Flow regimes during fluid-filled crack propagation in elastic media: Applications to volcanology and magma flow within dykes

Scaled analogue experiments were conducted to explore the effect of magma flow regimes, characterised by the Reynolds number (Re), on the transit of magma through the lithosphere via fractures. An elastic, transparent gelatine solid (the crust analogue) was injected by a fluid (magma analogue) to create a thin, vertical, and penny-shaped crack that is analogous to a magma-filled crack (dyke). A vertical laser sheet fluoresced passive-tracer particles suspended in the injected fluid, and particle image velocity (PIV) was used to map the location, magnitude, and direction of flow within the growing dyke from its inception to its surface rupture. Experiments were conducted using water, hydroxyethyl cellulose (HEC) or xanthan gum (XG) as the magma analogue. The results suggest that Re has significant impact on the direction of fluid flow within propagating dykes: Re > 0.1 (jet-flow) is characterised by a rapid central rising fluid jet and downflow at the dyke margin, whereas Re < 0.1 (creeping flow) is characterised by broadly uniform velocities across the dyke plane. Re may be underestimated by up to two orders of magnitude if tip velocity rather than internal fluid velocity is used. In nature, these different flow regimes would affect the petrological, geochemical, geophysical, and geodetic measurements of magma movement, key information upon which reconstructions of volcanic plumbing system architectures and their growth are based.

Mass Advection-Diffusion in Creeping Flow Through an Orifice Plate: A Model for Nanoporous Atomically Thin Membranes

Continuum transport equations are commonly applied to nanopores in atomically thin membranes for simple modeling. Although these equations do not apply for nanopores approaching the fluid or solute molecule size, they can be reasonably accurate for larger nanopores. Relatively large graphene nanopores have applications in small particle filtration and appear as unwanted defects in large-area membranes. Solute transport rates through these nanopores determine the rejection performance of the membrane. Atomically thin membranes commonly operate in a regime where advection and diffusion both contribute appreciably to transport. Solute mass transfer rates through larger nanopores have previously been modeled by adding continuum estimates for pure diffusion and pure advection through an infinitesimally thick orifice plate, as if the separate contributions were independent. We show here that estimating the transport rate in this way is accurate to within 30%. We further derive an expression for the net mass transfer rate in advection-diffusion through an infinitesimal thickness orifice plate at low Reynolds numbers that is accurate to within 1% for positive Peclet numbers (where diffusion is in the same direction as advection) and applies for negative Peclet numbers as well. Based on our expression, we devise an equation for the net mass transfer rate in creeping flow through orifice plates of arbitrary thickness that matches finite volume calculations to within 3% for positive Peclet numbers. These simple but accurate analytical equations for mass transfer rates in creeping flow through an orifice plate are useful tools in constructing approximate transport models.

Development Length of Fluids Modelled by the gPTT Constitutive Differential Equation

In this work, we present a numerical study on the development length (the length from the channel inlet required for the velocity to reach 99% of its fully-developed value) of a pressure-driven viscoelastic fluid flow (between parallel plates) modelled by the generalised Phan–Thien and Tanner (gPTT) constitutive equation. The governing equations are solved using the finite-difference method, and, a thorough analysis on the effect of the model parameters α and β is presented. The numerical results showed that in the creeping flow limit (Re=0), the development length for the velocity exhibits a non-monotonic behaviour. The development length increases with Wi. For low values of Wi, the highest value of the development length is obtained for α=β=0.5; for high values of Wi, the highest value of the development length is obtained for α=β=1.5. This work also considers the influence of the elasticity number.

Investigation of Drift Phenomena at the Pore Scale during Flow and Transport in Porous Media

This paper reports an analytical study conducted to investigate the behaviour of tracers undergoing creeping flow between two parallel plates in porous media. A new coupled model for the characterisation of fluid flow and transport of tracers at pore scale is formulated. Precisely, a weak-form solution of radial transport of tracers under convection–diffusion-dominated flow is established using hypergeometric functions. The velocity field associated with the radial transport is informed by the solution of the Stokes equations. Channel thickness as a function of velocities, maximum Reynolds number of each thickness as a function of maximum velocities and concentration profile for different drift and dispersion coefficients are computed and analysed. Analysis of the simulation results reveals that the dispersion coefficient appears to be a significant factor controlling the concentration distribution of the tracer at pore scale. Further analysis shows that the drift coefficient appears to influence tracer concentration distribution but only after a prolonged period. This indicates that even at pore scale, tracer drift characteristics can provide useful information about the flow and transport properties of individual pores in porous media.

