Some remarks on the difference between a bounding surface and a material surface

ht. 1. Engng Sci.

Vol. 28, NO.8, pp. 793-7%, 1990 Printedin GreatBritain.All rightsreserved

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SOME REMARKS ON THE DIFFERENCE BETWEEN A BOUNDING SURFACE AND A MATERIAL SURFACE R. R. HUILGOL School of Mathematical Sciences, The Flinders University of South Australia, Bedford Park, S.A. 5042, Australia Abstract-It was claimed by Dussan [I. Fluid Me& 75, 609 (1976)] that in order for a bounding surface not to be a material surface either the speed of propagation of the surface is not equal to the normal speed of the particle situate upon it or more than one deformation can be associated with the velocity field. These conditions are shown to be insufficient through examples drawn from the theory of dynamical systems which also suggests the correct way to separate a bounding surface from a material surface.

1. INTRODUCTION

In a paper published some years ago, Dussan [l] discussed the differences between a bounding surface S and a material surface. According to her, both these surfaces are closed, connected surfaces. The bounding surface S separates one part of a continuous body Bdlfrom another part $?&of the same substance or another substance (see Fig. 1) in such a way that a material particle belonging to the one part of the body cannot move over to the other part by traversing the surface. Also, according to her, a material surface which consists of the same particles always is ipso facto a bounding surface. Dussan showed by one example that on a bounding surface which was not a material surface, the condition II - n = c failed; here u(x, t) is the spatial velocity field on the surface, II is the unit normal to the bounding surface and c is its speed of propagation. By a second example, Dussan showed that the condition II - n = c held but the bounding surface was not again a material surface; her conclusion from this example was that the reason for this lay in the non-uniqueness of the motion associated with the given velocity field. The third example, due to Truesdell [2], was recalled by Dussan to demonstrate that while the condition u - II = c held, the surface in question was neither a bounding surface nor a material surface; here again, the non-uniqueness of the motion associated with the velocity field was the presumed culprit. The major result of Dussan [l], as claimed by her, was contained in Theorem 2 of her paper, which is quoted below. “THEOREM 2. If the surface S possesses a continuous unit normal vector n and speed of propagation c, if u *n = c on S, if the velocity field is continuous throughout the entire domain contained within W, and if the motion associated with the velocity field is unique, then the surface S must always consist of the same material.” REMARK. Here W is a closed set which contains both the bodies S?$and $, or W = ~23~ u ?i&in a set theoretical sense and S = C?& n &. See Fig. 1 for a visual presentation of these symbols.

By a simple appeal to dynamical systems theory, it will be shown that the above theorem is not correct; again, use of dynamical systems theory provides the correct answer to the difference between a bounding surface and a material surface and vindicates the earlier claim by Truesdell [2] that the density of the material on a bounding surface plays a crucial role in separating it from a material surface. It must be noted that in this note, we are judging Dussan’s work on a mathematical level because in her paper [l] she criticised the works of Lagrange, Poisson, Kelvin and Truesdell on mathematical grounds and claimed that her theorem put the question of the difference between a bounding surface and a material surface to rest. Because some readers may consider the four examples below to be artificial from the point of real continuum mechanics by claiming that infinite time spans and/or zero and infinite densities are irrelevant, it is our contention that these objections have no value in an examination of Dussan’s theorem. 793

R. R. HUILGOL

794

Fig. 1

2. THE

(Y- AND

w-LIMIT

SETS

Suppose that the velocity field u(x, t) is continuous t~ou~out the domain W and that the motion associated with it is unique. That is, the system of differential equations i = u(x, t)

(I) has the unique solution x(t) = M(X, t) such that the reference coordinate X = M(X, 0). It is well known that in this case, the solution M is a dynamical system (Hirsch and Smale [3]) on the set W. Now, although not every dynamical system has an ty- and an o-limit set associated with it, there are many that do so. While one may consult, say Hirsch and Smale (3) for precise definitions and properties of these sets, one can summarise them heuristically as follows: (i) The a-limit set is the set of points from the neighbourhood of which the trajectories of the motion originate and thus the a-limit set could become a set of zero density. Note that y E W is an o-limit point of X E W if there is a sequence fn -+ -m such that lim n -+ UJM(X, t,) = y. (ii) The w-limit set is the set of points to which the particles converge and hence the o-limit set could become a set of infinite density. Thus, y E W is an o-limit point of X E W if there is a sequence f,, --$)+m such that lim II * CQM(X, f,,) = y. (iii) These two limit sets are closed; that is, a fluid particle cannot escape, strictly speaking, from each set. However, if in a motion the bounding surface S is the o-limit set of the dynamical system, then it is clear that all fluid particles will eventually find themselves on S and S could become a surface of infinite density. We now consider four velocity fields which exist in bounded or unbounded domains, and are defined for finite or infinite time spans and or steady or unsteady. First of all, let i = ei’“, j = 0,

