On the stability of fully nonlinear hydraulic-fall solutions to the forced water-wave problem

Two-dimensional free-surface flow over localised topography is examined with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a dip, respectively. Steady hydraulic-fall solutions to the full incompressible, irrotational Euler equations are computed, and their linear and nonlinear stability is analysed by computing eigenspectra of the pertinent linearised operator and by solving an initial value problem. The computations are carried out numerically using a specially developed computational framework based on the finite element method. The Hamiltonian structure of the problem is demonstrated and stability is determined by computing eigenspectra of the pertinent linearised operator. It is found that a hydraulic-fall flow over a bump is spectrally stable. The corresponding flow over a dip is found to be linearly unstable. In the latter case, time-dependent simulations show that the flow ultimately settles into a time-periodic motion that corresponds to an invariant solution in an appropriately defined phase space. Physically, the solution consists of a localised large amplitude wave that pulsates above the dip while simultaneously emitting nonlinear cnoidal waves in the upstream direction and multi-harmonic linear waves in the downstream direction

Continuation methods for time-periodic travelling-wave solutions to evolution equations

A numerical continuation method is developed to follow time-periodic travelling-wave solutions of both local and non-local evolution partial differential equations (PDEs). It is found that the equation for the speed of the moving coordinate can be derived naturally from the governing equations together with the condition that breaks the translational symmetry. The derived system of equations allows one to follow the branch of travelling-wave solutions as well as solutions that are time-periodic in a frame of reference travelling at a constant speed. Finally, we show as an example the bifurcation and stability analysis of single and double-pulse waves in long-wave models of electrified falling films

Nonlinear dynamics of two-layer channel flow with soluble surfactant below or above the critical micelle concentration

The nonlinear stability of an inertialess two-layer surfactant-laden Couette flow is considered. The two fluids are immiscible and have different thicknesses, viscosities and densities. One of the fluids is contaminated with a soluble surfactant whose concentration may be above the critical micelle concentration, in which case micelles are formed in the bulk of the fluid. A surfactant kinetic model is adopted that includes the adsorption and desorption of molecules to and from the interface, and the formation and breakup of micelles in the bulk. The lubrication approximation is applied and a strongly nonlinear system of equations is derived for the evolution of the interface and surfactant concentration at the interface, as well as the vertically averaged monomer and micelle concentrations in the bulk (as a result of fast vertical diffusion). The primary aim of this study is to determine the influence of surfactant solubility on the nonlinear dynamics. The nonlinear lubrication model is solved numerically in periodic domains and saturated travelling waves are obtained at large times. It is found that a sufficiently soluble surfactant can either destabilise or stabilise the interface depending on certain fluid properties. The stability behaviour of the system depends crucially on the values of the fluid viscosity ratio and thickness ratio in reference to the boundary. If the surfactant exists at large concentrations that exceed the critical micelle concentration, then long waves are stable at large times, unless density stratification effects overcome the stabilising influence of micelles. Travelling wave bifurcation branches are also calculated and the impact of various parameters (such as the domain length or fluid thickness ratio) on the wave shapes, amplitudes and speeds is examined. The mechanism responsible for interfacial (in)stability is explained in terms of the phase difference between the interface deformation and concentration waves, which is shifted according to the sign of the crucial factor and the strength of the surfactant solubility

Absolute and convective instabilities in non-local active-dissipative equations arising in the modelling of thin liquid films

Absolute and convective instabilities in a non-local model that arises in the analysis of thin-film flows over flat or corrugated walls in the presence of an applied electric field are discussed. Electrified liquid films arise, for example, in coating processes where liquid films are deposited onto a target surfaces with a view to producing an evenly coating layer. In practice, the target surface, or substrate, may be irregular in shape and feature corrugations or indentations. This may lead to non-uniformities in the thickness of the coating layer. Attempts to mitigate film-surface irregularities can be made using, for example, electric fields. We analyse the stability of such thin-film flows and show that if the amplitude of the wall corrugations and/or the strength of the applied electric field is increased the convectively unstable flow undergoes a transition to an absolutely unstable flo

