Dimensionless Shear Rate Research Articles

Stability of plane Couette flow of Carreau fluids past a deformable solid at arbitrary Reynolds numbers

The linear stability of the plane Couette flow of both power-law and Carreau fluids past a deformable, neo-Hookean solid is analyzed at arbitrary Reynolds numbers. An algebraic error in the mathematical formulation of the earlier studies (for the power-law fluid) is corrected and is shown to result in quantitative differences in the predictions for the stability of the flow. Due to the lack of a proper (zero-shear) viscosity scale and a time scale for the onset of shear thinning in the power-law model, we show that the stability analysis of the flow yields vastly different scalings for the unstable mode depending on the way the problem is scaled to render it dimensionless. When the deformable solid properties are used to non-dimensionalize, we show that for the unstable modes (the so-called “wall modes” at high Re) Γc∝Re−1(2n+1), while when flow properties are used to non-dimensionalize, Γc∝Re−13 much akin to a Newtonian fluid, where Γ=Vm*η*/G*R* is the dimensionless shear rate in the flow, and Γc denotes the minimum value required for instability. Here, Vm* is the velocity of the top plate, G* is the shear modulus of the solid, R* is the fluid thickness, and η* is the (arbitrary) viscosity scale in the power-law model. Within the framework of the power-law model, it is not possible to discriminate between the two predicted scalings. To resolve this in an unambiguous manner, we used the Carreau model to account for shear thinning and to study its role on the stability of flow past deformable solid surfaces. The Carreau model has a well-defined zero-shear viscosity η0* as well as a time scale λ* that characterizes the onset of shear thinning. For fixed λ*η0*/(ρ*R*2), we show that the unstable wall modes scale as Γc∼Re(1−2n)3 at high Re, thus providing a resolution to the ambiguity in the results obtained using the power-law model. The present work thus shows that, at moderate to high Re, shear thinning has a strongly stabilizing effect on the wall mode instability in flow past deformable solid surfaces.

Impact of a single bubble rising near a wall on the wall-to-liquid heat flux

This paper presents a numerical study of the heat transfer improvement around a bubble rising near a wall in a shear flow. The Navier Stokes equations and energy equation are solved in a frame of reference following the bubble to examine the dependence of the wall to liquid heat transfer on flow patterns around the bubble. The front tracking method is used to track the interface of non-condensable gas bubbles, and the finite volume method is used to solve for temperature and momentum. Our simulations show an enhancement and a diminishment of heat transfer in the downstream and upstream regions of the bubble, respectively. The enhancement and reduction of heat transfer are attributed to the reversal of flow that occurs near the wall due to the presence of a bubble. The heat transfer rate non-monotonically varies with the dimensionless shear rate γ̇‾=γ̇dg. In the range of 0.3⩽γ̇‾≤2.25, the heat transfer rate first increases, attains a maximum value and then decreases. The optimum γ̇‾ increases as the Archimedes number is increased. The heat transfer rate is found to decrease as the Laplace number is increased. The flow reversal near the wall is characterized by two parameters, namely the ‘reversal height’ and ‘reversal width’. A control volume analysis is performed for a simplified 2D version of the problem in order to relate heat transfer enhancement to the reversal height and width. The functional dependence of heat transfer enhancement on reversal height and width from our 3D simulations match the result of the control volume analysis. The analytical solution of an inviscid shear flow over a cylinder is used for qualitative insights into flow reversal mechanism. Our simulations of transient heat transfer showed that the fractional improvement in heat transfer compared to a single phase flow attains a constant value after a certain amount of time. Finally, we report the results of heat transfer enhancement due of an array of bubbles rising near a wall.