Stokesian motion of a spherical particle near a right corner made by tangentially moving walls

The slow motion of a small buoyant sphere near a right dihedral corner made by tangentially sliding walls is investigated. Under creeping-flow conditions the force and torque on the sphere can be decomposed into eleven elementary types of motion involving simple particle translations, particle rotations and wall movements. Force and torque balances are employed to find the velocity and rotation of the particle as functions of its location. Depending on the ratio of the wall velocities and the gravitational settling velocity of the sphere, different dynamical regimes are identified. In particular, a non-trivial line attractor/repeller for the particle motion exists at a location detached from both the walls. The existence, location and stability of the corresponding two-dimensional fixed point are studied depending on the wall velocities and the buoyancy force. The impact of the line attractors/repellers on the motion of small particles in cavities and its relevance for corner cleaning applications are discussed.

2D Hydrodynamics of a Plate: From Creeping Flow to Transient Vortex Regimes

Based on the numerical and experimental visualization methods, the flow patterns around a uniformly moving plate located at an arbitrary angle of attack are studied. The study is based on the fundamental equations of continuity, momentum and stratifying substance transport for the cases of strong and weak stratified fluids, as well as potential and actually homogeneous ones. The visualization technique and computation codes were compiled bearing in mind conditions of internal waves, vortices, upstream, and downstream wakes registration, as well as the resolution of ligaments in the form of thin interfaces in schlieren flow images. The analysis was carried out in a unified mathematical formulation for a wide range of plate motion parameters, including slow diffusion-induced flows and fast transient vortex flows. The patterns of formation and subsequent evolution of the basic structural components, such as upstream disturbances, downstream wake, internal waves, vortices, and ligaments, are described both at start of motion and subsequent uniform movement of the plate. Calculations of forces acting on the obstacle in the flow were carried out to study effects of variations in fluid properties, flow conditions and plate parameters on the dynamic characteristics of the obstacle. The numerical and experimental results on the flow patterns around a plate are in a good agreement with each other for different flow regimes.

Leveraging Elasticity to Uncover the Role of Rabinowitsch Suspension through a Wavelike Conduit: Consolidated Blood Suspension Application

The present work presents a mathematical investigation of a Rabinowitsch suspension fluid through elastic walls with heat transfer under the effect of electroosmotic forces (EOFs). The governing equations contain empirical stress-strain equations of the Rabinowitsch fluid model and equations of fluid motion along with heat transfer. It is of interest in this work to study the effects of EOFs, which are rigid spherical particles that are suspended in the Rabinowitsch fluid, the Grashof parameter, heat source, and elasticity on the shear stress of the Rabinowitsch fluid model and flow quantities. The solutions are achieved by taking long wavelength approximation with the creeping flow system. A comparison is set between the effect of pseudoplasticity and dilatation on the behaviour of shear stress, axial velocity, and pressure rise. Physical behaviours have been graphically discussed. It was found that the Rabinowitsch and electroosmotic parameters enhance the shear stress while they reduce the pressure gradient. A biomedical application to the problem is presented. The present analysis is particularly important in biomedicine and physiology.

Velocity Dependence and Tracer Dispersion in Newtonian Fluids Undergoing Creeping Flow

The dynamics of tracer particles in a viscous Newtonian fluid is studied analytically and numerically through channels of varying thickness for fluids undergoing creeping flow. Exact analytical solutions of mass conservation equations of tracer particles including consideration for pressure forces are obtained. Results of the analysis indicates that Stokes velocity is an indispensable parameter and is dependent on parameters such as channel thickness (height), viscosity of the fluid, pressure gradient driven the fluid and Reynolds number corresponding to the channel thickness. The accuracy of the solution obtained is verified by comparing its velocity profiles with those obtained from finite-element-based numerical simulation studies.