An Analytic Study of the Reversal of Hartmann Flows by Rotating Magnetic Fields

Abstract

The effects of a background uniform rotating magnetic field acting in a conducting fluid with a parallel flow are studied analytically. The stationary version with a transversal magnetic field is well known as generating Hartmann boundary layers. The Lorentz force includes now one term depending on the rotation speed and the distance to the boundary wall. As one intuitively expects, the rotation of magnetic field lines pushes backwards or forwards the flow. One consequence is that near the wall the flow will eventually reverse its direction, provided the rate of rotation and/or the magnetic field are large enough. The configuration could also describe a fixed magnetic field and a rotating flow.

1. Introduction

In [1], a generalization of the Hartmann flow was announced. The Hartmann flow is a parallel one influenced by a strong transversal magnetic field [2, 3] and has been studied intensively for its role in many aspects of plasma physics, such as liquid metal pumping [4, 5], plasma convection [6, 7], and geophysics [8]. We now will allow the background magnetic field to rotate while keeping its spatial uniformity. Our aim in this paper is to analyze in depth the action of this field upon the flow through the Lorentz force, emphasizing the possibility of flow reversal at some time near the walls confining the fluid. Given the Galilean invariance of the relevant equations, we could change to a rotating reference frame and interpret the problem as consisting in a fixed magnetic field and a rotating flow; this configuration could perhaps be more easily constructed and have wider applications.

Let us begin by recalling briefly the MHD equations appropriate for our configuration. The flow will be assumed two dimensional, and the background magnetic field will be uniform in space but rotating in time at a rate given by the angle 𝜃(𝑡): 𝐁(𝑡)=𝐵(cos𝜃(𝑡),sin𝜃(𝑡)). (1.1)

Although this field will create a secondary one 𝐛, it is assumed that all the terms in 𝐛 may be neglected when an analogous one for 𝐁 is present; thus, |𝐮×𝐛|≪|𝐮×𝐁|, |𝐛̇|≪|𝐁̇|. This does not hold for the current density, as 𝐉=∇×𝐁=𝟎, 𝐣=∇×𝐛≠𝟎. The approximate induction equation becomes 𝜕𝐁,𝜕𝑡=∇×−𝜂𝐣+𝐮×𝐁 (1.2) where 𝐮 is the flow velocity and 𝜂 the resistivity. By substituting 𝐁 by its value in (1.1), uncurling (1.2) to find 𝐣 and taking it to the Lorents force 𝐣×𝐁, we may write the momentum equation as 𝜕𝑢()𝜕𝑡+𝐮⋅∇𝐮=𝜈Δ𝐮+𝜎𝐮×𝐁×𝐁−𝜎𝐵²𝜃̇𝑦,0+∇𝑓, (1.3) where 𝜎=𝜂⁻¹ is the conductivity, 𝜈 the viscosity, and 𝑓 is the sum of the pressure and the gauge obtained by uncurling (1.2). Detailed calculations may be found in [1]. If in analogy with the classical Hartmann flow we assume a horizontal flow 𝐮(𝑥,𝑦)=(𝑢(𝑦),0), (1.3) reduces to 𝜕𝐮𝜕𝜕𝑡=𝜈²𝑢𝜕𝑦²−𝜎𝐵²𝑢sin²𝜃−𝜎𝐵²𝜃̇𝑦+𝐶, (1.4) where 𝐶 includes all the conservative forces upon the fluid, such as the pressure and a possible electrostatic field. The interpretation is rather easy; as the magnetic field lines rotate at angular velocity 𝜃, their linear velocity grows linearly with the radius 𝑦. This has the undesirable effect of providing infinite energy to the flow, but this should not deter us from studying the model. Other classical flows, such as Karman's, are the object of a vast literature in relation with swirling flows [9, 10] and also possess infinite energy. Their local properties are generalizable to many realistic situations.

We assume that the fluid is bounded either by one no-slip wall or two. In the first case, initial and boundary conditions for (1.4) are 𝑢()𝑢𝑦,0,𝑡=0,𝑦,0=𝜙 (1.5) whereas for the second one, if the walls are separated by a distance 𝐿, 𝑢()𝑢()𝑢𝑦.0,𝑡=0,𝐿,𝑡=0,𝑦,0=𝜙 (1.6) Obviously 𝜙 must satisfy at the walls the same conditions as 𝑢.