Effects of surfactant on lift coefficients of bubbles in linear shear flows

Effects of surfactant on the lateral motion of bubbles in linear shear flows are numerically investigated by using an interface tracking method. Ranges of the bubble Reynolds number Re, Eötvös number Eo, Morton number M and dimensionless shear rate Sr are 2 < Re < 70, 0.6 < Eo < 5, 10−6 < M < 10−4 and 0 ≤ Sr < 1, respectively. The main conclusions obtained are as follows: (1) the lift coefficient, CL, of a bubble decreases as the Langmuir number increases, (2) the surfactant decreases the critical values of Eo and Re, at which CL changes its sign, (3) the reduction in CL of bubbles of large Hatta numbers due to the presence of surfactant can be understood from the surface tension reduction, and (4) the Marangoni force affects CL of small Hatta number bubbles, and therefore, the surface immobilization effect of the Marangoni force should be taken into account to model CL of contaminated bubbles.

Lift force acting on single bubbles in linear shear flows

Lift coefficients, CL, of single bubbles in linear shear flows are measured to investigate effects of the bubble shape, the liquid velocity gradient and the fluid property on CL. The range of the Morton number, M, tested is from logM = − 6.6 to − 3.2. The shapes of bubbles are spherical and ellipsoidal. A correlation of bubble aspect ratio for single bubbles in infinite stagnant liquids proposed in our previous study can give good evaluations for bubbles in the linear shear flows. The CL of spherical bubbles at low bubble Reynolds numbers, Re, depend on the dimensionless shear rate Sr and Re and decrease with increasing Re. These characteristics agree with the Legendre-Magnaudet correlation. The use of a single dimensionless group such as Re, the Eötvös number, the Weber number and the Capillary number cannot correlate CL of non-spherical bubbles. The trend of the critical Re for the reversal of the sign of CL is the same as that for the onset of oscillation of bubble motion, which supports the mechanism proposed by Adoua et al., at least within the range of −6.6 ≤ logM ≤ −3.2. An experimental database of CL is provided for validation of available CL models and CFD.

An orientational order transition in a sheared suspension of anisotropic particles

Under Stokesian conditions, a neutrally buoyant non-Brownian spheroid in simple shear flow rotates indefinitely in any of a one-parameter family of closed (Jeffery) orbits characterized by an orbit constant $C$. The limiting values, $C=0$ and $C=\infty$, correspond to spinning and tumbling modes respectively. Hydrodynamics alone does not determine the distribution of spheroid orientations across Jeffery orbits in the absence of interactions, and the rheology of a dilute suspension of spheroids remains indeterminate. A combination of inertia and stochastic orientation fluctuations eliminates the indeterminacy. The steady-state Jeffery-orbit distribution arising from a balance of inertia and thermal fluctuations is shown to be of the Boltzmann equilibrium form, with a potential that depends on $C$, the particle aspect ratio ($\unicode[STIX]{x1D705}$), and a dimensionless shear rate ($Re\,Pe_{r}$, $Re$ and $Pe_{r}$ being the Reynolds and rotary Péclet numbers), and therefore lends itself to a novel thermodynamic interpretation in $C{-}\unicode[STIX]{x1D705}{-}Re\,Pe_{r}$ space. In particular, the transition of the potential from a single to a double-well structure, below a critical $\unicode[STIX]{x1D705}$, has similarities to a thermodynamic phase transition, and the small-$C$ and large-$C$ minima are therefore identified with spinning and tumbling phases. The hysteretic dynamics within the two-phase tumbling–spinning envelope renders the rheology sensitively dependent on the precise shear rate history, the signature in simple shear flow being a multivalued viscosity at a given shear rate. The tumbling–spinning transition identified here is analogous to the coil–stretch transition in the polymer physics literature. It should persist under more general circumstances, and has implications for the suspension stress response in inhomogeneous shearing flows.

