1 Introduction
Different aspects of theory of Navier–Stokes equations were developed in numerous publications (see [1], [2], [3], [4], [5] and references therein). During the past 190 years, many exact solutions to the Navier–Stokes equations (1823) were derived. There are several reviews devoted to the exact solutions possessing different symmetries [6], [7], [8], [9], [10].
We introduce in this article new axisymmetric time-dependent exact solutions to the Navier–Stokes equations. The solutions are studied in the cylindrical coordinates r,z,φ and depend on variables r, z, and time t. We construct an infinite-dimensional space of solutions for which fluid velocity V(r,z,t) is analytic in the whole space ℝ3 and is defined for all moments of time t. Inner transformations acting on the space of exact solutions are presented that generate from any exact solution an infinite sequence of new ones.
We study the bifurcations of the instantaneous (for t=t0) phase portraits of the viscous fluid flows for the new exact solutions. Namely, we investigate dynamics of the vortex blobs and vortex rings, which are the maximal compact domains invariant (for any fixed moment of time t0) with respect to the vorticity vector field ∇×V(r,z,t0). As known, for the ideal incompressible fluid, the vorticity field is frozen into the fluid flow. Therefore, the vortex blobs and vortex rings are transported with the ideal fluid flow; their volume is constant. We show that for the constructed exact solutions to the Navier–Stokes equations, the vortex blobs and vortex rings are not frozen into the viscous fluid flow and collapse and disappear as t → ∞.
For the new exact solutions, we study the behaviour of the volume Vm(t) of the vortex blob. We show that despite the analyticity of exact solutions the function Vm(t) is not even continuous. The function Vm(t) is a discontinuous monotonously decreasing function of time t that has jumps down and infinite derivatives at an infinite sequence of moments of time −∞<⋯<tk<⋯<t3<t2<tm, where tk→−∞ when k → ∞. The volume function Vm(t) has its minimal value at t=tm. Function Vm(t) is defined for t∈(−∞,tm). Here, tm is the maximal time when the vortex blob exists; it does not exist for t>tm.
2 Infinite-Dimensional Space of Exact Solutions
I. In this article, we derive and study new exact solutions to the Navier–Stokes equations
(1)∂V∂t+(V⋅∇)V=−1ρ∇p+∇Ψ+νΔV,∇⋅V=0,
where V(x,t) is the fluid velocity; p(x,t), the pressure; ρ, a constant density; Ψ(x,t), an arbitrary gravitational potential; ν(t), the kinematic viscosity that is an arbitrary piece-wise continuous nonnegative function of time t; and Δ, the Laplace operator.
Theorem 1The Navier–Stokes equations (1) have exact z-axisymmetric solutions
(2)V(r,z,t)=αξre^φ+2ξe^z+f(t)B(r,z),
(3)p(r,z,t)=ρ[C+Ψ(r,z,t)+α2ξ2r2+α2ξf(t)ψ(r,z)−12|V(r,z,t)|2],
where α, ξ, and C are arbitrary constant parameters; e^r, e^z, e^φ are the unit vector fields tangent to the cylindrical coordinates r,z,φ. The function of time f(t) is
(4)f(t)=exp(−α2∫0tν(τ)dτ).
The z-axisymmetric vector fields B(r,z) are steady and have the form
(5)B(r,z)=−1r∂ψ∂ze^r+1r∂ψ∂re^z+αψre^φ,
where the streamfunction ψ(r,z) is an arbitrary solution to the equation
(6)∂2ψ∂r2−1r∂ψ∂r+∂2ψ∂z2=−α2ψ.
The vector fields B(r,z) (5)–(6) satisfy the Beltrami equation
(7)∇×B(x)=αB(x).
If for some period of time c≤t≤d viscosity ν(t)=0, then solution (2), (3) becomes a steady solution for c≤t≤d to the Euler equations for ideal incompressible fluid dynamics.
Proof.(a) The vorticity field for the vector field (5) has the form
(8)∇×B(r,z)=−αr∂ψ∂ze^r+αr∂ψ∂re^z−1r(∂2ψ∂r2−1r∂ψ∂r+∂2ψ∂z2)e^φ.
Substituting here (6), we arrive at the Beltrami equation (7).
As a consequence of (7), we get ∇⋅B=0. As re^φ=−ye^x+xe^y, the vector fields V(r,z,t) (2) also satisfy the incompressibility equation ∇⋅V=0.
(b) Let us show that vector fields V(r,z,t) (2) satisfy equation
(9)(∇×V)×V=∇[−α2ξ2r2−α2ξf(t)ψ(r,z)].
Indeed, using equation
(10)∇×[αξre^φ+2ξe^z]=2αξe^z
and (7), we find for the vector fields V(r,z,t) (2):
(11)∇×V(r,z,t)=2αξe^z+αf(t)B(r,z)=αV(r,z,t)−α2ξre^φ.
The equation proves that vector fields V(r,z,t) (2) for ξ ≠ 0 are not the Beltrami fields. Equation (11) yields
(12)(∇×V)×V=(αV−α2ξre^φ)×V=−α2ξre^φ×V.
