On Lie Reduction of the MHD Equations to Ordinary Differential Equations

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1 Symmetry in Nonlinear Mathematical Physics 1997, V.1, On Lie Reduction of the MHD Equations to Ordinary Differential Equations Victor POPOVYCH Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivs ka Str., Kyiv 4, Ukraine Abstract The MHD equations describing flows of a viscous homogeneous incompressible fluid of finite electrical conductivity are reduced to ordinary differential equations by means of Lie symmetries. The MHD equations (the MHDEs) describing flows of a viscous homogeneous incompressible fluid of finite electrical conductivity have the following form: u t +( u ) u u + p + H rot H = 0, H t rot ( u H) ν m H = 0, div u =0, div H =0. (1) System (1) is very complicated and the construction of new exact solutions is a difficult problem. Following [1], in this paper we reduce the MHDEs (1) to ordinary differential equations by means of three-dimensional subalgebras of the maximal Lie invariance algebra of the MHDEs. In (1) and below, u = {u a (t, x)} denotes the velocity field of a fluid, p = p(t, x) denotes the pressure, H = {H a (t, x)} denotes the magnetic intensity, ν m is the coefficient of magnetic viscosity, x = {x a }, t = / t, a = / x a, = { a }, = is the Laplacian. The kinematic coefficient of viscosity and fluid density are set equal to unity, permeability is done (4π) 1. Subscript of a function denotes differentiation with respect to the corresponding variables. The maximal Lie invariance algebra of the MHDEs (1) is an infinite-dimensional algebra A(MHD) with the basis elements (see [2]) t, D = t t x a a 1 2 ua u a 1 2 Ha H a p p, J ab = x a b x b a + u a u b u b u a + H a H b H b H a,a<b, R( m) =R( m(t)) = m a a + m a t u a m a ttx a p, Z(η) =Z(η(t)) = η p, (2) where m a = m a (t) andη = η(t) are arbitrary smooth functions of t (for example, from C ((t 0,t 1 ), R) ). We sum over repeated indices. The indices a, b take values in {1, 2, 3} and the indices i, j in {1, 2}. The algebra A(MHD) is isomorphic to the maximal Lie invariance algebra A(NS) of the Navier-Stokes equations [3, 4, 5]. Besides continuous transformations generated by operators (2), the MHDEs admit discrete transformations I b of the form t = t, x b = x b, x a = x a, p = p, ũ b = u b, Hb = H b, ũ a = u a, Ha = H a,a b, where b is fixed.

2 228 V. Popovych We construct a complete set of A(MHD)-inequivalent three-dimensional subalgebras of A(MHD) and choose those algebras from this set which can be used to construct ansatzes for the MHD field. The list of the classes of these algebras is given below. 1. A 3 1 = D, t,j A 3 2 = D κj 12, t,r(0, 0, 1), where κ 0. Here and below, κ, σ ij, ε i, µ, andν are real constants. 3. A 3 3 = D, J 12 + R(0, 0,ν t 1/2 ln t )+Z(νε 2 t 1 ln t + ε 1 t 1 ), R(0, 0, t σ+1/2 )+Z(ε 2 t σ 1 ), where σν =0, σε 2 =0, ε 1 0, and ν A 3 4 = t,j 12 + R(0, 0,νt)+Z(νε 2 t + ε 1 ),R(0, 0,e σt )+Z(ε 2 e σt ), where σν =0, σε 2 =0, σ { 1; 0; 1}, and, if σ =0, the constants ν, ε 1,andε 2 satisfy one of the following conditions: ν =1,ε 1 0; ν =0,ε 1 =1,ε 2 0; ν = ε 1 =0,ε 2 {0; 1}. 