Session Q18: General Fluid Dynamics: Viscous Flows

Spreading of a viscous fluid between an elastic sheet and a porous substrate

The dynamics of a thin film of viscous fluid spreading between an elastic sheet and a porous substrate are controlled by the deformation of the elastic sheet, the hydrostatic pressure gradient in the flow, and the viscous resistance of the flow in the gap and in the pores. We present experiments where fluid is injected between a rigid porous substrate and an overlying elastic sheet. First, fluid invades the pores, and after reaching a critical pressure, the elastic sheet is peeled and uplifted from the porous substrate forming a thin fluid-filled gap. Further injection causes the fluid front in the pores and the peeling front of the elastic sheet to advance radially. We study experimentally and theoretically the spreading dynamics in various regimes which are a function of the permeability of the porous substrate, bending rigidity of the elastic sheet, and injection rate.   

The evolution of a viscous gravity current in a confined geometry

We present a study of a viscous gravity current confined to the space between two horizontal parallel plates. Fluid is released at a constant flux from a source located at the center of the lower plate. Two regions of flow develop: an inner region where the fluid is in contact with both plates and an outer annular region where the fluid has not yet reached the upper plate. We develop a simple theoretical model within the context of lubrication theory with the aim of determining the height profile of the free surface, and the evolution equations for the leading edge of the outer annular region as well as the interface position between the two regions. We then perform a series of experiments for various fluxes and gap spacings and compare the results with the theoretical model. A mismatch between the experimental data and the theoretical model is observed and then reduced by incorporating the effect of surface tension at the interface between the two regions into the theoretical model.   

Viscocapillary lift force near a fluid interface

The force exerted on an object moving in a viscous fluid, and its modification induced by the presence of a nearby boundary, have been abundantly studied in fluid mechanics for the past century. In the last decades, soft gels and elastomers have been of growing interest, and have been shown to break the symmetry of nearby low-Reynolds-number flows, thus generating a force normal to the substrate. Such a lift force depends in particular on the material properties of the substrate and the intervening fluid. Here, we revisit this problem, by addressing the novel situation where the soft substrate is replaced by a fluid interface. We analyze in details the lubrication flow between a moving object and the interface. Using a combination of analytical and numerical treatments, we derive an expression of the viscocapillary lift force at leading order in compliance.

    

A spectral method for axisymmetric deformed sphere in Stokes flow

We present a general algebraic spectral solution to axisymmetric Stokes flow within a non-perturbative framework for a radially highly deformed sphere. Spectra of harmonic and biharmonic Stokes flow modes are used. The method is demonstrated on two problems: hydrodynamic radii of radially deformed spheres for a range of strength and angular dependence of the deformation, with results in good agreement with the boundary element method; and self-phoretic velocity of spheroids with surface flux of the driving field in a source/sink or source/inert configuration.   

Experimental investigation of the hydrodynamics of a slipping cylinder in a viscous shallow water flow.

In the context of atmospheric re-entry flows, viscous effects cause a thick boundary layer around the object, which plays an important role in the aerodynamics of the object. The flows encountered can be at low Reynolds number and supersonic conditions. Using the analogy between rarefied supersonic flows and viscous shallow water flows, it is possible to study the aerodynamic behaviour. This work focuses on the hydrodynamics of a cylinder in a viscous shallow water flow. Slip at the cylinder wall is synthetically added with a super-hydrophobic treatment to add slip and achieve analogy with the slip regime of rarefied flows. Different cylinder diameters and configurations of the shallow water hydraulic test bench were used to vary the Reynolds number between 10 and 500 for a Froude number of 2. A random dot pattern in the background of the flow is imaged with a high-resolution camera, which allows the reconstruction of the water free surface height by backward oriented schlieren technics. The drag coefficient is thus obtained using a momentum budget on the reconstructed heights and velocity field.   