Applying to (12) the identities e^φ×e^z=e^r, e^φ×e^r=−e^z, e^φ×e^φ=0 and (2), (5), we find
(13)(∇×V)×V=(−α2ξre^φ)×[αξre^φ+2ξe^z+f(t)(−1r∂ψ∂ze^r+1r∂ψ∂re^z+αψre^φ)]=−2α2ξ2re^r−α2ξf(t)[∂ψ∂re^r+∂ψ∂ze^z]=∇[−α2ξ2r2−α2ξf(t)ψ(r,z)].
(c) Using the well-known identity
(V⋅∇)V=(∇×V)×V+∇(12|V|2)
and (13), we present the Navier–Stokes equations (1) in the form
(14)∂V∂t=−∇[1ρp−Ψ−α2ξ2r2−α2ξf(t)ψ(r,z)+12|V|2]+ν(t)ΔV.
Applying to the identity ΔB=∇(∇⋅B)−(∇×)(∇×B) the Beltrami equation (7) and equation ∇⋅B=0, we derive ΔB=−α2B. Formula re^φ=−ye^x+xe^y implies Δ(αξre^φ+2ξe^z)=0. Therefore, for the vector field V(r,z,t) (2), we find
(15)ΔV=Δ[αξre^φ+2ξe^z+f(t)B]=f(t)ΔB=−α2f(t)B.
Substituting formulas (2) and (15) into the Navier–Stokes equation (14), we transform it to the form
(16)∂[αξre^φ+2ξe^z+f(t)B]∂t=−∇[1ρp−Ψ−α2ξ2r2−α2ξf(t)ψ(r,z)+12|V|2]−α2ν(t)f(t)B.
Inserting here formula (3) for the pressure p(r,z,t), we find that (16) is reduced to equation
(17)df(t)dt=−α2ν(t)f(t)
that is identically satisfied by the function f(t) (4).
(d) If ν(t)=0 for c≤t≤d, then Navier–Stokes equations (1) become Euler equations for ideal incompressible fluid. Equation (17) yields f(t)=const for c≤t≤d. Hence, solutions (2), (3) for c≤t≤d become steady solutions to the Euler equations.
Analogously, if viscosity ν(t)=0 on two intervals of time c≤t≤d and c1≤t≤d1 and ν(t)>0 for d<t<c1, then solution (2), (3) describes transition of viscous fluid between two steady flows of inviscid fluid for c≤t≤d and c1≤t≤d1. ⊡
Remark 1The new exact solutions (2), (3) depend on the infinite-dimensional family of axisymmetric Beltrami fields B(r,z) (5) and (6) and on two arbitrary parameters α and ξ. Therefore, Theorem 1 presents an infinite-dimensional space ℒα of exact solutions to the Navier–Stokes equations (1).
Remark 2After changing parameter α to (−α) in the exact solution (2), (3), one gets the exact viscous flow having the opposite rotation around the axis z.
Remark 3Using results of our article [11], we get that the z-axisymmetric Beltrami vector fields B(r,z) (5) to (7) admit the integral representation
(18)B(x)=∫S2[sin(αk⋅x)T(k)+cos(αk⋅x)k×T(k)]dσ.
Here, T(k) is an arbitrary z-axisymmetric differentiable vector field tangent to the unit sphere S2: k⋅k=1, and dσ is the standard Euclidean measure on the sphere S2. Indeed, in [11], we proved that the general nonsymmetric solution to the Beltrami equation (7) has form (18), where T(k) is an arbitrary vector field tangent to the sphere S2, and dσ is an arbitrary measure on S2. The Beltrami field B(x) (18) evidently becomes z-axisymmetric if the vector field T(k) and the measure dσ are z-axisymmetric. As we have shown in [11], the absolute value |B(x)| decreases as C/|x| when |x|→∞; see also [12].
Remark 4The solutions (2), (3) exist for all moments of time t∈(−∞,∞). Below we assume that viscosity ν(t)≥ν0>0; for this case, function f(t) (4) monotonously decreases and f(t)→∞ at t→−∞ and f(t)→0 at t → ∞. Therefore, (2) yields that the exact solutions at t → ∞ tend to the steady flow
(19)V~(r,z)=αξre^φ+2ξe^z,
that according to (10) has constant vorticity ∇×V~(r,z)=2αξe^z. Therefore, solutions (2) describe a relaxation of the axisymmetric flows (2) to the steady flow (19) with constant vorticity 2αξe^z.
Solutions (2) at t→−∞ have the leading term f(t)B(r,z) which describes a Beltrami flow with the streamfunction f(t)ψ(r,z).
3 Infinite-Dimensional Family of Inner Transformations
Theorem 2If two axisymmetric vector fields V1(r,z,t) and V2(r,z,t) are solutions of form (2) to the Navier–Stokes equations (1), then vector fields
(20)VMN(r,z,t)=∑n=1Ma1nV1(r,z+u1n,t)+∑n=1Ma2nV2(r,z+u2n,t)+αξre^φ+2ξe^z+∑k=1N∑n=1M[bkn∂nV1(r,z+zkn,t)∂zn+ckn∂nV2(r,z+z~kn,t)∂zn]
also are solutions to the Navier–Stokes equations (1). The constants ajn, ujn(j=1,2), bkn, ckn, zkn, z~kn are arbitrary, n=1,⋯,M, k=1,⋯,N.