A study of the rheology of planar granular flow of dumbbells using discrete element method simulations

Granular materials handled in industries are typically non-spherical in shape and understanding the flow of such materials is important. The steady flow of mono-disperse, frictional, inelastic dumbbells in two-dimensions is studied by soft sphere, discrete element method simulations for chute flow and shear cell flow. The chute flow data are in the dense flow regime, while the shear cell data span a wide range of solid fractions. Results of a detailed parametric study for both systems are presented. In chute flow, increase in the aspect ratio of the dumbbells results in significant slowing of the flow at a fixed inclination and in the shear cell it results in increase in the shear stress and pressure for a fixed shear rate. The flow is well-described by the μ-I scaling for inertial numbers as high as I = 1, corresponding to solid fractions as low as ϕ = 0.3, where μ is the effective friction (the ratio of shear stress to pressure) and I is the inertial number (a dimensionless shear rate scaled with the time scale obtained from the local pressure). For a fixed inertial number, the effective friction increases by 60%-70% when aspect ratio is increased from 1.0 (sphere) to 1.9. At low values of the inertial number, there is little change in the solid fraction with aspect ratio of the dumbbells, whereas at high values of the inertial number, there is a significant increase in solid fraction with increase in aspect ratio. The dense flow data are well-described by the Jop-Forterre-Pouliquen model [P. Jop et al., Nature 441, 727–730 (2006)] with the model parameters dependent on the dumbbell aspect ratio. The variation of μ with I over the extended range shows a maximum in the range I ∈ (0.4, 0.5), while the solid fraction shows a faster than linear decrease with inertial number. A modified version of the JFP model for μ(I) and a power law model for ϕ(I) is shown to describe the combined data over the extended range of I.

Solution of the complete Curtiss-Bird model for polymeric liquids subjected to simple shear flow.

The complete kinetic theory model for concentrated polymer solutions and melts proposed by Curtiss and Bird is solved for shear flow: (a) analytically by providing a solution for the single-link (or configurational) distribution function as a real basis spherical harmonics expansion and then calculating the materials functions in shear flow up to second order in the dimensionless shear rate and, (b) numerically via the execution of Brownian dynamics simulations. These two methods are actually complementary to each other as the former is accurate only for small dimensionless shear rates where the latter produces results with increasingly large uncertainties. The analytical expansions of the material functions with respect to the dimensionless shear rate reduce to those of the extensively studied, simplified Curtiss-Bird model when ε' = 0, and to the rigid rod when ε' = 1. It is known that the power-law behavior at high shear rates is very different for these two extremal cases. We employ Brownian dynamics simulation to not only recover the limiting cases but to find a gradual variation of the power-law behaviors at large dimensionless shear rates upon varying ε'. The fact that experimental data are usually located between these two extremes strongly advocates the significance of studying the solution of the Curtiss-Bird model. This is exemplified in this work by comparing the solution of this model with available rheological data for semiflexible biological systems that are clearly not captured by the original Doi-Edwards or simplified Curtiss-Bird models.

On a suspension of nearly spherical colloidal particles under large-amplitude oscillatory shear flow

The dynamics of a dilute, monodisperse suspension of nearly spherical particles that undergo Brownian rotations in an oscillatory simple shear flow is quantified, as a paradigm for large-amplitude oscillatory shear (LAOS) rheology of complex fluids. We focus on the ‘strongly nonlinear’ regime of LAOS, defined by ${\it\beta}\gg 1$ and ${\it\beta}/{\it\alpha}\gg 1$, where ${\it\beta}$ is a dimensionless shear rate (or Weissenberg number) and ${\it\alpha}$ is a dimensionless oscillation frequency (or Deborah number). We derive an asymptotic solution for the long-time periodic orientation probability density function of the particles. Our analysis reveals that the orientation dynamics consists of ‘core’ regions of rapid oscillation (on the time scale of the inverse of the shear-rate amplitude), separated by comparatively short ‘turning’ regions of slow evolution when the imposed flow vanishes. Uniformly valid approximations to the shear stress and normal stress differences (NSDs) of the suspension are then constructed: the non-Newtonian contribution to the shear stress, first NSD and second NSD, decays as ${\it\beta}^{-3/2}$, ${\it\beta}^{-1}$ and ${\it\beta}^{-1/2}$, respectively, at large ${\it\beta}$. These stress scalings originate from the orientation dynamics at the turning regions. Therefore, it is the occasions when the flow vanishes that dominate the rheology of this paradigmatic complex fluid under LAOS.