2. Flow with a Single Wall

We will solve (1.4), (1.5) for 𝑡≥0, 𝑦≥0. To simplify calculations and highlight the contributions of the different terms, we will consider separately the following three problems: 𝜕𝑢₁𝜕𝜕𝑡=𝜈²𝑢₁𝜕𝑦²−𝜎𝐵²𝑢₁sin²𝑢𝜃+𝐶,₁()𝑢0,𝑡=0,₁𝑦,0=0,𝜕𝑢₂𝜕𝜕𝑡=𝜈²𝑢₂𝜕𝑦²−𝜎𝐵²𝑢₂sin²𝜃−𝜎𝐵²̇𝑢𝜃𝑦,₂()𝑢0,𝑡=0,₂𝑦,0=0,𝜕𝑢₃𝜕𝜕𝑡=𝜈²𝑢₃𝜕𝑦²−𝜎𝐵²𝑢₃sin²𝑢𝜃,₃()𝑢0,𝑡=0,₃𝑦,0=𝜙𝑦. (2.1) Then, 𝑢=𝑢₁+𝑢₂+𝑢₃ is the desired solution. From now on, we will denote 𝐹(𝑡)=𝜎𝐵^2𝑡₀sin²𝜃(𝑠)𝑤𝑑𝑠,_𝑗𝑦,𝑡=𝑒^𝐹(𝑡)𝑢_𝑗𝑦,𝑡. (2.2) Notice that 𝑤_𝑗 satisfies the same equation as 𝑢_𝑗, except that the term in 𝜎𝐵²𝑢_𝑗sin²𝜃 disappears, and the forcing is multiplied by 𝑒^𝐹(𝑡). Although the general analysis is valid for any function 𝜃(𝑡), in order to obtain explicit results we will consider a constant rotation 𝜃(𝑡)=𝜆𝑡; the rotation is counterclockwise if 𝜆>0, clockwise otherwise. Then, 𝐹(𝑡)=𝜎𝐵²𝑡2−sin2𝜆𝑡4, (2.3) so that for large 𝑡𝑒^𝐹(𝑡)∼𝑒^{𝜎𝐵2𝑡/2},𝑒^{−𝐹(𝑡)}∼𝑒^{−𝜎𝐵2𝑡/2}. (2.4)

2.1. Effect of the Constant Forcing

Let erf denote the error function, (𝑥)=2erf√𝜋^𝑥₀𝑒^−𝑡2𝑑𝑡. (2.5) Then the solution of 𝜕𝑤₁𝜕𝜕𝑡=𝜈²𝑤₁𝜕𝑦²+𝐶𝑒^𝐹(𝑡)𝑤₁()𝑤0,𝑡=0₁𝑦,0=0 (2.6) is 𝑤₁𝑦,𝑡=^𝑡₀𝑦erf4𝜈(𝑡−𝑠)𝐶𝑒^𝐹(𝑠)𝑑𝑠, (2.7) so that 𝑢₁𝑦,𝑡=^𝑡₀𝑦erf4𝜈(𝑡−𝑠)𝐶𝑒^{−𝐹(𝑡)+𝐹(𝑠)}𝑑𝑠. (2.8) This implies 𝜕𝑢₁=𝜕𝑦2𝐶√𝜋^𝑡₀1()−𝑦4𝜈𝑡−𝑠exp²()(𝑡)(𝑠)4𝜈𝑡−𝑠−𝐹+𝐹𝑑𝑠. (2.9)

Let us study the limit of this function when 𝑡→∞ in the case of constant rotation, 𝜃(𝑡)=𝜆𝑡, 𝜆>0. It is an easy consequence of Lebesgue's theorem that, for any fixed 𝑡₀>0, lim_𝑡→∞^𝑡0₀1()−𝑦4𝜈𝑡−𝑠exp²()(𝑡)(𝑠)4𝜈𝑡−𝑠−𝐹+𝐹𝑑𝑠=0. (2.10)