Proof.Let vector fields Vj(r,z,t) (2) (j=1,2) are as follows:
(21)Vj(r,z,t)=αξjre^φ+2ξje^z+f(t)Bj(r,z),
where vector fields Bj(r,z) have the form (5) and satisfy the Beltrami equation (7). Formulae (20), (21) yield
(22)VMN(r,z,t)=αξ¯re^φ+2ξ¯e^z+f(t)[∑n=1Ma1nB1(r,z+u1n)+∑n=1Ma2nB2(r,z+u2n)]+f(t)∑k=1N∑n=1M[bkn∂nB1(r,z+zkn)∂zn+ckn∂nB2(r,z+z~kn)∂zn],
where parameter ξ¯=ξ+∑n=1M(a1nξ1+a2nξ2). Here, vector fields ∂nBj(r,z+zkn)/∂zn (5) (j=1,2) correspond to the streamfunctions ∂nψj(r,z+zkn)/∂zn. The latter together with the streamfunctions ψj(r,z) for Bj(r,z) evidently satisfy (6) because it is invariant under arbitrary differentiations ∂n/∂zn and translations z→z+zkn. Therefore, all vector fields ∂nBj(r,z+zkn)/∂zn and Bj(r,z+ujn) satisfy the Beltrami equation (7). Hence, vector field VMN(r,z,t) (22) has the form
(23)VMN(r,z,t)=αξ¯re^φ+2ξ¯e^z+f(t)BMN(r,z),
where vector field BMN(r,z) is the linear combination of all steady Beltrami fields in (22), having the common factor f(t). As the Beltrami equation (7) is linear, we get that vector field BMN(r,z) also is a Beltrami field. Hence, vector fields VMN(r,z,t) (20), (23) have the form (2) and therefore by Theorem 1 define exact solutions to the Navier–Stokes equations (1). The corresponding pressure pMN(r,z,t) is defined by the formula (3) with the new parameter ξ¯=∑n=1M(a1nξ1+a2nξ2). ⊡
Remark 5Theorem 2 proves that the space of exact solutions ℒα for a fixed parameter α and variable parameter ξ is linear with respect to the vector fields V(r,z,t) (2) and is nonlinear with respect to the pressure p(r,z,t) (3).
Corollary 1The infinite-dimensional space Lα of exact solutions (2)–(3) is invariant under the transformations:
(24)V(r,z,t)→FMN(V(r,z,t))=αξre^φ+2ξe^z+∑n=1ManV(r,z+un,t)+∑k=1N∑n=1Mbkn∂nV(r,z+zkn,t)∂zn.
Here, an,un,bkn,zkn are arbitrary parameters.
Proof.Applying Theorem 2 for the case V2(r,z,t)=αξre^φ+2ξe^z, we get that transformations (24) are special cases of transforms (20). The transformations (24) commute with each other because the differentiations ∂n/∂zn commute with arbitrary translations z→z+uk. ⊡
4 Backlund Transforms between the Axisymmetric Helmholtz Equation and the Linear Case of the Grad-Shafranov Equation
As known, the Helmholtz equation
(25)ΔF(x)=−α2F(x)
for the z-axisymmetric functions F(r, z) has the form
(26)Frr+1rFr+Fzz=−α2F.
Consider two cases of the Grad–Shafranov equation [13], [14]
(27)ψrr−1rψr+ψzz=−r2dPdψ−GdGdψ,
corresponding to (a) P(ψ)=0, G(ψ)=αψ and (b) P(ψ)=0, G(ψ)=β2+α2ψ2. For both cases, (27) becomes
(28)ψrr−1rψr+ψzz=−α2ψ.
Equations (26) and (28) describe absolutely different physical phenomena. Therefore, the closeness in form of these equations is striking.
We introduce the new Backlund transforms between the axisymmetric Helmholtz equation (26) and the linear case (28) of the Grad–Shafranov equation, which coincides with (6). The Backlund transforms are used in Section 6 below.
Lemma 1(a) Backlund transform
(29)ψ(r,z)=r∂F(r,z)∂r
maps any solution of (26) into a solution to (28).
(b) Backlund transform
(30)F(t,z)=1r∂ψ(r,z)∂r
maps any solution of (28) into a solution to (26).
Proof.(a) Rewrite the Helmholtz equation (26) in the form r−1(rFr)r+Fzz=−α2F. Denoting here rFr=ψ and differentiating with respect to r, we find r−1ψrr−r−2ψr+Frzz=−α2Fr. Multiplying this equation with r and putting rFr=ψ, we get (28).
(b) Represent the linear case (28) of the Grad–Shafranov equation in the form r(r−1ψr)r+ψzz=−α2ψ and denote r−1ψr(r,z)=F(r,z). After differentiation with respect to r, we get rFrr+Fr+ψrzz=−α2ψr. Multiplying with r−1 and putting r−1ψr=F, we get (26). ⊡
Remark 6The composition of Backlund transforms (29) and (30) is
(31)F~(r,z)=Frr(r,z)+r−1Fr(r,z).
By Lemma 1, the mapping (31) is auto-Backlund transform of the axisymmetric Helmholtz equation (26); it has also the form F~=−Fzz−α2F.