5. A 3 5 = D + κj 12, R( t 1/2 f ij (t)â(t) e j)+z( t 1 f ij (t)ε j ),i=1, 2, 6. A 3 6 = t + κj 12,R(f ij (t)ǎ(t) e j)+z(f ij (t)ε j ),i=1, 2, Here, (f 1j (t),f 2j (t)),j =1, 2, are solutions of the Cauchy problems fτ ij = σ ik f kj, f ij (0) = δ ij, where τ =ln t in the case of the algebra A 3 5 and τ = t in the case of the algebra A3 6. e i = const : e i e j = δ ij,δ ij is the Kronecker delta, σ ii (σ 12 σ 21 2κK e 1 e 2 )=0, ε i (σ ik σ kj e j 2κσ ij K e j )= 0, σ 12 + σ 21 =0. cos ζ sin ζ Â(t) =A(κ ln t ), A(ζ) = sin ζ cos ζ 0, K = 1 0 0, Ǎ(t) =A(κt) κε l K e k e l =0ifσ kj =0andε k (2κK e k e l σ kl )=0ifσ kj = σ ll =0, where k and l take fixed values from {1; 2}, k l. To simplify parameters in A 3 5 and A3 6, one can also use the adjoint actions generated by I b,d,j 12, and, if κ =0,J 23 and J A 3 7 = J 12 + R(0, 0,η 3 ),R(η 1,η 2, 0), R( η 2,η 1, 0), where η a are smooth functions of t, η i η i 0,η 3 0,ηttη 1 2 η 1 ηtt 2 = 0. The algebras A 3 7 (η1,η 2,η 3 ) and A 3 7 ( η1, η 2, η 3 ) are equivalent if E i R\{0}, δ R, (b ij ) O(2): η i ( t) =E 2 b ij η i (t), η 3 ( t) =E 1 η 3 (t), where t = E1 t 2 + δ. 8. A 3 8 = R( ma ),a= 1, 3, where m a are smooth functions of t, rank ( m 1, m 2, m 3 )=3, m a tt m b m a m b tt =0. The algebras A 3 8 ( m1, m 2, m 3 ) and A 3 8 ( m 1, m 2, m 3 ) are equivalent if E 1 R\{0}, δ R, B O(3), and (d ab ) : det(d ab ) 0 such that m a ( t) =d ab B m b (t), where t = E 2 1 t + δ.

3 On Lie Reduction of the MHD Equations to Ordinary Differential Equations 229 The way of obtaining (and the form of writing down) the algebras A 3 1 A3 8 differs slightly from the one used in [1]. By means of subalgebras A 3 1 A3 8, one can construct the following ansatzes that reduce the MHDEs to ODEs: 1. u 1 = x 1 R 2 ϕ 1 x 2 (Rr) 1 ϕ 2 + x 1 x 3 r 1 R 2 ϕ 3, u 2 = x 2 R 2 ϕ 1 + x 1 (Rr) 1 ϕ 2 + x 2 x 3 r 1 R 2 ϕ 3, u 3 = x 3 R 2 ϕ 1 rr 2 ϕ 3, p = R 2 h, where ω = arctan r/x 3, the expressions for H a are obtained by means of the substitution of ψ a for ϕ a in the expressions for u a. Here and below, ϕ a = ϕ a (ω), ψ a = ψ a (ω), h = h(ω), R = (x x2 2 + x2 3 )1/2, r =(x x2 2 )1/2. The numeration of ansatzes and reduced systems corresponds to that of the algebras above. All the parameters satisfy the equations given for these algebras. 2. u 1 = r 2 (x 1 ϕ 1 x 2 ϕ 2 ), u 2 = r 2 (x 2 ϕ 1 + x 1 ϕ 2 ), u 3 = r 1 ϕ 3, p = r 2 h, where ω = arctan x 2 /x 1 κ ln r, the expressions for H a are obtained by means of substituting ψ a for ϕ a in the expressions for u a. 3. u 1 = r 2 (x 1 ϕ 1 x 2 ϕ 2 )+ 1 2 x 1t 1, u 2 = r 2 (x 2 ϕ 1 + x 1 ϕ 2 )+ 1 2 x 2t 1, u 3 = t 1/2 ϕ 3 +(σ )x 3t 1 + ν t 1/2 t 1 arctan x 2 /x 1, H 1 = r 2 (x 1 ψ 1 x 2 ψ 2 ), H 2 = r 2 (x 2 ψ 1 + x 1 ψ 2 ), H 3 = t 1/2 ψ 3, p = t 1 h x ax a t σ2 x 2 3 t 2 + ε 1 t 1 arctan x 2 /x 1 + ε 2 x 3 t 3/2, where ω = t 1/2 r. 4. u 1 = r 2 (x 1 ϕ 1 x 2 ϕ 2 ), u 2 = r 2 (x 2 ϕ 1 + x 1 ϕ 2 ), u 3 = ϕ 3 + σx 3 + ν arctan x 2 /x 1, H 1 = r 2 (x 1 ψ 1 x 2 ψ 2 ), H 2 = r 2 (x 2 ψ 1 + x 1 ψ 2 ), H 3 = ψ 3, p = h 1 2 σ2 x ε 1 arctan x 2 /x 1 + ε 2 x 3, where ω = r. 5. u = t 1/2 t 1 Â(t) v xt 1 κk xt 1, H = t 1/2 t 1 Â(t) G, p = t 1 q x ax a t κ2 x i x i t 2, (3) where y = t 1/2 Â(t) T x. 6. u = Ǎ(t) v κk x, H = Ǎ(t) G, p = q κ2 x i x i, (4) where y = Ǎ(t)T x.