Take 𝑡₀ large enough that we may substitute 𝐹(𝑡) by 𝜎𝐵²𝑡/2. It is enough to consider 2𝐶√𝜋^𝑡_𝑡01()−𝑦4𝜈𝑡−𝑠exp²()−4𝜈𝑡−𝑠𝜎𝐵²(𝑡−𝑠)2=𝑑𝑠2𝐶√𝜋^𝑡−𝑡0₀1√−𝑦4𝜈𝑠exp²−4𝜈𝑠𝜎𝐵²𝑠2𝑑𝑠, (2.11) whose limit when 𝑡→∞ is 2𝐶√𝜋^∞₀1√−𝑦4𝜈𝑠exp²−4𝜈𝑠𝜎𝐵²𝑠2𝑑𝑠, (2.12) which is obviously a finite integral. It reaches its maximum at 𝑦=0, where its value is 𝐶√𝜋𝜈^∞₀𝑠^−1/2𝑒^{−𝜎𝐵2𝑠/2}𝐶𝑑𝑠=√𝜋𝜈2𝜎𝐵^2∞₀𝑟^−1/2𝑒^−𝑟=𝐶𝑑𝑟√𝜋𝜈2𝜎𝐵²Γ12=𝐶√𝜋𝜈2𝜎𝐵²√𝜋=𝐶2𝜈𝜎𝐵²=𝐶√2𝐵𝑃_𝑚, (2.13) where 𝑃_𝑚 is the magnetic Prandtl number. Therefore, 𝑢₁ also has a limit when 𝑡→∞ for every 𝑦>0, given by the integral of (2.12) with respect to 𝑦.

2.2. Effect of the Lorentz Force

𝑤₂ satisfies 𝜕𝑤₂𝜕𝜕𝑡=𝜈²𝑤₂𝜕𝑦²−𝜎𝐵²̇𝑤𝜃𝑦,₂()𝑤0,𝑡=0,₂𝑦,0=0. (2.14) Although there exists an appropriate Green function for this problem, in this particular case it is simpler to assume that the solution is linear in 𝑦 (which is correct). Setting first 𝑤₂𝑦,𝑡=ℎ(𝑡)𝑦,ℎ(0)=0, (2.15) and recovering 𝑢₂ from it, one finds 𝑢₂𝑦,𝑡=−𝑦𝜎𝐵^2𝑡₀exp(−𝐹(𝑡)+𝐹(𝑠))𝜃̇(𝑠)𝑑𝑠. (2.16) Therefore, 𝜕𝑢₂𝜕𝑦𝑦,𝑡=−𝜎𝐵^2𝑡₀exp(−𝐹(𝑡)+𝐹(𝑠))𝜃̇(𝑠)𝑑𝑠. (2.17) This function also has a limit when 𝑡→∞. Its value may be found by an argument similar to the previous one, although simpler; in this case lim_𝑡→∞−𝜆𝜎𝐵^2𝑡_𝑡0exp−𝜎𝐵²(𝑡−𝑠)2𝑑𝑠=−2𝜆. (2.18) Hence, 𝑢₂(𝑦,𝑡) tends to −2𝜆𝑦 when 𝑡→∞. Notice how the Lorentz force pushes the fluid backwards, the more rapidly the higher the distance from the wall.

2.3. Effect of the Initial Condition

𝑤₃ satisfies 𝜕𝑤₃=𝜕𝜕𝑡²𝑤₃𝜕𝑦²𝑤₃()𝑤0,𝑡=0₃𝑦,0=𝜙𝑦. (2.19)𝜙 must satisfy 𝜙(0)=0. The solution is found by the method of images: 𝜙 is extended to (−∞,∞) as an odd function. Then, 𝑤₃=1𝑦,𝑡√4𝜋𝜈𝑡^∞−∞exp−𝑦−𝑧²𝜙(𝑧)4𝜈𝑡𝑑𝑧, (2.20) so that 𝑢₃=𝑒𝑦,𝑡^{−𝐹(𝑡)}√4𝜋𝜈𝑡^∞−∞exp−𝑦−𝑧²𝜙(𝑧)4𝜈𝑡𝑑𝑧. (2.21) Therefore, provided 𝜙 is differentiable, 𝜕𝑢₃=𝑒𝜕𝑦𝑦,𝑡^{−𝐹(𝑡)}√4𝜋𝜈𝑡^∞−∞exp−𝑦−𝑧²𝜙4𝜈𝑡^′(𝑧)𝑑𝑧. (2.22) All this supposes that 𝜙 and 𝜙^′ satisfy certain reasonable conditions: for this example. if both are bounded. In that case, given that 1√4𝜋𝜈𝑡^∞−∞exp−𝑦−𝑧²4𝜈𝑡𝑑𝑧=1, (2.23) both 𝑢₃ and 𝜕𝑢₃/𝜕𝑦 are bounded by a function of the form 𝑀exp(−𝐹(𝑡)), which means that they tend to zero as 𝑡→∞.