Stability of waves on fluid of infinite depth with constant vorticity

The stability of periodic travelling waves on fluid of infinite depth is examined in the presence of a constant background shear field. The effects of gravity and surface tension are ignored. The base waves are described by an exact solution that was discovered recently by Hur and Wheeler (J. Fluid Mech., vol. 896, 2020). Linear growth rates are calculated using both an asymptotic approach valid for small-amplitude waves and a numerical approach based on a collocation method. Both superharmonic and subharmonic perturbations are considered. Instability is shown to occur for any non-zero amplitude wave

Electromagnetic navigation in total knee arthroplasty—a single center, randomized, single-blind study comparing the results with conventional techniques

We report on the results of a randomized study (n = 200) to compare total knee arthroplasty performed using conventional instrumentation or electromagnetic computer assisted surgical technique. 92% of navigated and 85% of conventional knees were implanted within ± 3° from neutral mechanical alignment; there was no statistically significant difference between these proportions. There was also no difference in femoral or tibial rotation assessed by CT scan. At 1 year follow up there was no statistical difference between the two groups in American Knee Society Score, Oxford Knee Scores, patient satisfaction, quality of life, hospital length of stay, complication rates or other adverse events. Tourniquet time in the navigated group was longer. Proving value for navigation in total knee arthroplasty surgery remains a challenge

Robotic arm-assisted bi-unicompartmental knee arthroplasty maintains natural knee joint anatomy compared with total knee arthroplasty : a prospective randomized controlled trial

Aims The aim of this study was to compare robotic arm-assisted bi-unicompartmental knee arthroplasty (bi-UKA) with conventional mechanically aligned total knee arthroplasty (TKA) in order to determine the changes in the anatomy of the knee and alignment of the lower limb following surgery. Methods An analysis of 38 patients who underwent TKA and 32 who underwent bi-UKA was performed as a secondary study from a prospective, single-centre, randomized controlled trial. CT imaging was used to measure coronal, sagittal, and axial alignment of the knee preoperatively and at three months postoperatively to determine changes in anatomy that had occurred as a result of the surgery. The hip-knee-ankle angle (HKAA) was also measured to identify any differences between the two groups. Results The pre- to postoperative changes in joint anatomy were significantly less in patients undergoing bi-UKA in all three planes in both the femur and tibia, except for femoral sagittal component orientation in which there was no difference. Overall, for the six parameters of alignment (three femoral and three tibial), 47% of bi-UKAs and 24% TKAs had a change of < 2° (p = 0.045). The change in HKAA towards neutral in varus and valgus knees was significantly less in patients undergoing bi-UKA compared with those undergoing TKA (p < 0.001). Alignment was neutral in those undergoing TKA (mean 179.5° (SD 3.2°)) while those undergoing bi-UKA had mild residual varus or valgus alignment (mean 177.8° (SD 3.4°)) (p < 0.001). Conclusion Robotic-assisted, cruciate-sparing bi-UKA maintains the natural anatomy of the knee in the coronal, sagittal, and axial planes better, and may therefore preserve normal joint kinematics, compared with a mechanically aligned TKA. This includes preservation of coronal joint line obliquity. HKAA alignment was corrected towards neutral significantly less in patients undergoing bi-UKA, which may represent restoration of the pre-disease constitutional alignment (p < 0.001)

Robotic arm-assisted bi-unicompartmental knee arthroplasty maintains natural knee joint anatomy compared with total knee arthroplasty : a prospective, randomised controlled trial

Unicompartmental knee arthroplasty (UKA) is associated with faster postoperative recovery than total knee arthroplasty (TKA), sparing cruciate ligaments and normal joint surfaces. Although registry data indicate that revision rates are higher following UKA than TKA, robotic arm-assistance can improve accuracy of implant positioning and restoration of normal kinematics, when compared to manual implantation. Bi-unicompartmental knee arthroplasty (Bi-UCKA) with manual implantation has been demonstrated to maintain native joint obliquity better than TKA, but not as well as UKAs. There has been no RCT to date examining whether the benefits of UKA hold true for robotic-assisted Bi-UCKA compared to TKA