ON THE KINEMATICS OF 2+1-DIMENSIONAL MOTIONS OF A FIBRE-REINFORCED FLUID. INTEGRABLE CONNECTIONS

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1 ON THE KINEMATICS OF 2+1-DIMENSIONAL MOTIONS OF A FIBRE-REINFORCED FLUID. INTEGRABLE CONNECTIONS by W. K. SCHIEF, C. ROGERS and S. MURUGESH (School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia and Australian Research Council Centre for Mathematics and Statistics of Complex Systems) Summary Evolution of foliations of the plane is shown, under a condition of constant divergence, to be linked to the scattering problem for the integrable modified Korteweg de Vries hierarchy. This result is applied to a set of kinematic relations which arise in the theory of ideal fibre-reinforced fluids. In particular, it is established that the fibres, which are convected with the fluid, constitute generalized tractrices. 1. Introduction The link between privileged motions of inextensible curves and modern soliton theory may be said to have its origin in the work of Da Rios (1) concerning the spatial evolution of an isolated vortex filament in an unbounded viscous liquid. Therein, Da Rios invoked what has come to be known as the localized induction approximation to derive a pair of coupled evolution equations for the curvature and torsion of the vortex filament. These results were subsequently collated and extended in a survey by Levi-Civita (2). The Da Rios equations were rediscovered by Betchov (3). Later, Hasimoto (4) motivated by the geometric study of Betchov showed that the Da Rios equations may be combined to produce the celebrated nonlinear Schrödinger equation of modern soliton theory. The nonlinear Schrödinger equation may be generated in a purely geometric manner via privileged binormal motions of inextensible curves (see, for example, (5)). Binormal motions of curves of constant curvature or torsion likewise lead to solitonic equations which generically admit auto- Bäcklund transformations (6). Moreover, a Heisenberg spin equation, equivalent to the nonlinear Schrödinger equation, has been shown to arise, without approximation, out of a geometric formulation of spatial hydrodynamics subject to a vanishing divergence constraint (7, 8). This result applies mutatis mutandis in magnetohydrostatics and has been exploited to construct equilibrium configurations wherein the constant total pressure surfaces comprise nested tori foliated in accordance with the vanishing divergence condition (9). It is these connections between integrability, the motion of inextensible curves and geometries with constant divergence that motivates the present study. Here, it is established that time-dependent one-parameter families of curves which foliate the plane are governed by the AKNS scattering problem for the integrable modified Korteweg de Vries (mkdv) hierarchy (10) whenever the divergence of the unit tangent to the curves is constant. schief@maths.unsw.edu.au Q. Jl Mech. Appl. Math, Vol. 60. No. 1 c The author Published by Oxford University Press; all rights reserved. For Permissions, please journals.permissions@oxfordjournals.org Advance Access publication 27 January doi: /qjmam/hbl025

2 50 W. K. SCHIEF et al. Importantly, it is then shown that these 2+1-dimensional geometries are compatible with a system of kinematic relations which arise in a theory of ideal fibre-reinforced fluids (11). In fact, it is demonstrated that all kinematically admissible motions subject to the constant divergence condition are governed by the evolution of a single base curve whose associated one-parameter family of generalized tractrices represents the fibres convected by the fluid. As an illustration, the motions corresponding to a straight base curve are constructed explicitly. These may be used to generate iteratively motions corresponding to base curves of increasing complexity such as those associated with breather potentials in the AKNS scattering problem. Our account starts with a geometric decomposition of the above-mentioned kinematic conditions via a formalism originally introduced in hydrodynamics by Marris and Passman (12). In the physical context of the theory of ideal fibre-reinforced fluids, the geometric constraint of constant divergence of the fibre lines is seen to be both consistent and natural. In this connection, it is noted that the condition of vanishing divergence has been investigated by Spencer (13). 2. The kinematic conditions A kinematic study of the motion of ideal fibre-reinforced fluids has been conducted by Spencer (11). The ideal model, as investigated therein, consists of an incompressible viscous liquid which contains inextensible fibre lines occupying the volume of the fluid by which they are convected (14). Let t = t(r, t) denote the unit vector tangential to a generic fibre and q = q(r, t) be the fluid velocity. Then, the kinematic condition which encodes both the convection requirement and inextensibility of the fibres is given by t + (q )t = (t )q. (2.1) t Here, as in (11), it is augmented by the usual continuity equation div q = 0. (2.2) The geometric properties of the above kinematic conditions have been investigated in detail for steady plane motions in (15, 16). It was shown that the kinematic relations may be reduced to a single third-order nonlinear equation which, remarkably, admits an important reduction to a solitonic system. A novel duality principle as well as an auto-bäcklund transformation were recorded. As observed by Spencer (11), the kinematic conditions (2.1), (2.2) together imply the necessary condition ( ) t + q div t = 0 (2.3) so that div t is constant along particle lines. This constraint obtains, in particular, in the privileged case when this divergence is everywhere constant. 1 It is with this class of motions that we shall be concerned in the sequel dimensional geometries Henceforth, our attention will be restricted to time-dependent two-dimensional motions. This is, in part, motivated by the assertion (11) that two-dimensional flows of an ideal fibre-reinforced fluid 1 Indeed, Spencer (13) paid particular attention to the case div t = 0.

