Volume 24, Issue 7, July 2012

We report on numerical simulations of the detailed evolution of the single mode RayleighTaylor [Lord Rayleigh, Scientific Papers II (Cambridge University Press, Cambridge, 1900), p. 200;G. I. Taylor, “The instability of liquid surfaces when accelerated in a direction perpendicular to their plane,” Proc. R. Soc. London, Ser. A201, 192 (1950)10.1098/rspa.1950.0052;S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford, 1961)]instability to late times and high aspect ratios. In contrast to established potential flowmodels that predict a terminal velocity and a constant Froude number at low Atwood numbers, we observe a complex sequence of events that can be summarized in four stages: I. Exponential growth of imposed perturbations, II. Saturation to terminal velocity, III. Reacceleration to a higher Froude number, and IV. Chaotic mixing. The observed reacceleration away from the Froude number predicted by potential flowtheory is attributed to the appearance of secondary Kelvin–Helmholtz structures, and described with a modification to the potential flowmodel proposed by Betti and Sanz [R. Betti and J. Sanz, “Bubble acceleration in the ablative RayleighTaylor instability,” Phys. Rev. Lett.97, 205002 (2006)10.1103/PhysRevLett.97.205002]. The secondary KH instability is in turn sensitive to several parameters, and can be suppressed at large Atwood numbers, as well as viscosity (physical or numerical), with the bubble/spike velocity in each case reverting to the potential flow value. Our simulations delineate the change in dynamics of the primary and secondary instabilities due to changes in these flow parameters. When the flow is allowed to evolve to late times, further instability is observed, resulting in chaotic mixing which is quantified here. The increased atomic mixing due to smallscale structures results in a dramatic drop in the latetime Froude number. Spike behavior resembles bubbles at low A, while for large A, spikes approach freefall – thus, the notion of a terminal velocity appears not to be applicable to spikes at any density difference. We expect the results to be relevant to turbulent mix models that are based on bubble growth and interaction.
 ARTICLES

 Micro and Nanofluid Mechanics

Liquid flow retardation in nanospaces due to electroviscosity: Electrical double layer overlap, hydrodynamic slippage, and ambient atmospheric CO_{2} dissolution
View Description Hide DescriptionA theoretical investigation is performed into the electroviscousinduced retardation of liquid flows through finitely long nanochannels or nanotubes with large wells at either end. Given the assumption of equilibrium conditions between the ionic solution in the wells and that within the nanochannel or nanotube, an exact solution is derived for the overlapped electrical double layer(EDL) for the case where the concentrations of the positive and negative ions in the wells may be unequal. The ion concentrations in the wells are determined by the conditions of global electroneutrality and mass conservation. It is shown that the overlapped EDL model proposed by Baldessari and Santiago [J. Colloid Interface Sci.325, 526 (2008)10.1016/j.jcis.2008.06.007] is in fact the same as the “thick EDL model” (i.e., the traditional PoissonBoltzmann model) when the positive and negative ion concentrations in the large enough wells are both equal to the bulk concentration of the salt solution. Utilizing the proposed overlapped EDL analytical model, an investigation is performed to evaluate the effects of hydrodynamic slippage on the flow retardation caused by electroviscosity in nanochannels or nanotubes. Furthermore, exact and approximate solutions are derived for the electroviscosity in ionselective nanochannels and nanotubes. It is shown that in the absence of slip, the maximum electroviscosity in nanochannels and nanotubes containing a unipolar solution of simple monovalent counterions occurs at surface charge densities of hσ = 0.32 nm × C/m^{2} and aσ ≈ 0.4 nm × C/m^{2}, respectively. In addition, it is shown that the electroviscosity in a nanotube is smaller than that in a nanochannel. For example, given a LiCl solution, the maximum electroviscosites in a nonslip nanochannel and nonslip nanotube are η_{ a }/η ≈ 1.6 and 1.47, respectively. For both nanospaces, the electroviscosity is greatly increased when the liquid slip effect is taken into account. Significantly, under slip conditions, the electroviscosity in the nanotube is greater than that in the nanochannel. Finally, an investigation is performed into the effects of ambient atmospheric CO_{2} dissolution on the electroviscosities of salt/buffer solution and deionized (DI) water in silica nanochannels. The results show that the electroviscosity of CO_{2}saturated DI water (pH = 5.6) can be reasonably neglected in silica nanochannels with a height of less than 100 nm.
 Interfacial Flows