Thus, the limit when 𝑡→∞ of 𝑢(0,𝑡) is 𝐶√𝑘=2𝐵𝑃_𝑚−2𝜆. (2.24)

If 𝑘<0, we may be certain that the fluid inverts its direction and flows backwards from some point on. The factors contributing to this are a small (or negative) potential force 𝐶, which includes pressure and electric fields; a large background magnetic field; a large magnetic Prandtl number, which means that the viscosity dominates the resistivity; finally and most obviously, a large rate of rotation of the magnetic field.

3. Flow between Two Walls

When the fluid is stationary along two walls, the Lorentz force does not have so clear an effect as in the previous case, as it is stopped by the upper no-slip wall. We may again decompose the solution in the three components given by (2.1) but with the boundary and initial conditions of (1.6). The solution to these problems may be found again by the method of images, but it is equivalent and somewhat simpler to postulate a Fourier series expression of the form ^∞_𝑘=1𝑐_𝑘(𝑡)sin𝜋𝑘𝐿𝑦, (3.1) and find the coefficients 𝑐_𝑘.

3.1. Effect of the Constant Forcing

We have 𝜕𝑤₁𝜕𝜕𝑡=𝜈²𝑤₁𝜕𝑦²+𝐶𝑒^𝐹(𝑡),𝑤₁(0,𝑡)=𝑤₁()𝑤𝐿,𝑡=0,₁𝑦,0=0. (3.2) First we extend the forcing 𝐶𝑒^𝐹(𝑡) as an odd function of 𝑦 in the interval [−𝐿,𝐿]. Its Fourier series is 𝐶𝑒^{∞𝐹(𝑡)}_𝑘=04𝜋()𝜋()2𝑘+1sin2𝑘+1𝐿𝑦, (3.3) so that if we set 𝑤₁(𝑡)=^∞_𝑛=1𝑤₁(𝑛,𝑡)sin𝜋𝑛𝐿𝑦, (3.4) we have 𝜕𝑤₁()𝜋𝜕𝑡2𝑘,𝑡=−𝜈²(2𝑘)²𝐿²𝑤₁(2𝑘,𝑡),𝑤₁(2𝑘,0)=0,𝜕𝑤₁()𝜋𝜕𝑡2𝑘+1,𝑡=−𝜈²(2𝑘+1)²𝐿²𝑤₁()+42𝑘+1,𝑡𝜋(2𝑘+1)𝐶𝑒^𝐹(𝑡),𝑤₁(2𝑘+1,0)=0. (3.5) Hence, all the even terms 𝑤₁(2𝑘,𝑡) vanish and 𝑤₁()=42𝑘+1,𝑡𝜋()−()2𝑘+1exp𝜈𝜋2𝑘+1²𝑡𝐿²×^𝑡₀()exp𝜈𝜋2𝑘+1²𝑠𝐿²𝐶exp(𝐹(𝑠))𝑑𝑠. (3.6) If we define 𝐹_𝑚(𝑡)=𝜎𝐵^2𝑡₀sin²𝜃(𝑠)𝑑𝑠+𝜈𝜋²𝑚²𝑡𝐿², (3.7) we obtain 𝑢₁(2𝑘+1,𝑡)=4𝐶𝜋(2𝑘+1)exp−𝐹_2𝑘+1(𝑡)^𝑡₀exp𝐹_2𝑘+1(𝑠)𝑑𝑠. (3.8) Thus, 