simple shear flow
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2022 ◽  
Vol 934 ◽  
Author(s):  
E. Guilbert ◽  
B. Metzger ◽  
E. Villermaux

The interplay between chemical reaction and substrate deformation is discussed by adapting Ranz's formulation for scalar mixing to the case of a reactive mixture between segregated reactants, initially separated by an interface whose thickness may not be vanishingly small. Experiments in a simple shear flow demonstrate the existence of three regimes depending on the Damköhler number $Da=t_s/t_c$ where $t_s$ is the mixing time of the interface width and $t_c$ is the chemical time. Instead of treating explicitly the chemical cross-term, we rationalize these different regimes by globalizing it as a production term involving a flux which depends on the rate at which the reaction zone is fed by the reactants, a formulation valid for $Da>1$ . For $Da<1$ , the reactants interpenetrate before they react, giving rise to a ‘diffusio-chemical’ regime where chemical production occurs within a substrate whose width is controlled by molecular diffusion.


Soft Matter ◽  
2022 ◽  
Author(s):  
Kevin S. Silmore ◽  
Michael Strano ◽  
James W. Swan

We perform Brownian dynamics simulations of semiflexible colloidal sheets with hydrodynamic interactions and thermal fluctuations in shear flow. As a function of the ratio of bending rigidity to shear energy...


2021 ◽  
Vol 932 ◽  
Author(s):  
Kengo Deguchi

Nonlinear Hall-magnetohydrodynamic dynamos associated with coherent structures in subcritical shear flows are investigated by using unstable invariant solutions. The dynamo solution found has a relatively simple structure, but it captures the features of the typical nonlinear structures seen in simulations, such as current sheets. As is well known, the Hall effect destroys the symmetry of the magnetohydrodynamic equations and thus modifies the structure of the current sheet and mean field of the solution. Depending on the strength of the Hall effect, the generation of the magnetic field changes in a complex manner. However, a too strong Hall effect always acts to suppress the magnetic field generation. The hydrodynamic/magnetic Reynolds number dependence of the critical ion skin depth at which the dynamos start to feel the Hall effect is of interest from an astrophysical point of view. An important consequence of the matched asymptotic expansion analysis of the solution is that the higher the Reynolds number, the smaller the Hall current affects the flow. We also briefly discuss how the above results for a relatively simple shear flow can be extended to more general flows such as infinite homogeneous shear flows and boundary layer flows. The analysis of the latter flows suggests that interestingly a strong induction of the generated magnetic field might occur when there is a background shear layer.


2021 ◽  
Vol 931 ◽  
Author(s):  
Darren G. Crowdy

It is shown that shape anisotropy and intrinsic surface slip lead to equilibrium tilt of slippery particles in a creeping simple shear flow, even for nearly shape-isotropic particles with a cross-section that is close to circular provided the Navier-slip length is sufficiently large. We study a rigid particle with an elliptical cross-section, and of infinite extent in the vorticity direction, in simple shear. A Navier-slip boundary condition is imposed on its surface. When a Navier-slip length parameter $\lambda$ is infinite, an analytical solution is derived for the Stokes flow around a particle tilting in equilibrium at an angle $(1/2)\cos ^{-1}((1-k)/(1+k))$ to the flow direction where $0 \le k \le 1$ is the ratio of the semi-minor to semi-major axes of its elliptical cross-section. A regular perturbation analysis about this analytical solution is then performed for small values of $1/\lambda$ and a numerical continuation method implemented for larger values. It is found that an equilibrium continues to exist for any anisotropic particle $k < 1$ provided $\lambda \ge \lambda _{crit}(k)$ where $\lambda _{crit}(k)$ is a critical Navier-slip length parameter determined here. As the case $k \to 1$ of a circular cross-section is approached, it is found that $\lambda _{crit}(k) \to \infty$ , so the range of Navier-slip lengths allowing equilibrium tilt shrinks as shape anistropy is lost. Novel theoretical connections with equilibria for constant-pressure gas bubbles with surface tension are also pointed out.