-m
-m
--a3
i = 0,

(2)

whereas in the half space x 2 0, f = 0, jl = 0, i = 0. Clearly, the surface x = 0 is a bounding surface for the motion generated by (2), its speed of propagation is zero, the velocity field is continuous in iR3, the motion is unique and all the fluid particles will converge on to the surface x = 0 as t -+ ~0. Further, note that lim u(x, t) - B = 0. x-+0

(3)

Next, consider a spiral velocity field in two dimensions, say of the form (Guckenheimer Holmes ]4]) i = r(l -

r2),

e=1,

05r51.

and (4)

Given this velocity field, the c-limit set is the origin and the u-limit set is clearly the set r = 1 on which u. n = 0. Moreover, r = 1 is a bounding surface on which c = 0, although it is not a material surface. Examples similar to (4) can be constructed in a 3-dimensional setting as well.

separating surfaces

795

As the third example, let

in two dimensions. The velocity field is isochoric (i.e. div u = 0) and the bounding surface is the line y = 0, on which u *n = 0 and c = 0. It is not a material surface because all points on it, except at x = 0, move to the point at infinity as c--, 01. Indeed, as t-+ 03, the entire body of the material tends to the point at infinity, except that on the line n = 0. In this example, the point at infinity is both a source and sink whereas the origin, being a critical point, is both an m-limit and an o-limit point for itself as well as being the o-limit point for all of the points on the line x = 0. Lastly, let us consider a flow defined for a finite time with a moving left boundary: -ef’T(r-T) 5 x 5 0, i=-&, jl = 0,

-cO
whereas in the half plane x > 0, P = 0, j = 0. The unique motion corresponding

(6) to (6) is given

by x(t) = Xer’T(t--T),

-15x=zo,

y(t) = Y,

--c4
(7)

Since X 5 0, as t+ T-, all the particles will move to the line x = 0 which is the w-limit set. Again, n-n=0 and c = 0 on this line which is a bounding surface, but not a material surface. Physically speaking, the above examples point out that an ablating surface or a condensing surface (or a surface to which all particles migrate eventually) can be a bounding surface but not a material surface. Mathematically speaking, the above examples meet the conditions of Theorem 2 of Dussan [l] and show that a bounding surface need not be a material surface even if her requirements are satisfied. What must be added to the hypotheses of Theorem 2 is the additional statement to the effect that the whole of the bounding surface or point(s) on it should not be an a-limit or w-limit set. This, in essence, is what Truesdall[2] has stated for he demanded that on the bounding surface the density not become zero or infinity. In conclusion, we propose the following modified version of Dussan’s main theorem. THEOREM 3. If the surface S possesses a continuous unit normal vector n and speed of propagation c, if u *n = c on S, if the velocity field is continuous throughout the entire domain contained within W, and if the motion associated with the velocity field is unique, then the surface S must always consist of the same material provided S or point(s) on S do not belong to an a-limit or an w-limit set of the associated dynamical system.

The proof follows along the lines of that in [l] and is omitted.

3. CONCLUDING

REMARKS

Recently it has been shown (Huilgol [S]) that the theory of dynamical systems can be used to answer the question of the form taken by the material description of a motion associated with a steady velocity field. In this note, the theory has been adapted to solve a problem in the theory of bounding and material surfaces,

REFERENCES [I] E. B. DUSSAN V, 1. R&f Me&. 75,609 (1976). [Z] C. TRUESDELL, ButI. Zidz. Univ. Istanbul 3, 71 (1951).

R. R. HUILGOL

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(31 M. W. HIRSCH and S. SMALE, Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York (1974). [4j J. GUCKENHEIMER and P. HOLMES, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1986). [5] R. R. HUILGOL, 2. Angew. Math. Phys. 37, 270 (1986). (Received

13 December

1989; accepted 16 January 1990)