r/a, (5.14),

and .R is given by (1.3). The result (5.13) is plotted in figure 5 in the inviscid limit (R = oo) with coc as a parameter and in figures 6a and 6b with R as a parameter.

This work was supported by the National Science Foundation, under Grant GA-849, and by the Office of Naval Research, under Contract Nonr-2216(29). I am indebted to Dr H. E. Huppert for aid with the computations.

REFERENCES

BASSET, A. B. 1888 A Treatise on Hydrodynamics, vol. 2. New York: Dover Publications (1961 reprint).

_COLE, R. H. 1948 Underwater Explosions. New York: Dover Publications (1965 reprint). Corsow, E. T. 1965 Asymptotic Expansions. Cambridge University Press.

ERDELYI, A., MAaNus, W., OBEREETTINGER, F. & 'TRicomi, F. Cr. 1953 Tables of Integral Transforms, vol. 2. New York: McGraw-Hill.

LAMB, H. 1932 Hydrodynamics. Cambridge University Press.

SRETENSKII, L. 1941 Concerning waves on the surface of a viscous fluid (in Russian). Trudy Tsentrcd. Aero-Gi.drodinam. Inst. No. 541, 1-34.