Merging fluid and solid granular behavior.

Simple homogeneous shear flows of frictionless, deformable particles are studied by particle simulations at large shear rates and for differently soft, deformable particles. Particle stiffness sets a time-scale that can be used to scale the physical quantities; thus the dimensionless shear rate, i.e. the inertial number I (inversely proportional to pressure), can alternatively be expressed as inversely proportional to the square root of particle stiffness. Asymptotic scaling relations for the field variables pressure, shear stress and granular temperature are inferred from simulations in both fluid and solid regimes, corresponding to unjammed and jammed conditions. Then the limit cases are merged to unique constitutive relations that cover also the transition zone in the proximity of jamming. By exploiting the diverging behavior of the scaling laws at the jamming density, we arrive at continuous and differentiable phenomenological constitutive relations for stresses and granular temperature as functions of the volume fraction, shear rate, particle stiffness and distance from jamming. In contrast to steady shear flows of hard particles the (shear) stress ratio μ does not collapse as a function of the inertial number, indicating the need for an additional control parameter. In the range of particle stiffnesses investigated, in the solid regime, only pressure is rate independent, whereas shear stress exhibits a slight shear rate- and stiffness-dependency.

Deformation of a Capsule in a Power-Law Shear Flow.

An immersed boundary-lattice Boltzmann method is developed for fluid-structure interactions involving non-Newtonian fluids (e.g., power-law fluid). In this method, the flexible structure (e.g., capsule) dynamics and the fluid dynamics are coupled by using the immersed boundary method. The incompressible viscous power-law fluid motion is obtained by solving the lattice Boltzmann equation. The non-Newtonian rheology is achieved by using a shear rate-dependant relaxation time in the lattice Boltzmann method. The non-Newtonian flow solver is then validated by considering a power-law flow in a straight channel which is one of the benchmark problems to validate an in-house solver. The numerical results present a good agreement with the analytical solutions for various values of power-law index. Finally, we apply this method to study the deformation of a capsule in a power-law shear flow by varying the Reynolds number from 0.025 to 0.1, dimensionless shear rate from 0.004 to 0.1, and power-law index from 0.2 to 1.8. It is found that the deformation of the capsule increases with the power-law index for different Reynolds numbers and nondimensional shear rates. In addition, the Reynolds number does not have significant effect on the capsule deformation in the flow regime considered. Moreover, the power-law index effect is stronger for larger dimensionless shear rate compared to smaller values.

Correlations for shear-induced percolation segregation in granular shear flows

Discrete element method (DEM) computer simulations are used to develop correlations for the mean percolation segregation speed and diffusion coefficient in a steady shear flow of a bi-disperse mixture of cohesionless, spherical particles. The simulations span a range of size ratios, dimensionless shear rates, and dimensionless applied normal stresses. The volume concentrations of the smaller species are small, with values between 0.04% and 9.6%. For the investigated set of parameters, the dimensionless mean percolation speed has an exponential dependence on the particle size ratio and is a linear function of the large particle concentration, but is independent of the dimensionless normal stress, at least for dimensionless normal stresses exceeding a critical value. Furthermore, the dimensional mean percolation speed is also independent of the dimensional shear rate. The diffusion coefficient, however, does depend on the shear rate, but asymptotes to the self-diffusion coefficient as the particle size ratio approaches one. The proposed correlations fit the simulation data very well, with coefficients of determination exceeding 0.99 and 0.98 for the percolation speed and diffusion coefficient, respectively. These correlations are intended for use with higher-level models, such as continuum models for granular flow, in order to predict segregation in large systems.