3 MOTIONS OF A FIBRE-REINFORCED FLUID 51 are essentially determined by kinematic considerations in that pressure p and tension T in the fibre direction can always be determined such that the equations of motion are satisfied. Planar motion is accordingly especially privileged and the study of its kinematics assumes an added importance. Against this background, we here investigate certain time-dependent one-parameter families of inextensible curves (fibres) which foliate the plane. Let t and n denote the unit tangent and principal normal respectively to a generic fibre. Then, the associated Serret Frenet equations adopt the form δ δs ( t n ) = ( 0 κ κ 0 )( t n ), ( ) ( ) δ t 0 θ t =, δn n θ 0)( n where δ δs = t, δ = n (3.2) δn designate the directional derivatives in the respective tangential and principal normal directions. Accordingly, the gradient is given by =t δ δs + n δ δn. (3.3) In the above, κ is the curvature of the fibres while θ is the curvature of their orthogonal trajectories, the n-lines. In this connection, it is important to note that (3.1) θ = div t. (3.4) The commutator relation (17) [ δ δn, δ ] = δ2 δs δnδs δ2 δsδn = κ δ δs + θ δ (3.5) δn applied to system (3.1) now requires that δκ δn δθ δs = κ2 + θ 2. (3.6) The general solution of this compatibility condition is obtained by parametrizing the orthonormal pair (t, n) according to t = cos ϕ i + sin ϕ j, n = sin ϕ i + cos ϕ j (3.7) with i and j the usual unit vectors in the direction of the Cartesian x- and y-axes so that κ = δϕ δs, In what follows, curvilinear coordinates (s, n,τ)of the form θ = δϕ δn. (3.8) s = s(x, y, t), n = n(x, y, t), τ = t (3.9)

4 52 W. K. SCHIEF et al. are introduced with δ δs = 1 φ s, δ δn = 1 ψ n (3.10) so that t = τ + ρ s + σ (3.11) n and s and n parametrize the fibres and their orthogonal trajectories respectively. The functions φ(s, n,τ),ψ(s, n,τ)and ρ(s, n,τ),σ(s, n,τ)are determined by the requirement that the operators / x, / y, / t commute. Thus, on the one hand, the commutator relation (3.5) leads to ψ s = ϕ n φ, φ n = ϕ s ψ. (3.12) On the other hand, the parametrization (3.7) implies that ( ) ( )( ) t 0 µ t =, t n µ 0 n µ = ϕ t. (3.13) Hence, the commutativity property [ / t, ] = 0 yields [ ] [ ] t, t = µn, t, n that is, [ t, δ ] = µ δ δs δn, The latter commutator relations, on insertion of (3.10), (3.11), show that = µt, (3.14) [ t, δ ] = µ δ δn δs. (3.15) φ τ + ρφ s + σφ n + φρ s = 0, µφ + ψσ s = 0 (3.16) and ψ τ + ρψ s + σψ n + ψσ n = 0, φρ n µψ = 0. (3.17) The Cartesian coordinate vector r = xi + yj is obtained by integration of the equations r s = φt, r n = ψn (3.18) together with r τ = ρφt σψn (3.19) since r/ t (x,y) = 0. By construction, the corresponding compatibility conditions r sn = r ns, r sτ = r τs and r nτ = r τn are satisfied modulo the system (3.12), (3.16), (3.17). At any instant t, the oneparameter (c) family of fibres is given parametrically in terms of s via r(s, n = c,τ = t). 4. The constraint div t = 1. Generalized tractrices In the following, we assume that the divergence of the unit tangent vector field t is constant. Thus, without loss of generality, we impose the constraint div t = θ = 1. (4.1)