𝑢₁𝑦,𝑡=^∞_𝑘=04𝐶𝜋(2𝑘+1)exp−𝐹_2𝑘+1(𝑡)^𝑡₀exp𝐹_2𝑘+1(𝑠𝜋())𝑑𝑠×sin2𝑘+1𝐿𝑦,𝜕𝑢₁=𝜕𝑡𝑦,𝑡^∞_𝑘=04𝐶𝐿exp−𝐹_2𝑘+1(𝑡)^𝑡₀exp𝐹_2𝑘+1(𝑠)𝜋()𝑑𝑠×cos2𝑘+1𝐿𝑦. (3.9) In particular 𝜕𝑢₁()=𝜕𝑡0,𝑡^∞_𝑘=04𝐶𝐿exp−𝐹_2𝑘+1(𝑡)^𝑡₀exp𝐹_2𝑘+1(𝑠)𝑑𝑠,𝜕𝑢₁()𝜕𝑡𝐿,𝑡=−^∞_𝑘=04𝐶𝐿exp−𝐹_2𝑘+1(𝑡)^𝑡₀exp𝐹_2𝑘+1(𝑠)𝑑𝑠. (3.10)(3.11) To evaluate the limit of these expressions when 𝑡→∞, let us assume as before that 𝜃(𝑡)=𝜆𝑡, 𝜆>0. Analogously to (2.4), we have now 𝐹_𝑘(𝑡)∼𝜎𝐵²2+𝜈𝜋²𝑘²𝐿²𝑡,−𝐹_𝑘(𝑡)∼−𝜎𝐵²2+𝜈𝜋²𝑘²𝐿²𝑡, (3.12) for 𝑡 large. Thus, for 𝑡₀ fixed, lim_𝑡→∞exp−𝐹_𝑘(𝑡)^𝑡0₀exp𝐹_𝑘(𝑠)𝑑𝑠=0. (3.13) By using an argument similar to the one in (2.10) and glossing over the technicalities concerning the infinite terms in the sums of (3.10) and (3.11) (which hold anyway), we obtain that if we denote 𝛼_𝑘=lim_𝑡→∞exp−𝐹_𝑘(𝑡)^𝑡₀exp𝐹_𝑘(𝑠)𝑑𝑠, (3.14) then 𝛼_𝑘=𝜎𝐵²2+𝜋²𝜈𝑘²𝐿²⁻¹; (3.15) moreover 𝜕𝑢₁()=𝜕𝑡0,𝑡4𝐶𝐿^∞_𝑘=0𝛼_2𝑘+1,𝜕𝑢₁()𝜕𝑡𝐿,𝑡=−4𝐶𝐿^∞_𝑘=0𝛼_2𝑘+1. (3.16)(3.17) To find these sums, we denote 𝛾=𝐵𝐿𝜋𝜎.2𝜈 (3.18) Then, 𝛼_𝑘=2𝜎𝐵²𝛾²𝛾²+𝑘². (3.19) It may be found by elementary means (e.g, the residue theorem) that ^∞_𝑘=1𝛾²𝛾²+𝑘²1=−2+𝜋𝛾2cotanh𝜋𝛾, (3.20)

Therefore ^∞_𝑘=1𝛼_2𝑘=2𝜎𝐵²𝛾2^2∞_𝑘=11𝛾/2²+𝑘²=1𝜎𝐵²−1+𝜋𝛾2cotanh𝜋𝛾2_∞𝑘=0,𝛼_2𝑘+1=1𝜎𝐵²𝜋𝛾cotanh𝜋𝛾−𝜋𝛾2cotanh𝜋𝛾2. (3.21)

This expression may be written in another form, which is more convenient if we take 𝐿 and 𝜈 as fixed and consider that the variation of 𝛾 comes from 𝜎𝐵². This is ^∞_𝑘=0𝛼_2𝑘+1=𝐿²12𝜋𝜈𝛾cotanh𝜋𝛾−2cotanh𝜋𝛾2. (3.22)

Given the behavior of the hyperbolic cotangent, it is easy to see that this sum tends to ∞ when 𝛾→0 and to 0 when 𝛾→∞. Combining this with (3.16) and (3.17), we obtain lim_𝑡→∞𝜕𝑢₁()=𝜕𝑡0,𝑡2𝐶𝐿1𝜋𝜈𝛾cotanh𝜋𝛾−2cotanh𝜋𝛾2,lim_𝑡→∞𝜕𝑢₁()𝜕𝑡𝐿,𝑡=−2𝐶𝐿1𝜋𝜈𝛾cotanh𝜋𝛾−2cotanh𝜋𝛾2. (3.23)(3.24)