Characteristics of drag and lift forces on a hemisphere under linear shear

In this study, the flow characteristics over a hemisphere under a linear shear are numerically investigated. The Reynolds number (Re) based on the velocity ( uc ) at the center of the inlet flow and the base diameter of the hemisphere ( d ) is in the range of 100 ≤ Re ≤ 300, and dimensionless inlet shear rate defined as s =|∇u|d / uc is up to 0.1 where |∇u| the shear rate at inlet. In terms of the Reynolds number and the inlet shear rate, the vortical structures behind the hemisphere show various flow regimes such as steady axisymmetric, steady planar-symmetric, unsteady planar-symmetric and unsteady asymmetric. Associated with the change in the vortical structures, the drag force slightly increases with the inlet shear rate. However, the drag fluctuations show more complicated behavior in terms of the inlet shear rate, showing that they are more influenced by the change in vortical structures than the drag force itself. On the other hand, the magnitude and direction of the lift force change drastically depending on both the Reynolds number and the inlet shear rate. Inside the Reynolds number range (100 ≤ Re ≤ 300), the mean lift and lift fluctuations show maximum value at around Re = 190 and 210, respectively, which is different from the fact that for the sphere, they increase with increasing the Reynolds number in the range of 100 ≤ Re ≤ 300. This might be attributed to the fact that for the hemisphere, the change in the flow regime from steady to unsteady or from planarsymmetric to asymmetric occurs at lower Reynolds numbers than for the sphere. Also, the orientation characteristics of the lift force were investigated with the phase diagram (Cy, Cz), together with the vortical structures existing behind the hemisphere. In the absence of the shear, the lift force is generated along any arbitrary direction because the location where vortex shedding happens is not determined a priori. However, when the inlet shear is applied to the hemisphere, the location of the vortex-loop detachment is restricted to the high-velocity side. Thus the lift force acts from the high-velocity side to the low-velocity side. However at very small Reynolds numbers such as Re = 1 the direction of the lift force becomes opposite. For the hemisphere considered in this study, the Reynolds number corresponding to the lift reversal is much smaller than that for the sphere studied by Kurose and Komori (1999).

Laminar MHD flow and heat transfer of power-law fluids in square microchannels

The steady fully developed laminar flow and heat transfer of electrically conducting non-Newtonian fluids through square microchannels at the presence of transverse magnetic field are investigated with emphasis on viscous dissipation and joule heating effects. The governing continuity, momentum and energy equations are being discretized using a finite difference approach using Modified Power-Law (MPL) model and have been solved iteratively, through Alternating Direction Implicit (ADI) scheme with successive over-relaxation under the classical boundary conditions of no-slip velocity and uniform constant heat flux for flow and thermal fields.The velocity, temperature profiles, product of friction factor–Reynolds number and Nusselt number are computed for various values of flow index and dimensionless shear rate parameter of modified power-law fluids. Also, the effect of Hartmann number and Brinkman number on velocity, temperature distributions and Nusselt number are discussed. It is shown that the results are in good agreement with the previous studies.

Can we reduce debris flow to an equivalent one-phase flow?

A recent extension of the discrete element method is reported for the simulation of dense mixtures of non-colloidal particles and viscous fluids in the non-inertial regime. As an application, we examine the interplay between rate dependent dilatancy and hydro-mechanical coupling which can be expected in debris flow. The numerical model includes sphere-sphere contacts using a soft contact approach [2], short range hydrodynamic interactions defined by frame-invariant expressions of forces and torques in the lubrication approximation, and drag forces resulting from the poromechanical coupling computed with the DEM-PFV technique [3]. The bulk shear stress is decomposed into contact stress and hydrodynamic stress. Both contributions are shown to be increasing functions of a dimensionless shear rate Iv, in agreement with experimental results [4]. Statistics of microstructural variables highlight a complex interplay between solid contacts and hydrodynamic interactions. In contrast with a popular idea, the results suggest that lubrication may not necessarily reduce the contribution of contact forces to the bulk shear stress. The proposed model is general and applies directly to sheared satured granular media in which pore pressure feedback plays a key role. We argue that it can be the case for debris fow, especially during the triggering phase, when run-out include transitional phases, and when the flow is stopped. It is then concluded that debris cannot be computed by assuming solely the rheological properties of an equivalent mixture.