Remark 7The composition of Backlund transforms (30) and (29) is
(32)ψ~(r,z)=ψrr(r,z)−r−1ψr(r,z).
The mapping (32) by Lemma 1 is the auto-Backlund transform of the linear case (28) of the Grad–Shafranov equation (27). The transform has also the form ψ~=−ψzz−α2ψ.
5 Vortex Blobs and Vortex Rings
In view of (5), vector fields (2) have the form
(33)V1(r,z,t)=−1r∂ψ1∂ze^r+1r∂ψ1∂re^z+αψ1re^φ,
where ψ1(r,z,t) is the time-dependent streamfunction:
(34)ψ1(r,z,t)=ξr2+f(t)ψ(r,z).
Remark 8The inner transforms (24) correspond to the following transformations of the streamfunctions ψ1(r,z,t):
(35)ψ1(r,z,t)→FMN(ψ1(r,z,t))=ξr2+∑n=1Manψ1(r,z+un)+∑k=1N∑n=1Mbkn∂nψ1(r,z+zkn)∂zn.
Equation (33) implies that for any fixed moment of time t0 the surface ψ1(r,z,t0)=const (the angle φ∈𝕊1 is arbitrary) is an invariant submanifold for the vorticity vector field ∇×V1(r,z,t0). This follows from formula (11): ∇×V1=αV1−α2ξre^φ and the z-axisymmetry of the flow.
As this surface ψ1(r,z,t0)=const is z-axisymmetric, it is a disjoint union of either some spheres 𝕊2 or some tori 𝕋2=Cψ1(t)×𝕊1 or some cylinders ℂ2=Rψ1(t)×𝕊1. Here, Cψ1(t) and Rψ1(t) are the level curves ψ1(r,z,t)=const in the poloidal plane (r, z) for a fixed time t. The curves Cψ1(t)⊂(r,z) are closed, and the curves Rψ1(t)⊂(r,z) are infinite. The circle 𝕊1 corresponds to the angular variable φ: 0≤φ≤2π.
Assume that a surface ψ1(r,z,t0)=C1 bounds a compact connected domain D1. We call the domain D1 maximal and denote it Dm if it is not contained in any bigger compact connected domain Dm¯ bounded by a surface ψ1(r,z,t0)=C1¯. If such a maximal domain Dm intersects the axis of symmetry r = 0, then topologically it is a z-axisymmetric ball 𝔹m3, which we call a vortex blob because it is invariant with respect to the vorticity field ∇×V1(r,z,t0).
Remark 9Suppose that function ψ(r, z) in (34) is obtained by transform (29). Then on the axis of symmetry r = 0, we have ψ1(0,z,t)=0. As the vortex blobs Dm intersect the axis r = 0, the same is true for their boundaries defined by equation ψ1(r,z,t)=Cm. Putting here r = 0, we get Cm=0. Hence, the boundaries of the vortex blobs satisfy the equation
(36)ψ1(r,z,t)=0.
Equation (36) can define several connected components; see exact solutions in Section 6 and 8.
If Cm≠0, then the corresponding maximal compact connected domain Dm bounded by the surface ψ1(r,z,t0)=Cm≠0 does not intersect the axis of symmetry r = 0 because ψ1(0,z,t0)=0. Therefore, the domain Dm for Cm≠0 topologically is a 3-dimensional z-axisymmetric ring 𝔹m2(t0)×𝕊1, where 𝔹m2(t0)⊂(r,z) topologically is equivalent to a 2-dimensional ball in the poloidal plane (r, z). The boundary of the ring 𝔹m2(t0)×𝕊1 is a torus 𝕋2=Cψ1(t0)×𝕊1 where Cψ1(t0)=∂𝔹m2(t0) is a closed level curve ψ1(r,z,t0)=Cm≠0, φ = 0. As the ring 𝔹m2(t0)×𝕊1 is invariant with respect to the vorticity field ∇×V1(r,z,t0), we call it a vortex ring. In view of (11), the vortex blobs and vortex rings are invariant also with respect to the velocity field V1(r,z,t0).
As known, for an ideal incompressible fluid, the vorticity field ∇×V(x,t) is frozen into the fluid flow. Therefore, for the inviscid fluid (ν = 0), the vortex blobs and vortex rings are transported with the fluid flow. For a viscous fluid with ν(t)≠0, the vortex blobs and vortex rings are not frozen into the viscous fluid flow and undergo a more sophisticated dynamics and can collapse and disappear at some moments of time t.
Both vortex blobs and vortex rings are equivalently represented by their intersections with the poloidal plane (r, z), φ = 0. Below we study dynamics in time t of the poloidal sections of the vortex blobs and vortex rings for the concrete exact solutions derived in Section 6.
6 Bifurcations in Exact Solutions to the Navier–Stokes Equations
I. The Helmholtz equation (25) for the spherical functions F(R), R=r2+z2, has the form FRR+2FR/R=−α2F. This equation has an important exact solution F(R)=sin(αR)/R. The solution evidently is z-axisymmetric and therefore satisfies (26). Applying the Backlund transform (29), we get that function
(37)ψ(r,z)=−rα3∂F(r,z)∂r=−r2G2(αR)=−r2α2R2[cos(αR)−sin(αR)αR]
satisfies (28) [or (6)]. Therefore, the corresponding z-axisymmetric vector field B(r,z) (5) by Theorem 1 satisfies the Beltrami equation (7). Function G2(u) in (37), G2(u)=u−2(cosu−u−1sinu), is connected with the Bessel function J3/2(u) of order 3/2 by the relation
G2(u)=−π/2u3/2J3/2(u).