5 MOTIONS OF A FIBRE-REINFORCED FLUID 53 The Serret Frenet equations (3.1) and (3.18) 2 then imply that (r t) n = 0, whence r = t + R(s,τ). (4.2) This relation reveals that, for any fixed τ, the distance in tangential direction between any point on a fibre and the curve Ɣ 0 represented by R is equal to 1. Accordingly, by definition (18), the fibres constitute generalized tractrices associated with the base curve Ɣ 0. Since the condition (4.2) is equivalent to θ = 1, it turns out that the complete class of fibre distributions with div t = 1is encoded in the base curve Ɣ 0. Hence, the condition div t = 1 may be regarded as an equivalent definition of (a family of) generalized tractrices. Indeed, the relation (4.2) implies that, on the one hand, div r = div t + div R = div t + t δr (4.3) δs and, on the other hand, δr δs = δr δs. (4.4) Combination of these relations leads to div t = 1. Moreover, since [(r R) 2 ] n = 0, the trajectories orthogonal to the fibres constitute circles, the centres of which form the base curve Ɣ 0. Consequently, the following theorem obtains (cf. Fig. 1). THEOREM 4.1 Fibre distributions with div t = 1 are confined to a strip bounded by two curves Ɣ ± which are parallel and at unit distance from a given base curve Ɣ 0. The fibres constitute the generalized tractrices associated with the base curve and their orthogonal trajectories are circles of unit radius. Fig. 1 Generalized tractrices (solid) and an orthogonal trajectory (dotted) associated with a base curve Ɣ 0 and bounded by the two parallel curves Ɣ ± (dashed)

6 54 W. K. SCHIEF et al. 4.1 Evolution of the base curve Ɣ 0. The mkdv recursion operator It is now evident that the evolution of fibre distributions subject to div t = 1 is completely encapsulated in the evolution of the base curve Ɣ 0. Thus, if we choose s to be arc length along the base curve and T and N are the corresponding unit tangent and principal normal respectively then the derivatives of the position vector R(s,τ)are given by R s = T, R τ = gt + hn, (4.5) where the functions g(s,τ) and h(s,τ) are yet to be determined. Compatibility of the above pair and the Serret Frenet and time-evolution equations ( T N) ( T N) s τ ( 0 f = ), )( T f 0 N ( ) 0 l T =, l 0)( N wherein f (s,τ)constitutes the curvature of Ɣ 0, produces the pair (4.6) f τ = h ss + (gf) s, g s = fh (4.7) together with l = h s + gf. (4.8) Elimination of g in the system (4.7) leads to f τ = Rh, R = 2 s + f 2 + f s 1 s f, (4.9) where the operator R is nothing but the recursion operator for the mkdv hierarchy of soliton equations (10). In fact, if, instead of f, we prescribe the function h according to h = f s then (4.9) becomes the integrable mkdv equation f τ = f sss f 2 f s. (4.10) In general, if we set h = R N f s then we retrieve the mkdv equation of order 2N + 3, that is, f τ = R N+1 f s. (4.11) Thus, any member of the mkdv hierarchy may be associated with particular classes of timedependent foliations of the plane with div t = 1. Even though a link between the mkdv hierarchy and the motion of curves on the plane has been noted previously (see, for example, (19)), it is demonstrated below that, remarkably, in the current context, the occurrence of the mkdv recursion operator is accompanied by the appearance of the corresponding AKNS scattering problem. This fact is key to the explicit generation of fibre distributions by means of Darboux transformations. It is emphasised that this AKNS connection differs from that established in the context of the planar motion of curves (see, for example, (20)) in that it applies to the fibres rather than the base curve.

