Viscous heating can play an important role in the dynamics of fluids with strongly temperature-dependent viscosities because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local temperature increase near the tube walls with a consequent decrease of the viscosity and a strong stratification in the viscosity profile. The problem of viscous heating in fluids was investigated and reviewed by Costa & Macedonio (2003) because of its important implications in the study of magma flows. Because of the strong coupling between viscosity and temperature, the temperature rise due to the viscous heating may trigger instabilities in the velocity field, which cannot be predicted by a simple isothermal Newtonian model. When viscous heating produces a pronounced peak in the temperature profile near the walls, a triggering of instabilities and a transition to secondary flows can occur because of the stratification in the viscosity profile. In this paper we focus on the thermal and mechanical effects caused by viscous heating. We will present the linear stability equations and we will show, as in certain regimes, these effects can trigger and sustain a particular class of secondary rotational flows which appear organised in coherent structures similar to roller vortices. This phenomenon can play a very important role in the dynamics of magma flows in conduits and lava flows in channels and, to our knowledge, it is the first time that it has been investigated by a direct numerical simulation.
Viscous heating plays an important role in the dynamics of fluids with strongly temperature-dependent viscosity because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local temperature increase near the tube walls with a consequent decrease of the viscosity which may dramatically change the temperature and velocity profiles. These processes are mainly controlled by the Peclét number, the Nahme number, the flow rate and the thermal boundary conditions. The problem of viscous heating in fluids was investigated in the past for its prac…
A non-modal linear stability analysis (NMA) of the miscible viscous fingering in a porous medium is studied for a toy model of non-monotonic viscosity variation. The onset of instability and its physical mechanism are captured in terms of the singular values of the propagator matrix corresponding to the non-autonomous linear equations. We discuss two types of non-monotonic viscosity profiles, namely, with unfavorable (when a less viscous fluid displaces a high viscous fluid) and with favorable (when a more viscous fluid displaces a less viscous fluid) end-point viscosities. A linear stability …
We theoretically examine the widespread hypothesis of an electro-viscous origin of the increase in apparent viscosity observed in recent experiments on capillary filling of nanochannels. Including Debye-layer corrections to the hydraulic resistance we find that the apparent viscosity reaches a maximum in the mesoscopic regime where the channel height (or more generally the hydraulic radius) is comparable to the screening length. However, for realistic estimates of central parameters, we find that the electro-viscous contribution to the apparent viscosity is at most a 1% effect.
A hierarchy of mathematical models describing viscosity-stratified flow in a Hele-Shaw cell is constructed. Numerical modelling of jet flow and development of viscous fingers with the influence of inertia and friction is carried out. One-dimensional multi-layer flows are studied. In the framework of three-layer flow the interpretation of the Saffman--Taylor instability is given. A kinematic-wave model of viscous fingering taking into account friction between the fluid layers is proposed. Comparison with calculations on the basis of two-dimensional equations shows that this model allows to dete…
We study the instability development during a viscous liquid drop impacting a smooth substrate, using high speed photography. The onset time of the instability highly depends on the surrounding air pressure and the liquid viscosity: it decreases with air pressure with the power of minus two, and increases linearly with the liquid viscosity. From the real-time dynamics measurements, we construct a model which compares the destabilizing stress from air with the stabilizing stress from liquid viscosity. Under this model, our experimental results indicate that at the instability onset time, the tw…
In the above paper an analysis has been carried out to obtain results in the Couette flow of a Newtonian fluid with viscous dissipation and temperature dependent viscosity. The fluid viscosity is an exponential function of temperature. Exact analytical solutions are obtained for the calculation of velocity and temperature. However, there are some fundamental errors in this paper which are presented herein.