Remark 10In another form, Beltrami field B(r,z) [(5) and (37)] was first derived in 1899 in the pioneer article by W.M. Hicks [16] that is the historical precursor of many works on fluid and plasma equilibria. The Beltrami field B(r,z) (5), (37) was rediscovered in the theory of plasma equilibria in terms of Bessel functions J3/2(u) [19] by Chandrasekhar [20] and Woltjer [21] as a model of axisymmetric plasma equilibria and is called the spheromak field. The term “spheromak” was first introduced in [22]; see review [23]. Moduli spaces of vortex knots for the spheromak Beltrami field in different invariant domains were presented in [24] and for another Beltrami field in [25].
II. We will use in this article the following functions Gn(u) connected with the Bessel functions Jn−1/2(u):
(38)G0(u)=−cosu,G1(u)=duduG0(u)=sinuu=π/2u1/2J1/2(u),G2(u)=duduG1(u)=1u2[cosu−sinuu]=−π/2u3/2J3/2(u),G3(u)=duduG2(u)=1u4[(3−u2)sinuu−3cosu]=π/2u5/2J5/2(u).G4(u)=duduG3(u)=1u6[(6u2−15)sinuu−(u2−15)cosu]=−π/2u7/2J7/2(u).
All functions Gn(u) are analytic everywhere and have the nonzero values at u = 0:
(39)G1(0)=1,G2(0)=−1/3,G3(0)=1/15,G4(0)=−1/105.
The plot of function y1(u)=G2(u) is shown in Figure 1. The range of function G2(u) is the segment I∗=(−1/3,ξ1≈0.02872).
Functions Gn(u) (38) are even and satisfy the easily verifiable identities
(40)G0(u)+G1(u)+u2G2(u)=0,G1(u)+3G2(u)+u2G3(u)=0.
The general identity
(41)Gn(u)+(2n+1)Gn+1(u)+u2Gn+2(u)=0,
for Gk+1(u)=u−1dGk(u)/du follows from identities (40) by induction.
III. Vector field V1(r,z,t) (33) with the streamfunction
(42)ψ1(r,z,t)=r2[ξ−f(t)G2(αR)]
has the form
(43)V1(r,z,t)=α2rzf(t)G3(αR)e^r+[2ξ−f(t)(2G2(αR)+α2r2G3(αR))]e^z+αr[ξ−f(t)G2(αR)]e^φ,
where f(t) is the function (4). This vector field together with the pressure p(r, z, t) (3) defined by the formula
p(r,z,t)=ρ[C+Ψ(r,z,t)+α2r2ξ[ξ−f(t)G2(αR)]−12|V1(r,z,t)|2]
is the new exact solution to the Navier–Stokes equations (1).
Remark 11For the vanishing viscosity ν(t)=0, function f(t) (4) equals 1. Fluid flows (43) for f(t)=const and arbitrary parameters α, ξ are equivalent to the steady solutions to Euler equations for the ideal incompressible fluid studied in [26].
Remark 12For exact solutions (42), (43), we find ψ1(0,z,t)=0 for r = 0. Hence, the boundary of a vortex blob is defined by equation ψ1(r,z,t)=0 (36); see Remark 9 above. Therefore, on the boundary, we have ξ=f(t)G2(αR). Hence, the vortex blob is a ball 𝔹ak3 of radius ak defined by the equation
(44)G2(αak)=ξ/f(t)=ξexp(α2∫0tν(τ)dτ)
and its boundary is the sphere 𝕊ak2 of radius R=ak.
IV. Function G2(u)→0 when u → ∞ and has infinitely many oscillations, see its formula in (38). Therefore, from Figure 1, it becomes evident that equation G2(u)=ξ/f(t) (44) for ξ ≠ 0, ξ/f(t)∈I∗=(−1/3,ξ1≈0.02872), has a finite number N(t) of roots and N(t)→∞ when ξ/f(t)→0. That means the vector field V1(r,z,t) in the whole space ℝ3 can have for ξ ≠ 0 a finite number N(t) of invariant spheroids 𝔹ai3, and it has infinitely many invariant spheroids when ξ = 0.
The velocity field V1(r,z,t) (43) does not have any invariant spheroids 𝔹c3(R≤c) for time t satisfying condition ξ/f(t)∉I∗ because for this case (44) has no solutions; see Figure 1.
Dynamical system defined by the vector field V1(r,z,t) (33), (43) has the form
(45)r˙=α2rzf(t)G3(αR),z˙=2ξ−f(t)[2G2(αR)+α2r2G3(αR)],
(46)φ˙=α[(ξ−f(t)G2(αR)].
Dynamics of fluid vanishes on the spheres R=Rk, where G3(αRk)=0 at the moments of time tk defined by equation f(tk)=ξ/G2(αRk).