Reduced equations of motion of the interface of dielectric liquids in vertical electric and gravitational fields
View Description Hide DescriptionThe dynamics of the interface between two dielectric fluids in the presence of vertical electric and gravitational fields is studied theoretically. It is shown that, in the particular case where the rate of change of the electric field is proportional to the effective gravitational acceleration, a special flow regime can be realized for which the velocity and electric potentials are linearly dependent functions. This means that there exists a frame of reference in which liquids move along the electric field lines. We derive and analyze the corresponding reduced equations of motion of a liquidliquid interface. For small density ratio, they turn into the equations describing the Laplacian growth. In the case of two spatial dimensions, we show that these equations determine the asymptotic behavior of the system. For arbitrary density ratios, the Laplacian growth equations adequately describe the initial (weakly nonlinear) stage of the interface instability development. The integrability of these equations makes it possible to investigate the evolution of nonlinear waves at the boundary and, in particular, to demonstrate the tendency to the formation of singularities (cusps).

Buckling instability of squeezed droplets
View Description Hide DescriptionMotivated by recent experiments, we consider theoretically the compression of droplets pinned at the bottom on a surface of finite area. We show that if the droplet is sufficiently compressed at the top by a surface, it will always develop a shape instability at a critical compression. When the top surface is flat, the shape instability occurs precisely when the apparent contact angle of the droplet at the pinned surface is π, regardless of the contact angle of the upper surface, reminiscent of a past work on liquid bridges and sessile droplets as first observed by Plateau. After the critical compression, the droplet transitions from a symmetric to an asymmetric shape. The force required to deform the droplet peaks at the critical point then progressively decreases the indicative of catastrophic buckling. We characterize the transition in droplet shape using illustrative examples in two dimensions followed by perturbative analysis as well as numerical simulation in three dimensions. When the upper surface is not flat, the simple apparent contact angle criterion no longer holds, and a detailed stability analysis is carried out to predict the critical compression.

Stability of a viscous pinching thread
View Description Hide DescriptionWe consider the dynamics of a fluid thread near pinchoff, in the limit that inertial effects can be neglected. There exists an infinite hierarchy of similarity solutions corresponding to pinchoff. Only one of the similarity solutions (the “ground state”) is stable, all other solutions are linearly unstable to perturbations, and thus cannot be observed. Eigenvalues and eigenfunctions are calculated analytically.

Theory of the forced wetting transition
View Description Hide DescriptionWe consider a solid plate being withdrawn from a bath of liquid which it does not wet. At low speeds, the meniscus rises below a moving contact line, leaving the rest of the plate dry. At a critical speed of withdrawal, this solution bifurcates into another branch via a saddlenode bifurcation: two branches exist below the critical speed, the lower branch is stable, the upper branch is unstable. The upper branch eventually leads to a solution corresponding to film deposition. We add the local analysis of the upper branch of the bifurcation to a previous analysis of the lower branch. We thus provide a complete description of the dynamical wetting transition in terms of matched asymptotic expansions.

Thermocapillary effects on steadily evaporating contact line: A perturbative local analysis
View Description Hide DescriptionThe evaporation process taking place close to the threephase contact line is considered and studied theoretically using a linear stability analysis approach. A domain perturbation method, taking into consideration thermocapillary effects and surface forces, is used to develop the higherorder solution in terms of series expansion about lubrication condition. A closedform solution is found for the film thickness, the pressure jump across the liquidvapor interface and the evaporative flux in the vicinity of the contact line. The key novelty in this work is considering thermocapillary instability for very thin films (∼10 nm) accounting for surface forces. For (quasi) flatverythin films, the analysis shows no instability, which is consistent with general knowledge in this field. However, for films extending from a meniscus, as encountered in wetting configurations, it is found that the competition between London–van der Waals, capillary, and thermocapillary forces leads to contact line instability and behavior revealed for the first time. According to the sign of the Marangoni number, the instability can enhance (if positive) or reduce (if negative) the evaporation rate by widening out or narrowing, respectively, the contact region and, in both cases, significantly modifies the vortical structure of the flow. If the Marangoni number is positive, the filminterface close to the contact line can exhibit corrugations. The occurrence of these latter is discriminated, in addition to the Marangoni number, by the value of three operating parameters, namely the film aspect ratio, the ratio of the film diffusive thermal resistance to evaporative heat transfer resistance, and the ratio of capillary pressure to disjoining pressure. By modifying the physical and operating parameters, it is also shown that the system can be optimized in order to suppress these corrugations.
 Viscous and NonNewtonian Flows