4 230 V. Popovych In (3) and (4) v, G, q, and ω are defined by means of the following formulas: v = ϕ(ω)+σ ij ( e i y) e j, G = ψ(ω), q = h(ω)+ε i ( e i y) 1 2 ( d i y)( e i y) 1 2 ( d i e 3 )ω( e i y), where ω =( e 3 y), e 3 = e 1 e 2, d i = σ ik σ kj e j 2κσ ij K e j. 7. u 1 = ϕ 1 cos z ϕ 2 sin z + x 1 θ 1 + x 2 θ 2, u 2 = ϕ 1 sin z + ϕ 2 cos z x 1 θ 2 + x 2 θ 1, u 3 = ϕ 3 + ηt 3 (η 3 ) 1 x 3, H 1 = ψ 1 cos z ψ 2 sin z, H 2 = ψ 1 sin z + ψ 2 cos z, H 3 = ψ 3, p = h 1 2 η3 tt(η 3 ) 1 x ηj tt ηj (η i η i ) 1 r 2, where ω = t, θ 1 = η i tη i (η j η j ) 1,θ 2 =(η 1 t η 2 η 1 η 2 t )(η j η j ) 1,z= x 3 /η u = ϕ + λ 1 ( n a x) m a t, H = ψ, p = h λ 1 ( m a tt x)( n a x)+ 1 2 λ 2 ( m b tt m a )( n a x)( n b x), where ω = t, m a tt m b m a m b tt =0,λ= m 1 m 2 m 3 0, n 1 = m 2 m 3, n 2 = m 3 m 1, n 3 = m 1 m 2. Substituting the ansatzes 1 8 into the MHDEs, we obtain the following systems of ODE in the functions ϕ a,ψ a, and h: 1. ϕ 3 ϕ 1 ω ψ 3 ψω 1 ϕ 1 ωω ϕ 1 ω cot ω ϕ a ϕ a 2h =0, ϕ 3 ϕ 2 ω ψ 3 ψω 2 ϕ 2 ωω ϕ 2 ω cot ω + ϕ 2 sin 2 ω +(ϕ 3 ϕ 2 ψ 3 ψ 2 )cotω =0, ϕ 3 ϕ 3 ω ψ 3 ψω 3 ϕ 3 ωω ϕ 3 ω cot ω + ϕ 3 sin 2 ω 2ϕ 1 ω + h ω + ψωψ a a ((ϕ 2 ) 2 (ψ 2 ) 2 )cotω =0, ϕ 3 ψω 1 ψ 3 ϕ 1 ω ν m (ψωω 1 + ψω 1 cot ω) =0, ϕ 3 ψω 2 ψ 3 ϕ 2 ω ν m (ψωω 2 + ψω 2 cot ω ψ 2 sin 2 ω)+2(ψ 1 ϕ 2 ψ 2 ϕ 1 )+ (ψ 3 ϕ 2 ψ 2 ϕ 3 )cotω =0, ϕ 3 ψω 3 ψ 3 ϕ 3 ω ν m (ψωω 3 + ψω 3 cot ω ψ 3 sin 2 ω +2ψω)+2(ψ 1 1 ϕ 3 ψ 3 ϕ 1 )=0, ϕ 3 ω + ϕ 3 cot ω + ϕ 1 =0, ψω 3 + ψ 3 cot ω + ψ 1 =0. 2. ϕ 2 ϕ 1 ω ψ 2 ψω 1 νϕ 1 ωω ϕ i ϕ i + ψ i ψ i 2 h κ h ω =0, ϕ 2 ϕ 2 ω ψ 2 ψω 2 νϕ 2 ωω 2(ϕ 1 ω + κϕ 2 ω)+ h ω =0, ϕ 2 ϕ 3 ω ψ 2 ψω 3 νϕ 3 ωω ϕ 1 ϕ 3 + ψ 1 ψ 3 2κϕ 3 ω ϕ 3 =0, ϕ 2 ψω 1 ψ 2 ϕ 1 ω νψωω 1 =0, ϕ 2 ψω 2 ψ 2 ϕ 2 ω νψωω 2 +2(ψ 1 ϕ 2 ψ 2 ϕ 1 ) 2ν m (ψω 1 + κψω)=0, 2 ϕ 2 ψω 3 ψ 2 ϕ 3 ω νψωω 3 + ψ 1 ϕ 3 ψ 3 ϕ 1 ν m (2κψω 3 + ψ 3 )=0, ϕ 2 ω =0, ψ2 ω =0,