Equilibrium points (at a fixed time t) of dynamical system (45) are defined by equations z = 0 and
(47)[2G2(u)+u2G3(u)]/2=ξ/f(t),
where u=αR. The second identity (40) yields 2G2(u)+u2G3(u)=−G1(u)−G2(u). Therefore, (47) takes the form
(48)y2(u)=−G1(u)+G2(u)2=−12[sinuu+1u2(cosu−sinuu)]=ξ/f(t).
The plot of function y2(u) (48) is shown in Figure 1. The range of function y2(u) is the segment (−1/3,ξ¯1≈0.11182). Thus, oscillations of function y2(u) are greater than those of function y1(u); see Figure 1. Function y2(u)→0 when u → ∞. Therefore, the number M(t) of roots of (48) is finite for all t and M(t)→∞ when ξ/f(t)→0.
The stream surfaces ψ1(r,z,t)=const for solutions (42), (43) are up-down symmetric and have different structure for ξ > 0 and ξ < 0. The poloidal contours of the stream surfaces for ξ > 0 are shown (for α = 1) in Figures 2–11 for a sequence of increasing moments of time t: −∞,t1<t2<⋯<t8,∞. In Figures 12–19, we show the poloidal contours of the stream surfaces for ξ < 0, α = 1 for a sequence of increasing moments of time t∗: −∞,t1∗<t2∗<⋯<t6∗,∞. The arrows in Figures 2–19 show the direction of the dynamics defined by system (45).
For the solutions (42), (43), all vortex blobs are the balls bounded by certain spheres R=ak (44); they are shown in blue. The roots uk=αRk of (48) for a given time t that are greater than all roots αaj of (44) define equilibria (r=Rk,z=0), which belong to the vortex rings that are shown in Figures 2–9 in pink. The roots uk=αRk (48) are extreme of function ψ1(r,z,t) (42); they are denoted in Figures 2–19 as cj, ai, and sk. Points cj are stable maxima or minima of function ψ1(r,z,t); points ai and sk are unstable saddles. The interiors of each vortex ball and vortex ring are filled with invariant tori 𝕋2=Cψ1(t)1×𝕊1 of dynamical system (45) to (46), where Cψ1(t)1⊂(r,z) is a closed curve defined by equation ψ1(r,z,t)=const (for the given moment of time t).
When ξ/f(t) satisfies the inequalities
(49)ξ1≈0.02872<ξ/f(t)<ξ¯1≈0.11182,ξ>0,
the vector field V1(r,z,t) (43) has finitely many vortex rings and no vortex balls; see Figures 7–9. Figures 2–19 illustrate dynamics of vortex balls and vortex rings. It is evident from these Figures that vortex balls and vortex rings as t → ∞ collapse and disappear.
V. Using (11), we find the corresponding to (43) vorticity field
(50)∇×V1(r,z,t)=α3rzf(t)G3(αR)e^r+α[2ξ−f(t)(2G2(αR)+α2r2G3(αR))]e^z−α2rf(t)G2(αR)e^φ.
In the Cartesian coordinates (x,y,z), the new solution V1(r,z,t) has the form
(51)V1(x,t)=[−αξy+f(t)(αyG2+α2xzG3)]e^x+[αξx+f(t)(−αxG2+α2yzG3)]e^y+[2ξ+f(t)(G1+G2+α2z2G3)]e^z,
where we substituted 2G2+α2r2G3=−(G1+G2+α2z2G3) [applying the second identity (40)]. Everywhere Gn=Gn(αR).
Remark 13The exact solutions (43), (51) for ξ > 0 and for ξ < 0 have the following important distinctions that follow from (44), (48):
(a) If the flow for ξ > 0 at a time t has a vortex blob, then it does have at least one vortex ring; see Figures 3–6. For ξ > 0, there is interval of time t satisfying inequalities (49) when the flow has vortex rings but does not have a vortex blob; see Figures 7–9.
(b) For ξ < 0, there is interval of time t satisfying inequalities
−1/3<ξ/f(t)<ξ¯2≈−0.0648,ξ<0,
when the flow has a vortex blob but does not have any vortex rings; see Figure 17. However, if for ξ < 0 the flow (43), (51) has a vortex ring, then it necessarily has a vortex blob; see Figures 13–16.
7 Discontinuous Volume Function Vm(t)
I. At any fixed time t=t0, the fluid velocity field V1(r,z,t0) (43) and vorticity field ∇×V1(r,z,t0) (50) are tangent to the surfaces of constant level of the streamfunction ψ1(r,z,t0) (42).
The zero level of function (42) at a fixed time t is the union of several spheres 𝕊ai(t)2 of radii ai(t) obeying equation G2(αai(t))=ξ/f(t) or
(52)G2(αai(t))=ξexp(α2∫0tν(τ)dτ).
Vector fields V1(r,z,t) (43) and ∇×V1(r,z,t) (50) on each sphere 𝕊ai(t)2 have the form
(53)V¯1(r,z,t)=α2rf(t)G3(αai(t))[ze^r−re^z],
(54)∇×V¯1(r,z,t)=α3rf(t)G3(αai(t))[ze^r−re^z]−α2rξe^φ.
It is evident from (53) to (54) that the spheres 𝕊ai(t)2 are invariant submanifolds for the flows V1(r,z,t) and ∇×V1(r,z,t). Therefore, the balls 𝔹ai(t)3 bounded by the spheres 𝕊ai(t)2 also are invariant under the vorticity field ∇×V1(r,z,t). Therefore, we call the ball 𝔹am(t)3 of the maximal radius am(t) a vortex blob.