Insights into symmetric and asymmetric vortex mergers using the core growth model
View Description Hide DescriptionWe revisit the two vortex merger problem (symmetric and asymmetric) for the NavierStokes equations using the core growth model for vorticity evolution coupled with the passive particle field and an appropriately chosen timedependent rotating reference frame. Using the combined tools of analyzing the topology of the streamline patterns along with the careful tracking of passive fields, we highlight the key features of the stages of evolution of vortex merger, pinpointing deficiencies in the lowdimensional model with respect to similar experimental/numerical studies. The model, however, reveals a far richer and delicate sequence of topological bifurcations than has previously been discussed in the literature for this problem, and, at the same time, points the way towards a method of improving the model.

Thermally excited vortical flow in a thin bidirectionally oriented nematic cell
View Description Hide DescriptionThe theoretical description of the reorientational dynamics in a thin bidirectionally aligned liquid crystal cell (BALC), where the nematic sample is confined by two horizontal and two lateral surfaces, under the influence of a temperature gradient ∇T is presented. We have carried out a numerical study of the system of hydrodynamic equations including director reorientation, fluid flow v, and the temperature redistribution across the BALC cell under the influence of ∇T, when the sample is heated both from below and from above. Calculations show that, due to interaction between the gradient of the director field and ∇T, the BALC sample settles down to a stationary bivortical flow regime. As for a nematogenic material, we have considered the BALC cell to be occupied by 4 − n − pentyl − 4^{′} − cyanobiphenyl, and investigated the effect of both and ∇T on the magnitude and direction of v, for a number of hydrodynamic regimes.

The effects of hydrodynamic interaction and inertia in determining the highfrequency dynamic modulus of a viscoelastic fluid with twopoint passive microrheology
View Description Hide DescriptionIn twopoint passive microrheology, a modification of the original onepoint technique, introduced by Crocker et al. [Phys. Rev. Lett.85, 888 (2000)]10.1103/PhysRevLett.85.888, the crosscorrelations of two micronsized beads embedded in a viscoelastic fluid are used to estimate the dynamic modulus of a material. The twopoint technique allows for the sampling of larger length scales, which means that it can be used in materials with a coarser microstructure. An optimal separation between the beads exists at which the desired length and time scales are sampled while keeping a desired signaltonoiseratio in the crosscorrelations. A large separation can reduce the effect of higher order reflections, but will increase the effects of medium inertia and reduce the signaltonoiseratio. The modeling formalisms commonly used to relate twobead crosscorrelations to G*(ω) neglect inertia effects and underestimate the effect of reflections. A simple dimensional analysis for a model viscoelastic fluid suggests that there exists a very narrow window of bead separation and frequency range where these effects can be neglected. Therefore, we consider both generalized data analysis and generalized Brownian dynamics (BD) simulations to examine the magnitude of these effects. Our proposed analysis relies on the recent analytic results of Ardekani and Rangel [Phys. Fluids18, 103306 (2006)]10.1063/1.2363351 for a purely viscous fluid, which are generalized to linear viscoelastic fluids. Implementation requires approximations to estimate Laplace transforms efficiently. These approximations are then used to create generalized BD simulation algorithms. The data analysis formalism presented in this work can expand the region of separation between the beads and frequencies at which rheological properties can be accurately measured using twopoint passive microrheology. Moreover, the additional physics introduced in the data analysis formalisms do not add additional significant computational costs.

Squeeze flow of a Carreau fluid during sphere impact
View Description Hide DescriptionWe present results from a combined numerical and experimental investigation into the squeeze flow induced when a solid sphere impacts onto a thin, ultraviscous film of nonNewtonian fluid. We examine both the sphere motion through the liquid as well as the fluid flow field in the region directly beneath the sphere during approach to a solid plate. In the experiments we use silicone oil as the model fluid, which is welldescribed by the Carreau model. We use highspeed imaging and particle tracking to achieve flow visualisation within the film itself and derive the corresponding velocity fields. We show that the radial velocity either diverges as the gap between the sphere and the wall diminishes (Z _{ tip } → 0) or that it reaches a maximum value and then decays rapidly to zero as the sphere comes to rest at a nonzero distance (Z _{ tip } = Z _{ min }) away from the wall. The horizontal shear rate is calculated and is responsible for significant viscosity reduction during the approach of the sphere. Our model of this flow, based on lubrication theory, is solved numerically and compared to experimental trials. We show that our model is able to correctly describe the physical features of the flow observed in the experiments.