5 On Lie Reduction of the MHD Equations to Ordinary Differential Equations 231 where ϕ 2 = ϕ 2 κϕ 1, ψ 2 = ψ 2 κψ 1, h = h ψa ψ a,ν=1+κ 2, ν = ν m (1 + κ 2 ) ω 1 (ϕ 1 ϕ 1 w ψ 1 ψω) 1 ϕ 1 ωω + ω 1 ϕ 1 ω ω 2 (ϕ i ϕ i ψ i ψ i )+ ω(h ω 2 ψ i ψ i (ψ3 ) 2 ) ω =0, ω 1 (ϕ 1 ϕ 2 w ψ 1 ψω) 2 ϕ 2 ωω + ω 1 ϕ 2 ω + ε 1 =0, ω 1 (ϕ 1 ϕ 3 w ψ 1 ψω) 3 ϕ 3 ωω ω 1 ϕ 3 ω + νεω 2 ϕ 2 + εσϕ 3 + ε 2 =0, ω 1 (ϕ 1 ψw 1 ψ 1 ϕ 1 ω) ν m (ψωω 1 ω 1 ψω) 1 εδψ 1 =0, ω 1 (ϕ 1 ψw 2 ψ 1 ϕ 2 ω) ν m (ψωω 2 ω 1 ψω)+2ω 2 2 (ψ 1 ϕ 2 ψ 2 ϕ 1 ) εδψ 2 =0, ω 1 (ϕ 1 ψw 3 ψ 1 ϕ 3 ω) ν m (ψωω 3 + ω 1 ψω) 3 νεω 2 ψ 2 (σ + δ)εψ 3 =0, ϕ 1 ω +(σ δ)εω =0, ψ1 ω =0, where ε =signt and δ = 1 in case 3 and ε =1andδ = 0 in case ϕ 3 ϕ ω ψ 3 ψω + σ ij ϕ i e j 2κK ϕ ε ϕ ωω +(h ω + ψ ψ ω ) e 3 + ε i e i +2κωσ ij (K e j e 3 ) e i = 0, ϕ 3 ψω ψ 3 ϕ ω σ ij ψ i e j ν m ε ϕ ωω δψ = 0, ϕ 3 ω + σ ii δ =0, ψ3 ω =0, where ϕ a = e a ϕ, ψ a = e a ψ; ε =signt and δ = 1 in case 5 and ε =1andδ = 0 in case ϕ 1 ω + ϕ 1 (θ 1 +(η 3 ) 2 )+ϕ 2 (θ 2 (η 3 ) 1 ϕ 3 )+ψ 3 (η 3 ) 1 ψ 2 =0, ϕ 2 ω ϕ 1 (θ 2 (η 3 ) 1 ϕ 3 )+ϕ 2 (θ 1 +(η 3 ) 2 ) ψ 3 (η 3 ) 1 ψ 1 =0, ϕ 3 ω + ηw(η 3 3 ) 1 ϕ 3 =0, ψω 1 + ψ 1 ( θ 1 + ν m (η 3 ) 2 ) ψ 2 (θ 2 +(η 3 ) 1 ϕ 3 )+ψ 3 (η 3 ) 1 ϕ 2 =0, ψω 2 + ψ 1 (θ 2 +(η 3 ) 1 ϕ 3 )+ψ 2 ( θ 1 + ν m (η 3 ) 2 ) ψ 3 (η 3 ) 1 ϕ 1 =0, ψw 3 ψ 3 ηw(η 3 3 ) 1 =0, 2θ 1 + ηw(η 3 3 ) 1 =0. 8. ϕ ω + λ 1 ( n a ϕ) m a ω = 0, ψω λ 1 ( n a ψ) m a ω = 0, n a m a ω =0. References [1] Fushchych W.I. and Popowych R.O., Symmetry reduction and exact solution of the Navier-Stokes equations, J. Nonlin. Math. Phys., 1994, V.1, N1, ; N2, [2] Nucci M.C., Group analysis for MHD equations, in: Atti Sem. Mat. Fis. Univ. Modena, 1984, V. XXXIII, [3] Danilov Yu.A., Group properties of the Maxwell and Navier-Stokes equations, Preprint, Acad. Sci. USSR, Kurchatov In-t of Atomic Energy, Moscow, [4] Bytev V.O., Group properties of the Navier-Stokes equations, in: Chislennye metody mekhaniki sploshnoy sredy, V.3, 13 17, Comp. Center, Siberian Dep. Acad. Sci. USSR, Novosibirsk, [5] Lloyd S.P., The infinitesimal group of the Navier-Stokes equations, Acta Mech., 1981, V.38, N1 2,