From Figure 1, it is evident that for ξ > 0 the number of solutions ai(t) to (52) is even, say equal to 2N(t). Therefore, inside the vortex blob 𝔹am(t)3, there are 2N(t)−1 invariant balls
(55)𝔹a1(t)3⊂𝔹a2(t)3⊂⋯𝔹a2N(t)−1(t)3⊂𝔹am(t)3.
For ξ < 0, the number of solutions ai(t) to (52) is odd, say equal to 2N(t)+1. Hence, inside the vortex blob 𝔹am(t)3, there are 2N(t) invariant balls
(56)𝔹a1(t)3⊂𝔹a2(t)3⊂⋯𝔹a2N(t)(t)3⊂𝔹am(t)3.
At t→−∞, we have exp(α2∫0tν(τ)dτ)→0. Hence, formula (52) and Figure 1 yield N(t)→∞ when t→−∞.
II. The fluid flow (53) becomes identically zero on the spheres 𝕊aℓ(t)2 defined by equation G3(αaℓ(t))=0. As G3(u)=u−1dG2(u)/du, the equation G3(uℓ)=0 means that the point uℓ=αaℓ(t) is a point of either local maximum or local minimum of function y1(u)=G2(u); see Figure 1. In view of (38), equation G3(u)=0 is equivalent to equation
(57)tanu=3u3−u2.
The first eight roots uℓ of (57) are
u1≈5.7635,u2≈9.0950,u3≈12.3229,u4≈15.5146,u5≈18.6890,u6≈21.8539,u7≈25.0128,u8≈28.1678.
The corresponding values ξℓ=G2(uℓ) are as follows:
(58)ξ1=G2(u1)≈0.02872,ξ2≈−0.0119,ξ3≈0.0065,ξ4≈−0.0041,ξ5≈0.0029,ξ6≈−0.0021,ξ7≈0.0016,ξ8≈−0.0013.
The positive values ξℓ>0 are local maxima of function G2(u); the negative values ξℓ<0 are local minima; see Figure 1.
The vortex blob 𝔹am(t)3 and invariant spheres 𝕊ai(t)2 exist if (52) has some roots ai(t). This is possible only if ξexp(α2∫0tν(τ)dτ) belongs to the range of function G2(u). The plot of function y1(u)=G2(u) in Figure 1 shows that the range of function G2(u) is the segment [−1/3,ξ1≈0.02872]. Here, ξ1 is the maximal value of function y1(u)=G2(u). It is attained at the point u1 satisfying equation G3(u)=u−1dG2(u)/du=0. The first root of equation G3(u)=0 (57) is u1≈5.7635. Hence, we calculate G2(u1)=ξ1≈0.02872.
III. Consider (52) for am(t) and differentiate it with respect to time t. Using equation dG2(u)/du=uG3(u) (38), we find
α2am(t)G3(αam(t))dam(t)dt=α2ν(t)ξexp(α2∫0tν(τ)dτ)=α2ν(t)G2(αam(t)).
Hence, we get
(59)dam(t)dt=ν(t)G2(αam(t))am(t)G3(αam(t)).
The volume Vm(t) of the vortex blob 𝔹am(t)3 is 4πam3(t)/3. Hence, from (59), we derive
(60)dVm(t)dt=4πν(t)am(t)G2(αam(t))G3(αam(t)).
Equation (60) shows that the speed of the change of the vortex blob volume Vm(t) is proportional to the kinematic viscosity ν(t) of the fluid.
IV. For ξ > 0, (52) yields that function G2(αam(t))>0. As um(t)=αam(t) is the maximal root of (52), we see from Figure 1 that dG2(u)/du<0 at u=um(t). Hence, G3(um(t))=um−1(t)dG2(um(t))/du<0. Therefore, from (60), we get dVm(t)/dt<0. Hence, function Vm(t) is monotonously decreasing.
Equation (60) shows that derivative dVm(t)/dt=−∞ at the moment of time t=tk when G3(αam(tk))=0. Function G2(u) has local maximum at u=uk=αam(tk), and for t=tk+ε, there is no invariant sphere of radius close to am(tk) because the two neighbouring spheres 𝔹a2N(t)−13 and 𝔹am(t)3 coincide at t=tk and then disappear. Therefore, the next ball 𝔹a2N(t)−23 in the nested sequence (55) becomes maximal. Hence, the radius am(tk) of the vortex blob jumps down to the value a2N(t)−2(tk) at the moment tk. The volume Vm(t)=4πam3(t)/3 jumps down correspondently. The jumps occur at the moments of time tk when function G2(αam(t)) takes the positive values ξk (58). From (52), we get equation for the corresponding times tk:
(61)∫0tkν(τ)dτ=α−2log(ξk/ξ),
where ξk=G2(uk)>0, and uk satisfies equation G3(uk)=0, am(tk)=uk/α.