Parametrically excited standing waves are observed on a cylindrical fluid filament. These are the cylindrical analog of Faraday instability in a flat surface or spherical droplet. Using the Floquet technique, linear stability analysis has been investigated on a viscous cylindrical fluid surface, which is subjected to a time-periodic radial acceleration. Viscosity has a significant impact on the critical forcing amplitude as well as the dispersion relation of the non-axis symmetric patterns. The effect of viscosity on onset parameters of the pattern with azimuthal wavenumber, $m=1$, has shown d…
The shear induced collective diffusivity in an emulsion of viscous drops, specifically as a function of viscosity ratio, was numerically computed. An initially randomly packed layer of viscous drops spreading due to drop-drop interactions in an imposed shear has been simulated. The shear induced collective diffusivity coefficient was computed using a self-similar solution of the drop concentration profile. We also obtained the collective diffusivity computing the dynamic structure factor from the simulated drop positions--an analysis typically applied only to homogeneous systems. The two quant…
The swimming of a deformable planar slab in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations. A continuum of plane wave displacements, symmetric on both sides of the slab and characterized by a polarization angle, allows optimization of the swimming efficiency with respect to polarization. The mean swimming velocity and mean rate of dissipation are calculated to second order in the amplitude of the stroke. The optimum efficiency depends on the ratio of viscosity and mass density of the fluid. For high viscosity a stroke is found with significantly higher ef…
The breakup of a jet of a viscous fluid with viscosity $μ_{1}$ immersed into another viscous fluid with viscosity $μ_{2}$ is considered in the limit when the viscosity ratio $λ=μ_{1}/μ_{2}$ is close to zero. We show that, in this limit, a transition from ordinary continuous selfsimilarity to discrete selfsimilarity takes place as $λ$decreases. The result being that instead of a single point breakup, the rupture of the inner jet occurs through the appearance of an infinite sequence of filaments of decreasing size that will eventually produce infinite sequences of bubbles of the inner fluid insi…
We study pressure-driven propagation of gas into a micron-scale gap between two linearly elastic substrates. Applying the lubrication approximation, the flow-field is governed by the interaction between elasticity and viscosity, as well as weak rarefaction and low-Mach-compressibility, characteristic to gaseous microflows. Several physical limits allow simplification of the governing evolution equation and enable solution by self-similarity. These limits correspond to different time-scales and physical regimes which include compressiblity-elasticity-viscosity, compressiblity-viscosity and elas…
In this article, we present some explicit solutions showing rough dependence upon initial data. We also studies viscous effects. An extension of a theorem of Cauchy to the viscous case is also presented.
The displacement of a more viscous fluid by a less viscous one in a quasi-two dimensional geometry leads to the formation of complex fingering patterns. This fingering has been characterized by a most unstable wavelength, $λ_c$, which depends on the viscosity difference between the two immiscible fluids and sets the characteristic width of the fingers. How the finger length grows after the instability occurs is an equally important, but previously overlooked, aspect that characterizes the global features of the patterns. As the lower viscosity fluid is injected, we show that there is a stable …
The hydrodynamic viscous fingering instability can be influenced by a simple viscosity changing chemical reaction of type A+B --> C, when a solution of reactant A is injected into a solution of B and a product C of different viscosity is formed. We investigate here numerically such reactive viscous fingering in the case of a reaction decreasing the viscosity to define the optimal conditions on the chemical and hydrodynamic parameters for controlling fingering. In particular, we analyze the influence of the injection flow rate or equivalently of the Peclet number (Pe) of the problem on the effi…
We investigate freely expanding liquid sheets made of either simple Newtonian fluids or solutions of high molecular water-soluble polymer chains. A sheet is produced by the impact of a drop on a quartz plate covered with a thin layer of liquid nitrogen that suppresses shear viscous dissipation thanks to an inverse Leidenfrost effect. The sheet expands radially until reaching a maximum diameter and subsequently recedes. Experiments indicate the presence of two expansion regimes: the capillary regime, where the maximum expansion is controlled by surface tension forces and does not depend on the …
An improved wetting boundary implementation strategy is proposed based on lattice Boltzmann color-gradient model in this paper. In this strategy, an extra interface force condition is demonstrated based on the diffuse interface assumption and is employed in contact line region. It has been validated by three benchmark problems: static droplet wetting on a flat surface and a curved surface, and dynamic capillary filling. Good performances are shown in all three cases. Relied on the strict validation to our scheme, the viscous fingering phenomenon of immiscible fluids displacement in a two-dimen…
