Viscosity gradient driven instability of `shear mode' in a strongly coupled plasma

The influence of viscosity gradient (due to shear flow) on low frequency collective modes in strongly coupled dusty plasma is analyzed. It is shown that for a well known viscoelastic plasma model, the velocity shear dependent viscosity leads to an instability of the shear mode. The inhomogeneous viscous force and velocity shear coupling supply the free energy for the instability. The combined strength of shear flow and viscosity gradient wins over any stabilizing force and makes the shear mode unstable. Implication of such a novel instability and its applications are briefly outlined.


Introduction
In complex physical systems, for example multispecies charged fluid (dusty plasma), various physical processes exist and interact simultaneously. The stability properties of such systems are complicated due to the presence of various free energy sources which ultimately lead to instabilities. Despite the large experimental and theoretical effort, it has not been possible to identify all the sources of free energies available in such a complex system. In this work we have identified a free energy source (viscosity gradient due to velocity shear) in a complex dusty plasma system and demonstrated a novel instability of the 'shear mode' due to the combined effect of viscosity gradient and velocity shear. The collective modes in dusty plasmas have been the subject of serious study in recent years due to their novel character and wide applications. Normally in a three-component plasmas, besides the electrons and the most abundant ion species, there is an additional species with a different mass and charge and whose abundance is not negligible compared to the other constituents. This additional heavy micron-size species, having a wide range of values for the mass-to-charge ratio, is referred to as 'dust' in the dusty-plasma literature [1]. The presence of the new species is expected to result in new effects in the collective-mode behavior in the plasma [2]. This is because the various species are mutually coupled through the electro-magnetic forces. In such a scenario the plasma is able to support many new modes as compared to those in a simple electron-ion plasma [3]. Due to the large amount of charge on a single dust particle, the dust fluid can also exhibit strong coupling behavior which can show strong viscous properties of the medium even leading to viscoelastic behavior [3]. The strongly coupled complex plasma has been realized in different experiments [4,5,6,7]. The strength of the coupling is characterized by the Coulomb coupling parameter Γ = q 2 d /(k B T d a) where q d is the charge on the dust grains, a(≃ n −1/3 d ) is the average distance between them for density n d , T d is the temperature of the dust component and k B is the Boltzmann constant [8]. In the regime of Γ from 1 to Γ c (a critical value beyond which system becomes crystalline) both viscosity and elasticity are equally important and this property together is known as visco-elasticity. When Γ > Γ c , viscosity disappears and only elasticity reigns over the system. Experiments [6] have also shown that as Γ increases, the dust components becomes strongly coupled and for large Γ values, the dust component becomes crystalline. This phenomenon, the plasma condensation is useful in studying phase transitions [7,9] and low frequency wave propagation [10,11]. It has been shown that the strong correlations make dusty plasma system rather rigid so that it can support a transverse shear mode [3]. This 'shear mode' has also been found experimentally [12] and its variant theoretically [13].
An interesting property observed in the case of complex dusty plasmas is the strong density dependence of the viscosity parameter [14] and owes its existence to the large amount of charge on each dust particle. Recent experiments [15] reveal that complex-plasma fluid has the signature of non-Newtonian property similar to other non-Newtonian fluids. Beyond some critical value of velocity shear rate, the medium shows shear-thinning property which means that the coefficient of viscosity decreases with the increase in shear rate. Based on experimental input, Ivlev et. al. [15] have shown the power law dependence of viscosity on velocity shear. The experiment has been done with gas induced shear flow for different discharge currents and also by applying laser beams of different power. Hence they have measured shear viscosity for a wide range of velocity shear rates and confirmed shear thinning property over a considerable range. Very recently similar experiment has also been reported by Gavrikov et al. [18] in a dusty plasma liquid. simulation work in this direction has also been reported [19].
Motivated by these experimental results, we have investigated the effect of equilibrium viscosity gradient on a shear mode in a strongly coupled plasma. Shear thinning property with generalized Oldroyd-B model [16,17]has been studied in neutral viscoelastic fluid. In this work, we have demonstrated indeed that a novel instability exists due to the coupling of shear flow to the velocity fluctuations via velocity shear induced viscosity gradient.