For ξ > 0, (52) yields that the maximal time t=tm when the fluid flow V1(r,z,t) (43) has a vortex blob 𝔹am(tm)3 is defined by the condition G2(αam(tm))=ξ1≈0.02872. At this moment, αam(tm)=u1≈5.7635. Hence, we get from (52) the equation for the time tm:
(62)∫0tmν(τ)dτ=α−2log(ξ1/ξ)≈α−2log(0.02872/ξ).
The last vortex blob (ball) 𝔹am(tm)3 has the minimal possible radius am(tm)=u1/α≈5.7635/α. Equation (53) yields that the fluid velocity V1(r,z,tm) is identically zero on the boundary sphere 𝕊am(tm)2. For all times t>tm, the fluid flow V1(r,z,t) (43) does not have any vortex blob.
The plot of the monotonously decreasing discontinuous function Vm(t) is shown in Figure 20.
V. Equation (52) for ξ < 0 yields that function G2(αam(t))<0. The range of negative values of G2(u) is the segment [−1/3,0]. As um(t)=αam(t) is the maximal root of (52), we see from Figure 1 that dG2(u)/du>0 at u=um(t). Hence, G3(um(t))=um−1(t)dG2(um(t))/du>0. Therefore, from (60), we get dVm(t)/dt<0. Hence, function Vm(t) is monotonously decreasing.
Equation (60) shows that derivative dVm(t)/dt=−∞ at the moment of time t=tk when G3(αam(tk))=0. Function G2(u) has local minimum at u=uk=αam(tk), and for t=tk+ε, there is no invariant sphere of radius close to am(tk) because the two neighbouring spheres 𝔹a2N(t)3 and 𝔹am(t)3 coincide at t=tk and then disappear. Therefore, the next ball 𝔹a2N(t)−13 in the nested sequence (56) becomes maximal. Hence, the radius am(tk) of the vortex blob jumps down to the value a2N(t)−1(tk) at the moment tk. The volume Vm(t)=4πam3(t)/3 jumps down correspondently. The jumps occur at the moments of time tk when function G2(αam(t)) takes the negative values ξk (58). The same (61) but with ξ < 0 and ξk<0 defines the corresponding times tk.
For ξ < 0, we get from (52) that the maximal time t=tm∗ when the flow V1(r,z,tm∗) (43) has a vortex blob 𝔹a(tm∗)3 is defined by the equation G2(αa(tm∗))=−1/3=minG2(u). Hence, (52) yields the equation for the time tm∗:
(63)∫0tm∗ν(τ)dτ=α−2log(−1/(3ξ)).
The last vortex blob 𝔹a(tm∗)3 has zero radius a(tm∗)=0. From (39), we get G2(0)=−1/3, G3(0)=1/15. Substituting this into (60), we get dVm(tm∗)/dt=0. Therefore, the plot of function Vm(t) for ξ < 0 differs from the plot in Figure 20 for ξ > 0 by its behaviour near point tm∗: the limit values of Vm(tm∗) and its derivative dVm(tm∗)/dt are both zeros; see Figure 21.
The fluid flow V1(r,z,t) (43) has no vortex blobs and invariant spheres for all times t>tm∗.
8 Conclusion
In this article, we presented an infinite-dimensional space of new exact time-dependent axisymmetric solutions (2) to (6) to the Navier–Stokes equations (1). The solutions are analytic in the whole space ℝ3 and exist for all times t; the velocity field and vorticity field for the solutions are not collinear and satisfy (11). The constructed space of exact viscous fluid flows V(r,z,t) is invariant under arbitrary shifts z→z+z0 and differentiations ∂k/∂zk, k=1,2,3,⋯. The iterations of these transforms generate infinite sequences of new exact solutions from any known one.
We studied solutions with velocity field V1(r,z,t) (33) having the streamfunction (42):
(64)ψ1(r,z,t)=r2[ξ−f(t)G2(αR)],f(t)=exp(−α2∫0tν(τ)dτ).
Applying transforms (35) to the streamfunction ψ1(r,z,t), we get an infinite sequence of streamfunctions
(65)ψn(r,z,t)=r2[ξ−f(t)∂n−1G2(αR)∂zn−1],
that define by formula (33) new exact solutions Vn(r,z,t) to the Navier–Stokes equations; here, n=2,3,⋯. The streamfunctions ψ1(r,z,t) (64) and ψ2k+1(r,z,t) (65) are up-down symmetric with respect to the reflection z→−z, as well as the corresponding velocity fields (33). The streamfunctions ψ2k(r,z,t) (65) and the related vector fields (33) are up-down asymmetric.
We presented Figures 2–19 describing the bifurcations of the instantaneous (for t=t0) phase portraits of the viscous fluid flows (43). As t → ∞, the derived exact solutions tend to the steady flow V(r,z)=αξre^φ+2ξe^z that has a constant vorticity ∇×V(r,z)=2αξe^z and hence has no vortex blobs and vortex rings. Therefore, for the constructed exact solutions to the Navier–Stokes equations, the vortex blobs and vortex rings collapse and disappear as t → ∞.
For the exact fluid flows (43), we studied the behaviour of the volume Vm(t) of the vortex blob. We proved that function Vm(t) is a discontinuous monotonously decreasing function of time t that has jumps down and infinite derivatives at an infinite sequence of moments of time −∞<⋯<tk<⋯<t3<t2<tm, where tm is the maximal time when the vortex blob exists.