A microfluidic model to study fluid dynamics of mucus plug rupture in small lung airways.

Fluid dynamics of mucus plug rupture is important to understand mucus clearance in lung airways and potential effects of mucus plug rupture on epithelial cells at lung airway walls. We established a microfluidic model to study mucus plug rupture in a collapsed airway of the 12th generation. Mucus plugs were simulated using Carbopol 940 (C940) gels at concentrations of 0.15%, 0.2%, 0.25%, and 0.3%, which have non-Newtonian properties close to healthy and diseased lung mucus. The airway was modeled with a polydimethylsiloxane microfluidic channel. Plug motion was driven by pressurized air. Global strain rates and shear stress were defined to quantitatively describe plug deformation and rupture. Results show that a plug needs to overcome yield stress before deformation and rupture. The plug takes relatively long time to yield at the high Bingham number. Plug length shortening is the more significant deformation than shearing at gel concentration higher than 0.15%. Although strain rates increase dramatically at rupture, the transient shear stress drops due to the shear-thinning effect of the C940 gels. Dimensionless time-averaged shear stress, T xy , linearly increases from 3.7 to 5.6 times the Bingham number as the Bingham number varies from 0.018 to 0.1. The dimensionless time-averaged shear rate simply equals to T xy /2. In dimension, shear stress magnitude is about one order lower than the pressure drop, and one order higher than yield stress. Mucus with high yield stress leads to high shear stress, and therefore would be more likely to cause epithelial cell damage. Crackling sounds produced with plug rupture might be more detectable for gels with higher concentration.

The motion induced by the orthogonal stretching and shearing of a membrane beneath a quiescent fluid

The flow induced above an impermeable membrane undergoing orthogonal linear stretching and orthogonal linear shearing is investigated. For an exact solution of the Navier–Stokes equations, the orthogonal shearing motions must be related through the constant σ = γ δ, where γ and δ are the dimensionless streamwise and transverse shear rates, respectively. The resulting similarity reduction leads to three nonlinearly coupled ordinary differential equations governed by σ and the ratio of membrane stretch rates β. All possible solutions of these equations are found either numerically or, in special cases, analytically. Features of the σ = 0 solutions at β = 0 and asymptotically as β → ∞ are found to be in excellent agreement with numerical calculations. An aside calculation shows that orthogonal shearing in the absence of any plate stretching cannot exist. However, shearing in one coordinate direction is possible as long as the membrane stretches in at least one direction with the caveat that there exists uniform suction through a porous membrane.

Microscopic origins of shear stress in dense fluid–grain mixtures

A numerical model is used to simulate rheometer experiments at constant normal stress on dense suspensions of spheres. The complete model includes sphere–sphere contacts using a soft contact approach, short range hydrodynamic interactions defined by frame-invariant expressions of forces and torques in the lubrication approximation, and drag forces resulting from the poromechanical coupling computed with the DEM-PFV technique. Series of simulations in which some of the coupling terms are neglected highlight the role of the poromechanical coupling in the transient regimes. They also reveal that the shear component of the lubrication forces, though frequently neglected in the literature, has a dominant effect in the volume changes. On the other hand, the effects of lubrication torques are much less significant. The bulk shear stress is decomposed into contact stress and hydrodynamic stress terms whose dependency on a dimensionless shear rate—the so called viscous number \(I_v\)—are examined. Both contributions are increasing functions of \(I_v\), contacts contribution dominates at low viscous number (\(I_v<0.15\)) whereas lubrication contributions are dominant for \(I_v> 0.15\), consistently with a phenomenological law infered by other authors. Statistics of microstructural variables highlight a complex interplay between solid contacts and hydrodynamic interactions. In contrast with a popular idea, the results suggest that lubrication may not necessarily reduce the contribution of contact forces to the bulk shear stress. The proposed model is general and applies directly to sheared immersed granular media in which pore pressure feedback plays a key role (triggering of avalanches, liquefaction).