Author not Attending

In this talk, we investigate the optimal shapes of the hydrodynamic resistance of a rigid body set in motion in a Stokes flow. In this low Reynolds number regime, the hydrodynamic drag properties of an object are encoded in a finite number of parameters contained in the grand resistance tensor. Considering these parameters as objective functions to be optimised, we use calculus of variations techniques to derive a general shape derivative formula, allowing to specify how to deform the body shape to improve the objective value of any given resistance tensor entry.

We then describe a practical algorithm for numerically computing the optimized shapes and apply it to several examples. Numerical results reveal interesting new geometries when optimizing the extra-diagonal inputs to the strength tensor, including the emergence of a chiral helical shape when maximising the coupling between the hydrodynamic force and the rotational motion. With a good level of adaptability to different applications, this work paves the way for a new analysis of the morphological functionality of microorganisms and for future advances in the design of microswimmer devices.   

Exact boundary integral solution for the Stokes traction on an active particle

Active particles produce fluid flow around them even when stationary and often this can lead to self-propulsion. Examples include microorganisms and autophoretic colloids. Activity, consisting of non-equilibrium processes at the surface of the particle (e.g., ciliary motion in microorganisms), can be modelled by a velocity boundary condition. The boundary integral formulation of Stokes flow has been used extensively in the dynamics of passive colloidal particles and, more recently, for active particles. It provides the traction on an active particle directly, obviating the need to solve for the fluid flow in the bulk. Using spectral expansions of the traction and the velocity boundary condition in a basis of tensorial spherical harmonics and Ritz-Galerkin discretisation the direct boundary integral equation can be reduced to an infinite-dimensional linear system. We have diagonalised this linear system exactly and obtained the solution for the traction in terms of the velocity boundary condition. We call these relations the generalised Stokes laws­_­. Apart from intrinsic theoretical interest, such a solution is of use in numerical solutions of the boundary integral equation for many particles, where numerical iterations can be initialized with the exact one-particle solution.   

The effect of temperature-dependent viscosity on the pressure drop in narrow channel flows

We investigate theoretically the effect of temperature-dependent viscosity on the pressure drop-flow rate relation in narrow pipe flows. Although seemingly a classical topic, we are not aware of previous results of this type. Different temperature boundary conditions at the wall alter the viscosity field under the same flow conditions, and we elucidate how this external heating affects the pressure drop across the pipe. We first use analytical and similarity solution methods to calculate the temperature distribution under constant temperature and constant heat flux boundary conditions, as well as assumed linear and other imposed polynomial temperature versus distance (along the flow) boundary conditions at the wall. We then employ the lubrication approximation and the Lorenz reciprocal theorem to derive an expression for the pressure drop across the channel for a temperature-dependent viscosity. Assuming a small fractional change in viscosity with temperature, we linearize the viscosity field and obtain an analytical expression for the pressure drop for a given flow rate. The results are reported as a function of the effective Peclet number for each boundary condition and the numerical results are compared with analytical predictions in the low and high Peclet number limits.

 

    

Banana Puree Wave Cycle in Twin-Fluid Atomizer

We present a numerical study of annular wave cycling inside a novel twin-fluid injector. Such waves provide mechanisms to enhance the atomization of viscous, non-Newtonian fluids. The working fluid is banana puree, which is both shear- and temperature-thinning and flows in an outer annulus surrounding a high-speed steam flow. The steam-puree mass ratio is 2.7%. As the puree is injected into the central steam flow, periodic waves cycle at a frequency of 1000 Hz. No wave train exits; rather, a new wave is birthed in the wake of a dying wave. Waves on average have a wave angle of 50° and a wavelength of 0.7 nozzle diameters. The Kelvin-Helmholtz instability (KHI) deforms the puree surface and dominates the wave formation process. Waves are characterized by a high blockage ratio, causing significant windward pressure buildup and transonic steam flow above the crest. Accelerated steam provides a lifting force, but pressure eventually overcomes inertia and surface tension at the nozzle exit, resulting in wave collapse and rupture. As the steam compresses and decompresses during wave formation and collapse, pressure, temperature, and velocity cycle with wave frequency. The result is bulk system pulsation, which provides feedback that influences wave cycling.