Direct numerical simulation of binary offcenter collisions of shear thinning droplets at high Weber numbers
View Description Hide DescriptionBinary droplet collisions, a prototype elementary subprocess inside sprays, are investigated by direct numerical simulations (DNS) based on an extended volume of fluid method. We focus on shearthinningdroplet collisions. In order to capture the dynamics of droplet collisions with different outcomes, we account for offcenter collisions at high Weber numbers. Such collision conditions lead to the formation of extremely thin fluid lamellae. It turns out that these thin lamellae determine the smallest length scales which must be resolved in a DNS. A stabilization algorithm is presented which prevents the lamellae from rupturing. It is validated by comparison with experimental data and applied for a droplet collision study of shearthinningliquids. The results show that, independent of the offset of the colliding droplets, a collision of Newtonian liquiddroplets with appropriately chosen viscosity can reproduce the collision dynamics of the shearthinningliquiddroplets. This includes temporal evolution of shapes and energy.
 Particulate, Multiphase, and Granular Flows

Mitigation of preferential concentration of small inertial particles in stationary isotropic turbulence using electrical and gravitational body forces
View Description Hide DescriptionParticles with a certain range of Stokes numbers preferentially concentrate due to action of turbulent motion and body forces such as gravity are known to influence this process. The effect of electric charge, residing on particles, upon the phenomenon of preferential concentration is investigated. We use direct numerical simulations of oneway coupled stationary isotropic turbulence over a range of particle Stokes numbers, fluid Taylor Reynolds numbers, and electrical and gravitational particle body force magnitudes, the latter characterized by nondimensional settling velocities, and , respectively. In contrast to the gravitational body force, the electrical analogue, acting on an electrically charged particle, is generated by an electric field, which is in turn a function of the degree of preferential concentration. Thus, the electrical body force is created by, and mitigates, preferential concentration. In the absence of gravity, it is estimated that ≈ 1.0 is sufficient to homogenise a preferentially concentrated particle distribution. It is seen that charging drastically reduces the radial distribution function values at Kolmogorov scale separations, which gravitational force does not. This implies that charging the particles is an efficient means to destroy small clusters of particles. On incorporating the gravitational force, the amount of charge required to homogenise the particle distribution is reduced. It is estimated that ≈ 0.6 is sufficient to homogenise particle distribution at = 2.0. This estimation is corroborated by several different indicators of preferential concentration, and the results also agree reasonably well with corresponding experiments reported in literature. Calculations also suggest that sprays generated by practical charge injection atomizers would benefit from this electrical dispersion effect.

Concentration instability of sedimenting spheres in a secondorder fluid
View Description Hide DescriptionThe slow sedimentation of a dilute suspension of spherical particles in a secondorder fluid is investigated using theory and numerical simulations. We first analyze the motion of a single isolated spherical particle sedimenting under gravity when placed in a linear flow field. In the limit of weak viscoelasticity (low Deborah number), the velocity of the particle is calculated, and the nonlinear coupling of the settling motion with the local flow field is shown to result in a lateral drift in a direction perpendicular to gravity. By the same effect, the mean flow driven by weak horizontal density fluctuations in a largescale suspension of hydrodynamically interacting particles will also result in a horizontal drift, which has the effect of reinforcing the fluctuations as we demonstrate using a linear stability analysis. Based on this mechanism, an initially homogeneous suspension is expected to develop concentration fluctuations, a prediction supported by previous experiments on sedimentation in polymeric liquids. We further confirm this prediction using largescale weakly nonlinear numerical simulations based on a pointparticle model. Concentration fluctuations are indeed found to grow in the simulations, and are shown to result in an enhancement of the mean settling speed and velocity fluctuations compared to the Newtonian case.