Physical damping, regarding the nonlinear Navier-Stokes viscous flow dynamics, refers to a tensorial turbulent dissipation term, attributed to adjacent moving macroscopic flow components. Mutual dissipation among these parts of fluid is described by a braking term in the momentum equation together with a heating term in the energy equation, both responsible of the damping of the momentum variation and of the viscous conversion of mechanical energy into heat. A macroscopic mixing scale length is currently the only characteristic length needed in the nonlinear modelling of viscous fluid dynamics…
We explore the behaviour of barotropic and irrotational fluids with a small viscosity under the effect of first-order acoustic perturbations. We discuss, following the extant literature, the difficulties in gleaning an acoustic geometry in the presence of viscosity. In order to obviate various technical encumbrances, when viscosity is present, for an extraction of a possible acoustic geometry, we adopted a method of double perturbations, whereby dynamical quantities such as the velocity field and potential undergo a perturbation both in viscosity and in an external acoustic stimulus. The resul…
We investigate both experimentally and numerically the impact of liquid drops on deep pools of aqueous glycerol solutions with variable pool viscosity and air pressure. With this approach we are able to address drop impacts on substrates that continuously transition from low-viscosity liquids to almost solids. We show that the generic corolla spreading out from the impact point consists of two distinct sheets, namely an ejecta sheet fed by the drop liquid and a second sheet fed by the substrate liquid, which evolve on separated timescales. These two sheets contribute to a varying extent to the…
The growth rate of the compressible Rayleigh-Taylor instability is studied in the presence of a background temperature gradient, $Θ$, using a normal mode analysis. The effect of $Θ$ variation is examined for three interface types corresponding to combinations of the viscous properties of the fluids (inviscid-inviscid, viscous-viscous and viscous-inviscid) at different Atwood numbers, $At$, and, when at least one of the fluids' viscosity is non-zero, as a function of the Grashof number. For the general case, the resulting ordinary differential equations are solved numerically; however, dispersi…
The stability of a thin liquid film bounded by two free surfaces is examined in the presence of insoluble surface active agents. The surface active agents not only cause gradients in surface tension, but could also render surface viscosity to be significant, and variable, as a function of their concentration. A set of three coupled nonlinear evolution equations are derived for the film height, concentration of surface active agents, and the horizontal liquid velocity which governs the dynamics of the free film. Suitable phenomenological models for variation of surface tension and surface visco…
In this technical note, we demonstrate the robustness of our numerical scheme of vorticity iteration in dealing with the dipole-wall interaction at small viscosity, with emphasis on mesh convergence, boundary vorticity as well as wall viscous dissipation. In particular, it is found that, among the four different dipole configurations, the processes of vortex-wall collision at no-slip surfaces are exceedingly complex and are functions of the initial conditions. The critical issue direct numerical simulations is to establish mesh convergence which appears to be case-dependent. Roughly speaking, …
We perform linear stability analyses (LSA) and direct numerical simulations (DNS) to investigate the influence of the dynamic viscosity on viscous fingering (VF) instability in miscible slices. Selecting the characteristic scales appropriately the importance of the magnitude of the dynamic viscosity of individual fluids on VF in miscible slice has been shown in the context of the transient interfacial tension. Further, we have confirmed this result for immiscible fluids and manifest the similarities between VF in immiscible and miscible slices with transient interfacial tension. In a more gene…
The exact solutions for MHD shock waves in an ideal gas are obtained taking into consideration only the viscosity of the gas. In view of an axial magnetic field, the analytical expressions for the particle velocity, temperature, pressure and change-in-entropy within the shock transition region are obtained. The flow variables are numerical analysed to explore the influence of static magnetic field, shock strength, specific heat ratio, initial pressure, initial density and coefficient of viscosity on the flow variables. The findings confirm that thickness of MHD shock front increases with incre…
Macro-textured superhydrophobic surfaces can reduce droplet-substrate contact times of impacting water droplets, however, surface designs with similar performance for significantly more viscous liquids are missing, despite their importance in nature and technology such as for chemical shielding, food staining repellency, and supercooled (viscous) water droplet removal in anti-icing applications. Here, we introduce a deterministic, controllable and up-scalable method to fabricate superhydrophobic surfaces with a 3D-printed architecture, combining arrays of alternating surface protrusions and in…
It is well known that the viscosity of a homogeneous electron liquid diverges in the limits of zero frequency and zero temperature. A nanojunction breaks translational invariance and necessarily cuts off this divergence. However, the estimate of the ensuing viscosity is far from trivial. Here, we propose an approach based on a Kramers-Krönig dispersion relation, which connects the zero-frequency viscosity, $η(0)$, to the high-frequency shear modulus, $μ_{\infty}$, of the electron liquid via $η(0) =μ_{\infty} τ$, with $τ$ the junction-specific momentum relaxation time. By making use of a simple…