3.2. Effect of the Lorentz Force

The Fourier series of the function 𝑦→−𝜎𝐵²𝜃̇𝑦 is 𝜎𝐵²𝜃̇^∞_𝑘=1()2𝐿−1^𝑘𝜋𝑘sin𝜋𝑘𝐿𝑦. (3.25) Hence, 𝑤₂ satisfies 𝜕𝑤₂()𝜋𝜕𝑡𝑘,𝑡=−²𝜈𝑘²𝐿²𝑤₂()+()𝑘,𝑡2𝐿−1^𝑘𝜋𝑘𝜎𝐵²𝜃̇(𝑡)exp(𝐹(𝑡))𝑤₂(𝑘,0)=0, (3.26) whose solution is 𝑤₂()=()𝑘,𝑡2𝐿−1^𝑘𝜋𝑘𝜎𝐵^2𝑡₀exp𝐹_𝑘(𝑠)𝜃̇(𝑠)𝑑𝑠. (3.27) Therefore, 𝑢₂()=()𝑘,𝑡2𝐿−1^𝑘𝜋𝑘𝜎𝐵^2𝑡₀exp−𝐹_𝑘(𝑡)+𝐹_𝑘(𝑠𝜃̇(𝑠)𝑢)𝑑𝑠,₂𝑦,𝑡=^∞_𝑘=12𝐿(−1)^𝑘𝜋𝑘𝜎𝐵^2𝑡₀exp−𝐹_𝑘(𝑡)+𝐹_𝑘(𝑠)𝜃̇(𝑠)𝑑𝑠sin𝜋𝑘𝐿𝑦, (3.28) which implies 𝜕𝑢₂𝜕𝑡𝑦,𝑡=2^∞_𝑘=1(−1)^𝑘𝜎𝐵^2𝑡₀exp−𝐹_𝑘(𝑡)+𝐹_𝑘(𝑠)𝜃̇(𝑠)𝑑𝑠cos𝜋𝑘𝐿𝑦. (3.29) In particular 𝜕𝑢₂()𝜕𝑡0,𝑡=2^∞_𝑘=1(−1)^𝑘𝜎𝐵^2𝑡₀exp−𝐹_𝑘(𝑡)+𝐹_𝑘(𝑠)𝜃̇(𝑠)𝑑𝑠,𝜕𝑢₂()𝜕𝑡𝐿,𝑡=2^∞_𝑘=1𝜎𝐵^2𝑡₀exp−𝐹_𝑘(𝑡)+𝐹_𝑘(𝑠)𝜃̇(𝑠)𝑑𝑠. (3.30)

To find the limit of this expression when 𝑡→∞ and 𝜃(𝑡)=𝜆𝑡, we use the same argument as in the previous paragraph. We find lim_𝑡→∞𝜕𝑢₂()𝜕𝑡0,𝑡=2𝜎𝐵²𝜆^∞_𝑘=1(−1)^𝑘𝛼_𝑘,lim_𝑡→∞𝜕𝑢₂()𝜕𝑡𝐿,𝑡=2𝜎𝐵²𝜆^∞_𝑘=1𝛼_𝑘. (3.31) In terms of 𝛾, lim_𝑡→∞𝜕𝑢₂()𝜕𝑡0,𝑡=4𝜆^∞_𝑘=1(−1)^𝑘𝛾²𝛾²+𝑘²,lim_𝑡→∞𝜕𝑢₂()𝜕𝑡𝐿,𝑡=4𝜆^∞_𝑘=1𝛾²𝛾²+𝑘². (3.32)(3.33) We know already (3.20) the value of the sum in (3.33). The one in (3.32) sums ^∞_𝑘=1(−1)^𝑘𝛾²𝛾²+𝑘²1=−2+𝜋𝛾2cosech𝜋𝛾. (3.34) This sum is a decreasing function of 𝛾, tending to 0 when 𝛾→0 and to −1/2 when 𝛾→∞.