Kinetictheorybased model of dense granular flows down inclined planes
View Description Hide DescriptionThis work extends a continuum model of sheared granular material comprising twodimensional disks [C. H. Lee and C. J. Huang, Phys. Fluids22, 043307 (2010)10.1063/1.3400203] to elucidate the dynamics of threedimensional spheres. The proposed model is applied to investigate dense granular flows down an inclined plane. In the model, stress has a static component and a kinetic component. The constitutive model for shear stress reduces to the Bagnold model when the diffusion of granular temperature is small. The predicted rheological characteristics are identical to those observed in the preceding experiments and numerical simulations, validating the present model. The predicted rheological characteristics reveal that dense granular flows down an inclined plane are characterized by three special angles that determine the phase diagram. The predicted thick granular flow on an inclined plane exhibits the Bagnold velocity profile and a uniform volume fraction throughout its depth. The governing equation of granular temperature is simplified and solved analytically. The proposed shear granular flowmodel is also solved completely using the finite volume method. The predicted velocity and volume fraction agree very well with previous discretely simulated results. This work also proposes an equation for determining the characteristic length of dense granular flows and shows that its static component is close to the stopping height.

Metal particle nucleation in laminar jets
View Description Hide DescriptionGas to particle conversion in the form of nucleation within various flow systems plays a significant role in a variety of industrial and natural processes. Recently developed surface tension models offer increased accuracy in the modeling of metal particle nucleation. These models facilitate the probing of the effects of fluid, scalar, and thermal transport on nucleation in an accurate manner. In this work we investigate the formation of metalnanoparticles in laminar flows. The flows consist of metal vapor diluted in argon issuing into a cooler argon stream. The fluid, thermal, and chemical fields are obtained by solving the Navier Stokes, enthalpy, and massfraction transport equations while nucleation is simulated via a homogeneous nucleation model with sizedependent surface tension. This approach is attractive in that it promises to be more accurate than the classical nucleation theory (CNT) while maintaining much of its simplicity when coupling with fluid dynamics. The results show that the sizedependent surface tensionnucleation model is more accurate than CNT and agrees well with physical data. Physically, the sensitivity of the saturation ratio to changes in temperature is shown to be greater than its sensitivity to mass fraction, highlighting the significance of differential molecular transport of energy and mass and the significance of nonunity Lewis numbers. More significantly, the sizedependent surface tension approach suggests that certain metals may have a maximum nucleation rate and further cooling—a strategy employed to increase particle nucleation rates—will actually decrease particle nucleation.

Anisotropy in pair dispersion of inertial particles in turbulent channel flow
View Description Hide DescriptionThe rate at which two particles separate in turbulent flows is of central importance to predict the inhomogeneities of particle spatial distribution and to characterize mixing. Pair separation is analyzed for the specific case of small, inertial particles in turbulent channel flow to examine the role of mean shear and smallscale turbulentvelocityfluctuations. To this aim an EulerianLagrangian approach based on pseudospectral direct numerical simulation (DNS) of fully developed gassolid flow at shear Reynolds numberRe _{τ} = 150 is used. Pair separation statistics have been computed for particles with different inertia (and for inertialess tracers) released from different regions of the channel. Results confirm that shearinduced effects predominate when the pair separation distance becomes comparable to the largest scale of the flow. Results also reveal the fundamental role played by particlesturbulence interaction at the small scales in triggering separation during the initial stages of pair dispersion. These findings are discussed examining Lagrangian observables, including the mean square separation, which provide prima facie evidence that pair dispersion in nonhomogeneous anisotropicturbulence has a superdiffusive nature and may generate nonGaussian number density distributions of both particles and tracers. These features appear to persist even when the effects of shear dispersion are filtered out, and exhibit strong dependency on particle inertia. Application of present results is discussed in the context of modelling approaches for particle dispersion in wallbounded turbulent flows.
 Laminar Flows

Eulerian indicators under continuously varying conditions
View Description Hide DescriptionIn this paper, we extend the notion of Eulerian indicators (EIs) for predicting Lagrangian mixing behavior previously developed for blinking flows to the continuous time setting. We apply the EIs to a study of mixing in a kinematicmodel of a timedependent doublegyre with five different time dependencies—sinusoidal, sawtooth, square wave, triangular, and noise (which is constructed so that it is also periodic in time). Each of the five velocity fields is described by two parameters; the strength of the time dependence (ε) and the period (T). Based on a trajectory based quality of mixing diagnostic (Danckwerts’ normalized variance of concentration) we find that noisy time dependence has the largest region of good mixing in the parameter space and triangular time dependence has parameter values corresponding to the most complete and fastest mixing. These Lagrangian based predictions are confirmed by the EIs (product of the transversality and mobility). Although not every feature of the mixing behavior is captured by EIs, we show that they do in general predict the regions in the parameter space under consideration that correspond to good mixing. Moreover, the EIs offer a factor of 100 computational advantage in exploring the parameter space in comparison with the trajectory based mixing diagnostic.