7 MOTIONS OF A FIBRE-REINFORCED FLUID Evolution of the fibres As already noted, for any given base curve Ɣ 0, the associated one-parameter (n) family of fibres with position vector r is obtained from the defining relation r(s, n,τ) = t(s, n,τ) + R(s,τ). (4.12) This constitutes an implicit differential relation between the fibres and the base curve since it is required to guarantee that t constitutes the unit tangent vector associated with r. Thus, if ω denotes the angle of rotation which relates the orthonormal pairs (T, N) and (t, n) then t = cos ω T + sin ω N, n = sin ω T + cos ω N (4.13) and the necessary and sufficient condition r s = φt becomes (φ cos ω)(cos ω T + sin ω N) = (ω s + f sin ω)( sin ω T + cos ω N) (4.14) so that φ = cos ω, ω s = sin ω f. (4.15) Accordingly, the following theorem obtains. THEOREM 4.2 Let f (s,τ) be the curvature of an inextensible curve Ɣ 0 : R = R(s,τ) moving on the plane and (T, N) be the corresponding Serret Frenet orthonormal frame. If ω(s, n,τ) obeys the first-order differential equation ω s = sin ω f (4.16) then the unit vector field t = cos ω T + sin ω N (4.17) has constant divergence div t = 1 and r = t + R (4.18) parametrizes the generalized tractrices associated with Ɣ 0. All time-dependent foliations of the plane r = r(s, n = c,τ = t) admitting this divergence property may be so generated. If T and N are parametrized according to T = cos i + sin j, N = sin i + cos j (4.19) then the Serret Frenet and time-evolution equations (4.6) translate into the pair s = f, τ = h s + gf (4.20) which is compatible modulo the system (4.7). Accordingly, the angles ϕ and are related by ϕ = ω +. (4.21) For completeness, it is noted that differentiation of (4.18) shows that the solution of the system (3.12), (3.16) and (3.17) is given by φ = cos ω, ψ = ω n, µ = h cos ω, (4.22) ρφ = hsin ω g cos ω, σψ = ω τ h s h cos ω + g(sin ω f ).

8 56 W. K. SCHIEF et al. 4.3 Integrable connections If we set then (4.16) becomes the Riccati equation ω = 2 arctanχ (4.23) which, in turn, admits the solution where y 1 and y 2 obey the linear system ) ( y1 χ s = χ f 2 (1 + χ 2 ) (4.24) χ = y 2 y 1, (4.25) = 1 ( )( 1 f y1 y 2 s 2 f 1 y 2 The latter constitutes the reduction q = r = f of the AKNS scattering problem (21) s = 1 2 ( iλ q r iλ ). (4.26) ) (4.27) associated with the mkdv hierarchy for the particular choice of the spectral parameter λ = i. Thus, it has been established that, in principle, the complete class of time-dependent foliations of the plane subject to div t = 1 may be obtained by arbitrarily specifying the function f and subsequently integrating the linear systems (4.7), (4.26). It is evident that, in practice, explicit integration of these linear systems may not be achieved for generic f. Infinite sequences of potentials for which integration is possible may, however, be generated iteratively by means of Darboux transformations (17) provided that the AKNS spectral problem (4.27) may be solved for a seed potential f and arbitrary spectral parameter λ. This is discussed in section 6. In summary, the complete class of time-dependent fibre distributions subject to div t = 1 may be obtained by arbitrarily specifying the motion of a base curve and subsequently determining the associated generalized tractrices. In what follows, the preceding results are applied to determine kinematically admissible non-steady planar motions of an ideal fibre-reinforced fluid. Remarkably, it turns out that the kinematic conditions do not impose any further constraints on the geometry of the fibres. 5. Kinematically admissible motions It is recalled that the kinematics of the two-dimensional motion of an ideal fibre-reinforced fluid is of particular significance. Here, it is shown that once a viable 2+1-dimensional geometry with div t = 1 has been constructed by the procedure of the previous section then associated motions of the fibre-reinforced fluid may be calculated explicitly. The constraints on these motions imposed by natural boundary conditions are derived. 5.1 The kinematic conditions On introduction of the planar decomposition q = vt + wn, (5.1)