Basic equations and equilibrium
In a standard fluid description of dusty plasma for studying low frequency (ω ≪ kv the , kv thi where v the,i correspond to the thermal velocities of electrons and ions respectively and k is a typical wave vector) phenomena normally we treat electrons and ions as a light fluid which can be modeled by a Boltzmann distribution neglecting their inertial effects in the momentum equations. This is justified because due to higher temperature and smaller electric charge compared to dust they can easily thermalize and give rise to Boltzmann distribution. The ion and electron densities can be written this way: n e(i) = n 0e(0i) exp [±eφ/T e(i) ], where n 0e(0i) , T e(i) correspond to the equilibrium densities and temperatures for the electrons (ions). Here, φ is the electrostatic potential.
The dust component on the other hand, can be described by generalized hydrodynamic (GH) equation described in Frenkel's book [21]. We follow the same procedure and write the generalized equation of motion of dust fluid in a viscoelastic where v is the dust fluid velocity, ρ = m d n d is the mass density of dust fluid, n d is corresponding number density, q is the charge on a dust particle, The parameter τ is the relaxation time of the medium [21] and viscosity tensor σ ij is given for an incompressible medium by Here η is the coefficient of shear viscosity. For a Newtonian fluid η is constant. However, for a non-Newtonian fluid, η depends on the scalar invariants of strain tensor. For an incompressible fluid, it has been shown that [20,22] the scalar invariant can be written as In this work we consider an incompressible plasma with constant mass density and since the medium is non-Newtonian, the viscosity coefficient can be considered to be a function of the scalar invariant. Hence, the viscosity parameter is taken to be of the form η(S) where S = I/2. It has been shown that in the kinetic limit τ ∂/∂t ≫ 1, linearized Eq.(1) gives rise to a 'shear wave' whose velocity is given by V s = η/ρτ , where η is a constant [3]. We would like to investigate the dynamics of this mode in presence of velocity shear dependent viscosity coefficient. In the kinetic limit, 1 can be neglected with respect to τ ∂/∂t in the above equation(1). We assume, that the equilibrium velocity is directed along y direction and has variation in x-direction, i.e. v 0 = v y0 (x)ê y , whereê y is a unit vector along y direction. It is clear that the left hand side of equation (1) will not contribute in equilibrium situation, so that the equilibrium is described by the equation where S 0 is the equilibrium value of the shear parameter. For small velocity fluctuations, we can write S = S 0 + S 1 , and it can be shown that S 0 = dv y0 /dx and S 1 = (∂v 1x /∂y + ∂v 1y /∂x). Recently, Ivlev et. al [15] have proposed a power-law model for the functional dependence of η(S 0 ), and performed an experiment to show the shear thinning property of dusty plasmas. In this paper, they have shown that η remains constant for low shear rate and after some critical value η decreases with increase of S 0 . Shear thinning behavior exists for a wide range of velocity shear and for some very high shear rate η increases with S 0 . A schematic sketch in fig.(1) represents the variation of shear viscosity with velocity shear rate. In the present work we concentrate in the shear thinning region (where η decreases with S 0 ) and use the same model consistent with the experiment. The functional form can be written as, (2), we find that dv y0 /dx is constant and hence we can write the equilibrium velocity as v y0 (x) = v ′ 0 x, where v ′ 0 is a constant having the dimension of frequency.