3.3. Effect of the Initial Condition

To deal with the problem 𝜕𝑤₃𝜕𝜕𝑡=𝜈²𝑤₃𝜕𝑦²,𝑤₃(0,𝑡)=𝑤₃()𝑤𝐿,𝑡=0,₃𝑦,0=𝜙𝑦, (3.35) we extend 𝜙 to [−𝐿,𝐿] as an odd function, with Fourier series 𝜙𝑦=^∞_𝑘=1𝜙(𝑘)sin𝜋𝑘𝐿𝑦. (3.36) Then, 𝜕𝑤₃()𝜋𝜕𝑡𝑘,𝑡=−²𝜈𝑘²𝐿²𝑤₃(𝑘,𝑡),𝑤₃(𝑘,0)=𝜙(𝑘), (3.37) whose solution is 𝑤₃()−𝜋𝑘,𝑡=exp²𝜈𝑘²𝐿²𝑡𝜙(𝑘). (3.38) Therefore, 𝑢₃𝑦,𝑡=^∞_𝑘=1exp−𝐹_𝑘(𝑡)𝜙(𝑘)sin𝜋𝑘𝐿𝑦,𝜕𝑢₃=𝜕𝑦𝑦,𝑡^∞_𝑘=1𝜋𝑘𝐿exp−𝐹_𝑘(𝑡)𝜙(𝑘)cos𝜋𝑘𝐿𝑦. (3.39) It is trivial to prove that both tend to zero when 𝑡→∞ for all 𝑦.

When 𝜕𝜙(0)𝜕𝑦>0,𝜕𝜙(𝐿)𝜕𝑦<0 (3.40) the fluid flows initially in the positive sense. If we show that under certain conditions lim_𝑡→∞𝜕𝑢()𝜕𝑡0,𝑡<0, (3.41) or lim_𝑡→∞𝜕𝑢()𝜕𝑡𝐿,𝑡>0, (3.42) the fluid reverses flow at least in one of the walls. Since 𝑢=𝑢₁+𝑢₂+𝑢₃ and we have found already these limits, we may combine first (3.23) and (3.34) to get lim_𝑡→∞𝜕𝑢()=𝜕𝑡0,𝑡2𝐶𝐿1𝜋𝜈𝛾cotanh𝜋𝛾−2cotanh𝜋𝛾2−1+4𝜆2+𝜋𝛾2cosech𝜋𝛾. (3.43) On the other hand, combining now (3.24), (3.20), and (3.33), we find lim_𝑡→∞𝜕𝑢()𝜕𝑡𝐿,𝑡=−2𝐶𝐿1𝜋𝜈𝛾cotanh𝜋𝛾−2cotanh𝜋𝛾2+2𝜆−1+𝜋𝛾cotanh𝜋𝛾. (3.44)

The expression in (3.43) tends to −2𝜆<0 when 𝛾→∞. Thus, for any 𝐶 and 𝐿, when 𝜆 and/or 𝜎𝐵² are large enough, there exists a flow reversal at this wall.

The limit of (3.44) when 𝛾→∞ is ∞. Since this becomes positive for 𝜎𝐵² or 𝜆 large enough, the fluid will eventually flow backwards at 𝑦=𝐿. Notice that given the respective size of the limits, this occurs much earlier than the flow reversal at 𝑦=0. This was to be expected, as the Lorentz force pushes backwards the flow the more rapidly the larger 𝑦 is; magnetic field lines move faster the higher the radius.

4. Conclusions

A generalization of the classical Hartmann flow is obtained when we pose a background uniform magnetic field rotating in time. Since the Lorentz force varies both in time and space, the resulting flow cannot be stationary. If we add one or two walls with the no-slip property, the interaction between the magnetic field and the boundary layers creates an interesting phenomenology. We study in particular the phenomenon of flow reversal near the walls for a magnetic field rotating at a constant rate; it is found that for large enough values of this rate, the conductivity of the fluid, or the size of the magnetic field, there exists always flow reversal at some instant. Forces opposing this include the kinetic pressure and a possible electrostatic field. This reversal makes excellent sense, as the Lorentz force (visualized through the rotation of magnetic field lines) pushes backwards the fluid if the sense of rotation of the field is contrary to the original flow.