9 MOTIONS OF A FIBRE-REINFORCED FLUID 57 the kinematic conditions yield, on use of (3.1) and (3.13), together with But, in two dimensions, div q = 0, t + (q )t = (t )q (5.2) t δv δw + vdiv t + + wdiv n = 0 (5.3) δs δn δv δs = wκ, δw µ + wθ = δs. (5.4) κ + div n = 0, (5.5) so that (5.3) reduces to δw + vθ = 0. (5.6) δn Accordingly, with the orthogonal parametrizations (3.10), the above kinematic conditions become v s = wφκ, w n = vψθ, µ + wθ = w s φ. (5.7) If attention is now restricted to geometries with constant divergence div t = 1 then the kinematic conditions reduce to the linear system w s = w cos ω h, together with the residual continuity requirement v = w n ω n (5.8) v s = w sin ω. (5.9) However, the relations (5.8) combine to show that (5.9) holds automatically. Hence, 2+1- dimensional geometries with div t = 1 are privileged in the present ideal fibre-reinforced fluid context in that the kinematic conditions reduce to two linear equations in the two velocity components v and w. This is summarized as follows. THEOREM 5.1 For any dimensional geometry with div t = 1,ifv and w be determined by w s = w cos ω h, v = w n, (5.10) ω n where ω and h are defined as in Theorem 4.2 and (4.5), then obeys the kinematic conditions q = vt + wn (5.11) div q = 0, t + (q )t = (t )q. (5.12) t

10 58 W. K. SCHIEF et al. On use of (3.16) 2, the differential equation (5.10) 1 may be brought into the form ( ) w ψ + σ = 0 (5.13) s so that w is given explicitly by w = w 0 (n,τ)ω n + ω τ + h s + h cos ω g sin ω + gf. (5.14) Thus, the determination of kinematically admissible motions of ideal fibre-reinforced fluids with div t = 1 has been reduced to a purely geometric problem. In fact, it is readily verified that in terms of r and arc length S = ln ω n (5.15) along the fibres, the fluid velocity as given by (5.10) 2, (5.11) and (5.14) may be brought into the following compact form. COROLLARY 5.2 Let r(s, n,τ)be a time-dependent foliation of (part of ) the plane with div t = 1 as constructed in Theorem 4.2. Then q = r τ (S τ + w 0 S n + w 0n )t + w 0 e S n, (5.16) where w 0 = w 0 (n,τ)and S = ln ω n, obeys the kinematic conditions (5.12). 5.2 The boundary conditions As stated in Theorem 4.1, at any instant t = τ, the fibres are bounded by the curves Ɣ ± : R ± (s,τ)= R ± N (5.17) which are parallel to the base curve Ɣ 0. Consequently, the fibres are perpendicular to the boundary Ɣ = Ɣ + Ɣ. Specifically t =±N on Ɣ ±. (5.18) Thus, relations (3.7), (4.19) and (4.21) imply that Ɣ is composed of the two level curves Ɣ ± : ω =± π 2. (5.19) The expression (4.22) 1 for φ reveals that this is equivalent to stating that so that the metric φ(ɣ) = 0 (5.20) I = φ 2 ds 2 + ψ 2 dn 2 (5.21) degenerates on the boundary. By assumption, the fibres are convected with the fluid and hence it is required that the flow through the boundary Ɣ vanishes. Here, we demand that the curves Ɣ ± are not self-intersecting. This implies that the curvature of the (continuous) base curve is constrained by f < 1.

11 MOTIONS OF A FIBRE-REINFORCED FLUID 59 The condition for the particles on the boundary to remain on Ɣ is given by (q R τ ) N = 0 on Ɣ (5.22) or, equivalently, η + (q )η = 0 t on Ɣ, (5.23) where the function η is such that its level curves include Ɣ + and Ɣ. Evaluation of (5.22) or (5.23) for η = sin ω is readily shown to produce v =±h on Ɣ ±. (5.24) Since the variable n labels the fibres, any function of n (and τ) may be regarded as being defined on the set of fibres. In this sense, the above boundary conditions involve functions which are defined on two disjoint sets of fibres. Indeed, for instance, if ω(s 0 ) = π/2 for some s 0 then ω s (s 0 )>0by virtue of the differential equation (4.16). Accordingly, π/2 <ω<3π/2 since ω s ( π/2) <0 and ω s (3π/2) <0. Thus, any fibre which is connected to Ɣ + cannot reach Ɣ. We therefore conclude that (5.24) serves to constrain consistently the function of integration in the general solution of the differential equation (5.10) 1. Specifically, differentiation of the expression (5.14) for w yields v = w 0n w 0 ω nn ω n ω τn ω n + h sin ω + g cos ω (5.25) so that the boundary conditions (5.24) become ω nn w 0n + w 0 ω + ω τn n Ɣ ω = 0. (5.26) n Ɣ In the following section, it is demonstrated how in the case of a straight but, in general, moving base curve, imposition of the above boundary condition determines the motion of the fibre-reinforced fluid. 6. Motions associated with straight base curves and their Darboux transforms 6.1 Straight base curves In order to illustrate the preceding analysis, we here consider a straight base curve Ɣ 0 which undergoes an arbitrary rigid motion. In this case, the position vector of Ɣ 0 adopts the form ( ) ( ) cos α R = st + P, T =, P =, (6.1) sin β where and α, β are functions of τ only. Differentiation with respect to τ yields ( ) sin R τ = s τ N + P τ, N = cos (6.2) so that g = P τ T, h = s τ + P τ N (6.3)