2021 ◽  
Author(s):  
◽  
Nathaniel Joseph Lund

<p>In this thesis, homogenization and perturbation methods are used to derive analytic expressions for effective slip lengths for Stokes flow over rough, mixed-slip surfaces, where the roughness is periodic, and the variation in slip length has the same period. If the classical no-slip boundary condition of fluid mechanics is relaxed, the slip velocity of the fluid at the surface is non-zero. For simple shear flow, the slip velocity is proportional to the shear rate. The constant of proportionality has dimensions of length and is known as the slip length. Any variation in the slip length over the surface will cause a perturbation to the flow adjacent to the surface. Due to the diffusion of momentum, at sufficient height above the surface, the flow perturbations have diminished, and flow is smooth and uniform. The velocity and shear rate at this height imply an effective slip length of the surface. The purpose of this thesis is to predict that effective slip length.  Homogenization is a technique for finding approximate solutions to partial differential equations. The essence of homogenization is to construct a mathematical model of a physical problem featuring some periodic heterogeneity, then generate a sequence of models such that the period in question reduces with each increment in the sequence. If the sequence is appropriately defined, it has a limit model in the limit of vanishing period, for which a solution can be found. The solution to the limit system is an approximation to the solutions of systems with a finite period.  We use homogenization to find the effective slip length of a system of Stokes flow over a periodically rough surface, described by periodic function h(x; y), with a local slip length b(x; y) varying with the same period. For systems where the period L is smaller than both the domain height P and typical slip lengths, the effective slip length bₑff is well-approximated by the harmonic mean of local slip lengths, weighted by area of contact between liquid and surface: [See 'Thesis' document below for equation.]  We further use a perturbation technique to verify the above expression in the special case of a flat surface, and to derive another effective slip length expression: For a flat surface with local slip lengths much smaller than the period and domain height, the effective slip length bₑff is well-approximated by the area-weighted average of local slip lengths: [See 'Thesis' document below for equation.]</p>


2021 ◽  
Author(s):  
◽  
Nathaniel Joseph Lund

<p>In this thesis, homogenization and perturbation methods are used to derive analytic expressions for effective slip lengths for Stokes flow over rough, mixed-slip surfaces, where the roughness is periodic, and the variation in slip length has the same period. If the classical no-slip boundary condition of fluid mechanics is relaxed, the slip velocity of the fluid at the surface is non-zero. For simple shear flow, the slip velocity is proportional to the shear rate. The constant of proportionality has dimensions of length and is known as the slip length. Any variation in the slip length over the surface will cause a perturbation to the flow adjacent to the surface. Due to the diffusion of momentum, at sufficient height above the surface, the flow perturbations have diminished, and flow is smooth and uniform. The velocity and shear rate at this height imply an effective slip length of the surface. The purpose of this thesis is to predict that effective slip length.  Homogenization is a technique for finding approximate solutions to partial differential equations. The essence of homogenization is to construct a mathematical model of a physical problem featuring some periodic heterogeneity, then generate a sequence of models such that the period in question reduces with each increment in the sequence. If the sequence is appropriately defined, it has a limit model in the limit of vanishing period, for which a solution can be found. The solution to the limit system is an approximation to the solutions of systems with a finite period.  We use homogenization to find the effective slip length of a system of Stokes flow over a periodically rough surface, described by periodic function h(x; y), with a local slip length b(x; y) varying with the same period. For systems where the period L is smaller than both the domain height P and typical slip lengths, the effective slip length bₑff is well-approximated by the harmonic mean of local slip lengths, weighted by area of contact between liquid and surface: [See 'Thesis' document below for equation.]  We further use a perturbation technique to verify the above expression in the special case of a flat surface, and to derive another effective slip length expression: For a flat surface with local slip lengths much smaller than the period and domain height, the effective slip length bₑff is well-approximated by the area-weighted average of local slip lengths: [See 'Thesis' document below for equation.]</p>