Stability analysis
We restrict our attention to two dimensional incompressible perturbations such that all variations are in the x − y plane. The incompressibility condition given by ∇ · v = 0 is consistent with the equilibrium flow. Now we perturb the system around this equilibrium flow writing v = v y0 (x)ê y + v 1 (x, y, t), and after a straightforward algebra we find that v 1x satisfies a differential equation which is given by where ∇ 2 ⊥ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 . We note here that if the velocity shear is absent in Eqs. (4), we get back the shear mode whose dispersion relation can be written as ω 2 = (k 2 x + k 2 y )V 2 s . For an inhomogeneous plasma the general solution of Eq. (4) including viscosity gradient can be obtained in various ways. The traditional starting point of an investigation of linear plasma stability is the eigenvalue analysis in which we assume the solution of the form v 1x = v 1 (x) exp(ik y y − iωt), where k y is the wave vector in y direction and ω is the frequency of the mode. We note here that Fourier type of solutions have been considered only in the y-direction, since inhomogeneity is present in the x-direction through velocity shear. The perturbed variable v 1 in general will satisfy a differential equation which is given by where ω 2 s = k 2 y η 0 /ρτ is the frequency of the shear wave in an inhomogeneous plasma. From the above differential equation first we can do the local analysis in which one uses the approximation kL ≫ 1. This implies that the perturbation wave length k −1 is much smaller than the inhomogeneity scale length L(= v 0 /v ′ 0 ). For the present problem, the local analysis is carried out by considering that the perturbed quantity v 1 has also an exponential variation in x i.e. v 1 ∼ exp(ik x x), where k x is the wave vector in x direction. Substituting in Eq. (5) the dispersion equation is obtained as where η ′ 0 = dη 0 /dv ′ 0 and ω ≫ k y v 0 . Now recalling the form of viscosity η 0 from Eq.
It is clear that the shear mode will be unstable if α > (k 2 x + k 2 y ) 2 /(k 2 y − k 2 x ) 2 and the growth rate is of the order of shear frequency. Next we consider the nonlocal analysis of Eq. (5) in which this eigenvalue equation may be solved to get well behaved solutions corresponding to unstable eigenvalues. Here we are looking for long radial (x) scale solution for the differential equation and therefore the fourth order derivative is subdominant compared with the second. Ignoring the fourth derivative in Eq. (5), we can reproduce the character of the mode with a very little change (this is apparent from the dispersion relation). This assumption simplifies the algebra without taking away the essential physics. The desired eigenvalue equation can be written as For the condition ωk y v ′ 0 /[ω 2 − 2ω 2 s (1 + α)] ≪ 1, which implies that, when the shear rate is small compared to the frequency of the mode, the above equation can be written in terms of the well known Weber equation which is given by The solution of Eq. (8) for the lowest order eigenmode is given by representing the existence of an unstable eigenmode. The condition for bounded solution is Re β 2 [(β 2 /β 1 ) − 1] 1/2 > 0. The behaviour of the eigenfunction v 1 at x → ±∞ is bounded and the typical mode width △ ∼ . The corresponding dispersion relation is given by where α = −η ′ 0 v ′ 0 /η 0 > 1 and ω 2 s = k 2 y η 0 /ρτ. In a homogeneous plasma i.e. when v ′ 0 = α = 0 we get back the shear mode i.e. ω 2 ∼ ω 2 s . In presence of velocity shear and velocity shear induced viscosity gradient we have solved Eq. (10) and found that for α < 2 there is one unstable root for real ω > 0. The growth rate for the instability for the given range of α can be seen in the figure 2. The shear mode is more unstable when velocity shear is stronger.

Summary
We have studied the effect of velocity shear induced viscosity gradient of low frequency shear waves in a viscoelastic dusty plasma. The dust dynamics has been modeled by including velocity shear dependent viscosity which is the main ingredient to drive a new instability in a complex plasma. The principal effect on the generation of the novel instability is the velocity dependence of viscosity that leads to a coupling between velocity fluctuations and equilibrium flow. The variation in the velocity is responsible for viscosity modulation which provides a feedback to the velocity through momentum equation. For a positive feedback of the velocity, an instability is triggered. This novel low frequency instability disappears when viscosity is uniform and we are left with a shear wave. We would like to point out that the instability theoretically investigated in this work has not been observed in real experiment as yet; its detailed experimental investigation is therefore of great interest. It would be of interest therefore to look for shear wave driven instabilities discussed in this model calculation which is based upon the experimental finding of a shear thinning region.