12 60 W. K. SCHIEF et al. in the decomposition (4.5) 2. On the other hand, since f = 0, the solution of the differential equation (4.16) reads ω = 2arctan[c(n,τ)e s ]. (6.4) Consequently, the unit tangent to the fibres is given by ( ) cos ϕ t =, ϕ = ω +, (6.5) sin ϕ leading to ( ) cos sin s + 1 c2 e 2s 1 + c r = 2 e 2s sin cos 2ce s + P. (6.6) 1 + c 2 e 2s Thus, the fibres comprise two families of classical tractrices (18) corresponding to c > 0 and c < 0 respectively which are separated by the straight fibre c = 0. The former have the generic shape ( ) s tanh s. (6.7) sech s In order to obtain a physically meaningful solution of the kinematic equations, we now consider the two families of semi-tractrices F ± represented by F + : s ln c, c > 0, F : s ln( c), c < 0, (6.8) and the straight fibre c = 0 denoted by F 0. The fibres F ± emanate from the boundaries Ɣ ± given by Ɣ + : s = ln c, Ɣ : s = ln( c) (6.9) respectively and, together with F 0, globally foliate the strip bounded by Ɣ ±. The fibres and the corresponding semi-circular orthogonal trajectories are displayed in Fig. 2. Fig. 2 Fibre distribution and orthogonal trajectories associated with a straight base curve

13 MOTIONS OF A FIBRE-REINFORCED FLUID 61 Integration of the differential equation (5.10) 1 (or evaluation of (5.14)) now results in w = q(n,τ)es 1 + c 2 e 2s + 1 c2 e 2s 1 + c 2 e 2s h + τ (6.10) so that v = q n + qce2s 2ces 2c n 1 + c 2 + e2s 1 + c 2 h. (6.11) e2s Since evaluation of the latter on the boundary s = ln c yields v Ɣ± =±h Ɣ± q n + q 2c n 2c, (6.12) the boundary conditions (5.24) show that it is required to set q = 2q 0 (τ)c. (6.13) Accordingly, v = q 0 cos ω + h sin ω, w = q 0 sin ω + h cos ω + τ (6.14) and hence q = q 0 T + hn + τ n. (6.15) In particular, in the steady case, the flow is uniform. 6.2 Base curves corresponding to breather potentials As indicated in section 4, motions of higher complexity may now be generated by means of Darboux transformations. Indeed, if the AKNS spectral problem (4.27) may be solved for a seed potential q = r = f and arbitrary spectral parameter λ then application of the Sym Tafel formula (22) to iterated Darboux transformations (see, for example, (17)) produces in a purely algebraic manner explicit expressions for both base curves and associated fibre distributions. Here, we summarize the results corresponding to a double Darboux transformation. These may be verified directly. Application of a single Darboux transformation to the AKNS spectral problem with potential f = 0 leads to a loop soliton potential f 1 (17) which cannot be used in the current physical context since the corresponding base curve exhibits a (self-intersecting) loop associated with a multi-valued flow. Two iterations of suitably chosen Darboux transformations give rise to the potential where λ 2 cosh χ 1 cos χ 2 λ 1 sinh χ 1 sin χ 2 f 12 = 4λ 1 λ 2, (6.16) χ 1 = λ 1 s + η 1, χ 2 = λ 2 s + η 2, = λ 2 2 cosh2 χ 1 + λ 2 1 sin2 χ 2 (6.17) and λ i = λ i (τ), η i = η i (τ), which may be regarded as a breather potential. This is due to the fact that if we make the choice η i = ( 1) i+1 λ i τ, λ λ2 2 = 1, λ i = const then the primitive ( ) λ1 sin χ 2 u 12 = 4arctan (6.18) λ 2 cosh χ 1