2021 ◽  
Author(s):  
◽  
Allan Raudsepp

<p>Shear banding, where a fluid spatially partitions into strain rate or shear bands in steadystate simple shear flow conditions, was first observed in wormlike micelles solutions and has since been observed in many other complex fluids. These solutions have been used extensively to explore the relationship between shear (or stress) banding and microstructure in complex fluids. This relationship is difficult to study because of its dynamic nature and there is still no clear consensus as to how banding relates to microstructural changes in wormlike micelles solutions. In this thesis, the rheology of a number of wormlike micelles solutions is examined using both conventional and novel techniques with the view to developing a better understanding of this relationship. The rheology of three wormlike micelles solutions composed of a surfactant cetylpyridinium chloride (CPCl) and counterion sodium salicylate in water with or without the salt sodium chloride were examined using mechanical rheometry and the rheo-optical techniques: homodyne photo-correlation spectroscopy (PCS), diffusing wave spectroscopy (DWS) and ellipsometry. Rheo-mechanical measurements were largely consistent with the predictions of the reptation-reaction model. While signi cant stress fluctuations were noted in one particular flow geometry, they were generally not observed in most rheomechanical measurements presented here, indicating that these fluctuations are not universal and that they are geometry dependent. Shear induced turbidity was directly observed in the cone-plate and parallel-plate geometries with turbid rings forming in samples that showed a stress plateau. The Poisson-renewal model, which extends the reptationreaction model to include the influence of high frequency modes on the linear rheology, was tested experimentally using mechanical rheometry, DWS microrheology and literature data. In most cases the data fitted the model behaviour quite well, giving a physically reasonable estimate of the average length of the micelles. DWS's spatial sensitivity to shear induced relative motion was then used to probe the flow behaviour of selected wormlike micelles solutions in the cylindrical-Couette, cone-plate and parallel-plate geometries. In the cylindrical-Couette, the  'flow-DWS' measurements were largely consistent with rheo-mechanical measurements and indicated that some wormlike micelles solutions were partitioning into apparently stable high and low strain rate bands in the vicinity of the stress plateau. While measurements in the cone-plate and parallel-plate geometries also suggested shear banding in samples that showed a stress plateau, the interpretation was less clear-cut. Homodyne PCS was combined with ellipsometry to examine the spatial relationship between strain rate and birefringence banding in selected wormlike micelles solutions in a cylindrical-Couette geometry. In contrast to the observations of previous workers, it was found here that the birefringence and strain rate bands did coincide. Furthermore, the high strain rate band was observed to be more turbid than the lower strain rate band suggesting a connection between strain rate, optical anisotropy and turbidity.</p>


2021 ◽  
Author(s):  
◽  
Allan Raudsepp

<p>Shear banding, where a fluid spatially partitions into strain rate or shear bands in steadystate simple shear flow conditions, was first observed in wormlike micelles solutions and has since been observed in many other complex fluids. These solutions have been used extensively to explore the relationship between shear (or stress) banding and microstructure in complex fluids. This relationship is difficult to study because of its dynamic nature and there is still no clear consensus as to how banding relates to microstructural changes in wormlike micelles solutions. In this thesis, the rheology of a number of wormlike micelles solutions is examined using both conventional and novel techniques with the view to developing a better understanding of this relationship. The rheology of three wormlike micelles solutions composed of a surfactant cetylpyridinium chloride (CPCl) and counterion sodium salicylate in water with or without the salt sodium chloride were examined using mechanical rheometry and the rheo-optical techniques: homodyne photo-correlation spectroscopy (PCS), diffusing wave spectroscopy (DWS) and ellipsometry. Rheo-mechanical measurements were largely consistent with the predictions of the reptation-reaction model. While signi cant stress fluctuations were noted in one particular flow geometry, they were generally not observed in most rheomechanical measurements presented here, indicating that these fluctuations are not universal and that they are geometry dependent. Shear induced turbidity was directly observed in the cone-plate and parallel-plate geometries with turbid rings forming in samples that showed a stress plateau. The Poisson-renewal model, which extends the reptationreaction model to include the influence of high frequency modes on the linear rheology, was tested experimentally using mechanical rheometry, DWS microrheology and literature data. In most cases the data fitted the model behaviour quite well, giving a physically reasonable estimate of the average length of the micelles. DWS's spatial sensitivity to shear induced relative motion was then used to probe the flow behaviour of selected wormlike micelles solutions in the cylindrical-Couette, cone-plate and parallel-plate geometries. In the cylindrical-Couette, the  'flow-DWS' measurements were largely consistent with rheo-mechanical measurements and indicated that some wormlike micelles solutions were partitioning into apparently stable high and low strain rate bands in the vicinity of the stress plateau. While measurements in the cone-plate and parallel-plate geometries also suggested shear banding in samples that showed a stress plateau, the interpretation was less clear-cut. Homodyne PCS was combined with ellipsometry to examine the spatial relationship between strain rate and birefringence banding in selected wormlike micelles solutions in a cylindrical-Couette geometry. In contrast to the observations of previous workers, it was found here that the birefringence and strain rate bands did coincide. Furthermore, the high strain rate band was observed to be more turbid than the lower strain rate band suggesting a connection between strain rate, optical anisotropy and turbidity.</p>


Author(s):  
David Chillingworth ◽  
M. Gregory Forest ◽  
Reiner Lauterbach ◽  
Claudia Wulff

AbstractWe use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits in response to steady shear flow. These are orbits for which the principal axis of orientation of the molecular field (the director) rotates out of the plane of shear and around the vorticity axis. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group $$\mathrm{SO}(3)$$ SO ( 3 ) that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of neutrally stable kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.


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