A numerical study of the laminar necklace vortex system and its effect on the wake for a circular cylinder
View Description Hide DescriptionLarge eddy simulation (LES) is used to investigate the structure of the laminar horseshoe vortex (HV) system and the dynamics of the necklace vortices as they fold around the base of a circular cylinder mounted on the flat bed of an open channel for Reynolds numbers defined with the cylinder diameter, D, smaller than 4460. The study concentrates on the analysis of the structure of the HV system in the periodic breakaway subregime, which is characterized by the formation of three main necklace vortices. Over one oscillation cycle of the previously observed breakaway subregime, the corner vortex and the primary vortex merge (amalgamate) and a developing vortex separates from the incoming laminar boundary layer (BL) to become the new primary vortex. Results show that while the classical breakaway subregime, in which one amalgamation event occurs per oscillation cycle, is present when the nondimensional displacement thickness of the incoming BL at the location of the cylinder is relatively large (δ*/D > 0.1), a new type of breakaway subregime is present for low values of δ*/D. This subregime, which we call the doublebreakaway subregime, is characterized by the occurrence of two amalgamation events over one full oscillation cycle. LES results show that when the HV system is in one of the breakaway subregimes, the interactions between the highly coherent necklace vortices and the eddies shed inside the separated shear layers (SSLs) are very strong. For the relatively shallow flow conditions considered in this study (H/D ≅ 1, H is the channel depth), at times, the disturbances induced by the legs of the necklace vortices do not allow the SSLs on the two sides of the cylinder to interact in a way that allows the vorticity redistribution mechanism to lead to the formation of a new wake roller. As a result, the shedding of largescale rollers in the turbulent wake is suppressed for relatively large periods of time. Simulation results show that the wake structure changes randomly between time intervals when largescale rollers are forming and are convected in the wake (von Karman regime), and time intervals when the rollers do not form. When the wake is in the von Karman regime, the shedding frequency of the rollers is close to that observed for flow past infinitely long cylinders.

Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit
View Description Hide DescriptionThis paper extends the mapping matrix formalism to include the effects of molecular diffusion in the analysis of mixing processes in chaotic flows. The approach followed is Lagrangian, by considering the stochastic formulation of advectiondiffusion processes via the Langevin equation for passive fluid particle motion. In addition, the inclusion of diffusional effects in the mapping matrix formalism permits to frame the spectral properties of mapping matrices in the purely convective limit in a quantitative way. Specifically, the effects of coarse graining can be quantified by means of an effective Péclet number that scales as the second power of the linear lattice size. This simple result is sufficient to predict the scaling exponents characterizing the behavior of the eigenvalue spectrum of the advectiondiffusion operator in chaotic flows as a function of the Péclet number, exclusively from purely kinematic data, by varying the grid resolution. Simple but representative model systems and realistic physically realizable flows are considered under a wealth of different kinematic conditions–from the presence of large quasiperiodic islands intertwined by chaotic regions, to almost globally chaotic conditions, to flows possessing “sticky islands”–providing a fairly comprehensive characterization of the different numerical phenomenologies that may occur in the quantitative analysis of mapping matrices, and ultimately of chaotic mixing processes.
 Instability and Transition

Secondary energy growth and turbulence suppression in conducting channel flow with streamwise magnetic field
View Description Hide DescriptionThe effects of a streamwise magnetic field on conducting channel flow are studied by analyzing secondary linear perturbations evolving on streamwise streaks and by direct numerical simulations of relaminarization. By means of an optimal perturbation approach, magnetic damping is found to increase the streamwise wavelength of the most amplified secondary perturbations and to reduce their amplification level. Complete suppression of secondary instability is observed at a critical magnetic interaction parameter that depends on the streak amplitude and on the Reynolds number when the transient evolution of the streaky basic flow is taken into account. Relaminarization in the direct numerical simulation occurs at lower values of the interaction parameter than the critical values from the stability computations for the streak amplitudes considered. The dependence of these threshold values of the interaction parameters on the Reynolds number is fairly similar between simulations and stability analysis. Relaminarization thresholds from the simulations are also in good agreement with experiments on pipe flow with streamwise magnetic field.