14 62 W. K. SCHIEF et al. of f 12 = u 12s constitutes the well-known breather solution of the sine-gordon equation u sτ = sin u. (6.19) This solution is of relevance to both the differential geometry of pseudospherical surfaces (23) and Frenkel and Kontorova s crystal dislocation theory (24). It turns out that the breather potential corresponds to well-defined motions of fibre-reinforced fluids provided that the time-dependent parameters λ i are chosen appropriately. The (time-dependent) base curve associated with the breather potential is of the form R 12 = Q(τ)R + P(τ), (6.20) where the matrix of rotation Q and the vector of translation P are arbitrary functions of τ and ( ) ( ) s R = 4λ 1λ 2 λ2 cosh χ 1 sinh χ 1 λ 1 cos χ 2 sin χ 2 0 (λ λ2 2 ). (6.21) λ 2 cosh χ 1 cos χ 2 + λ 1 sinh χ 1 sin χ 2 Moreover, the solution of the linear system (4.26) reads y 12,1 = (A + B)α(n,τ)e s/2 + (C + D)β(n,τ)e s/2, y 12,2 = (C D)α(n,τ)e s/2 + (A B)β(n,τ)e s/2 (6.22) Fig. 3 Fibre distribution associated with a breather potential corresponding to λ 1 = 1/8 and λ 2 = 1/2 Fig. 4 Fibre distribution associated with a breather potential corresponding to λ 1 = 1/8 and λ 2 = 1/6

15 MOTIONS OF A FIBRE-REINFORCED FLUID 63 with the coefficients A = 1 + (λ λ2 2 )λ2 2 cosh2 χ 1 λ 2 1 sin2 χ 2, λ 2 cosh χ 1 sinh χ 1 + λ 1 sin χ 2 cos χ 2 B = 2λ 1 λ 2, λ 2 cosh χ 1 cos χ 2 λ 1 sinh χ 1 sin χ 2 C = 2λ 1 λ 2, (6.23) The angle ω 12 is then given by D = 2λ 1 λ 2 (λ λ2 2 )cosh χ 1 sin χ 2. ω 12 = 2arctan y 12,2, (6.24) y 12,1 wherein we may make the identification α(n, τ)/β(n, τ) = n without loss of generality. Finally, the fibres are obtained from r 12 = Q(τ)r + P(τ), r = cos ω T + sin ω N + R (6.25) by setting n = const, τ= const. Here, the unit tangent T = R s and the principal normal N to the (reduced) base curve R are readily shown to be ( ) ( ) ( ) 1 cosh χ 1 sin χ 2 2λ1 λ 2 cosh χ 1 sin χ T = 4λ 1 λ λ 2 1 sin2 χ 2 λ 2, N = T. (6.26) 2 cosh2 χ The inequality [ ( )] 2 s 16λ 2 1 R 0 (λ (6.27) λ2 2 )2 now demonstrates that the fibre distributions associated with the breather potential f 12 may be regarded as deformations of those associated with a straight base curve in that the former converge to the latter uniformly as λ 1 0. In particular, once again, there exist two families of fibres F ± which emanate from the boundaries Ɣ ± corresponding to n > 0 and n < 0 respectively. These fibres asymptotically approach the remaining fibre Ɣ 0 obtained by setting n = 0. The grey curves in Figs 3 and 4 represent fibre distributions for two sets of parameters λ 1 and λ 2 and η i = 0. The black curves constitute the base curves Ɣ 0 and the parallel boundaries Ɣ ± at unit distance. The corresponding motions may be obtained explicitly from (5.16). References 1. L. S. Da Rios, Sul moto d un liquido indefinito con un filetto vorticoso, Rend. Circ. Mat. Palermo 22 (1906) T. Levi-Civita, Attrazione Newtoniana dei tubi sottili e vortici filiformi, Ann. R. Scuola. Norm. Sup. Pisa, Zanichelli, Bologna (1932). 3. R. Betchov, On the curvature and torsion of an isolated vortex filament, J. Fluid. Mech. 22 (1965)

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