Single particle motion in a sheared colloidal dispersion

The motion of a spherical Brownian “probe” particle addressed by an external force immersed in shear flow of a colloidal dispersion of spherical neutrally buoyant “bath” particles is quantified. The steady-state nonequilibrium microstructure of bath particles around the probe—induced by the applied force and ambient shear—is calculated to first order in the volume fraction of bath particles, ϕ. The distortion to the equilibrium microstructure caused by the moving probe is characterized by a Péclet number Pef (a dimensionless pulling force), and the distortion due to the shear flow is represented by another Péclet number Pes (a dimensionless shear rate). Matched asymptotic expansions are employed to quantify the microstructure at small Péclet numbers; specifically, within the distinguished limits Pes3/2≪Pef≪Pes1/2≪1. The nonequilibrium microstructure is subsequently utilized to compute the average rectilinear velocity of the probe through O(ϕPes3/2Us), for an arbitrary orientation of the external force to the shear flow. Here, Us is the Stokes velocity of the probe in a pure Newtonian fluid. It is also shown that to O(ϕPes3/2Us) the torque-free probe simply rotates with the ambient shear; a modification to the angular velocity of the probe is at most O(ϕPesPefUs). In particular, a probe forced along the flow axis of shear is demonstrated to experience a cross-streamline drift velocity of O(ϕPesUs), to leading order, which acts to propel the particle to streamlines of the ambient shear that move in the same direction as the external force. A mathematical connection between this result and cross-streamline drift of a particle in a Newtonian fluid at small, but nonzero, Reynolds numbers is drawn. The magnitude of the cross-streamline drift velocity is found to be sensitive to the degree of hydrodynamic interactions between the probe and bath particles, which are tuned via an excluded-annulus model. It is also demonstrated that a probe forced along the vorticity axis of the shear experiences a shear-driven enhancement in rectilinear velocity of O(ϕPes3/2Us), to leading order: This nonanalytic dependence originates from the microstructural deformation in the shear dominated (outer) region far from the probe. A connection of this finding to recent work on particle sedimentation in orthogonal shear flow of viscoelastic liquids is discussed.

Finite element solution of oblique stagnation-point flow of viscoelastic fluid and heat transfer with variable thermal conductivity

The oblique stagnation-point boundary layer flow of an incompressible viscoelastic fluid impinging on an infinite vertical plate and heat transfer characteristics with variable thermal conductivity is studied. The governing partial differential equations are transformed into non-dimensional, nonlinear coupled ordinary differential equations which are solved numerically by robust Galerkin finite element method. The streamline patterns and numerical results for the dimensionless shear rate and temperature profiles are displayed graphically for various physical parameters such as Prandtl number, Weissenberg number, thermal conductivity parameter and two parameters α and β, the later being a free parameter. The local Nusselt number is found to be the decreasing function of Weissenberg number whereas it increases with increasing values of Prandtl number. The present problem finds significant applications in cooling of nuclear reactors and electronic devices by fans, where the angle plays an important role in cooling.

Simulation of cohesive fine powders under a plane shear.

Three-dimensional molecular-dynamics simulations of cohesive dissipative powders under a plane shear are performed. We find the various phases depending on the dimensionless shear rate and the dissipation rate as well as the density. We also find that the shape of clusters depends on the initial condition of velocities of particles when the dissipation is large. Our simple stochastic model reproduces the non-Gaussian velocity distribution function appearing in the coexistence phase of a gas and a plate.