Abstract Calculation of the flux through a two-dimensional channel with one sinusoidal wall is presented using a perturbation expansion. This gives a series for the flux up to O ( ϵ 160 ) $\mathcal {O}(\epsilon ^{160})$ script upper O left parenthesis epsilon Superscript 160 Baseline right parenthesis , where ϵ $\epsilon $ epsilon is the amplitude of the wall undulation, ranging from 0 (flat wall) to 2 (channel pinched off). The series appears to diverge near ϵ = 0.95 $\epsilon =0.95$ epsilon equals 0.95 , but is continued through the use of Padé approximants. The divergence is investigated and found to be caused by pole singularities lying at ϵ ≈ ± 0.977 i $\epsilon \approx \pm 0.977i$ epsilon almost equals plus or minus 0.977 i on the imaginary axis. The continued series is validated against a purely numerical solution with excellent agreement for all values of the undulation amplitude.
Abstract Time-periodic flows play important roles in many practical problems, with a wide range of examples occurring in engineering, physiology and ocean dynamics, among others fields. While the stability of spatially developing fluid flows is a relatively well-studied topic of research, there is a relative paucity of corresponding results that discuss purely time-periodic flows. Here we examine the stability of the classical plane Stokes layer: this is the flow generated in a layer of fluid that lies above a bounding plate that oscillates harmonically in its own plane. It has proved surprisingly difficult to understand the linear stability of this type of fluid flow, and here we trace the steps that have been taken to achieve this goal. There is a significant difference between the theoretical description of the stability of a Stokes layer and the corresponding practical measurements. We discuss possible reasons for this discrepancy and its possible implications for more general time-periodic motions.
Abstract The problem concerning water wave scattering by a floating breakwater has been an interesting subject of study under linearized water wave theory in the past few decades due to the reason that not only do these structures allow better water circulation to have smaller impact on sediment transport and fish migration, but also their construction is cost effective compared with the bottom founded structures. Modern study observes that thick breakwaters, with their larger and bulkier structure, are more effective at reducing wave energy compared with thin breakwaters. However, recent research reveals that breakwaters that combine both thin and thick parts offer enhanced performance by providing effective wave attenuation and reducing the construction costs. We may mention that these problems are primarily studied when the depth of water is uniform. However, in practice, the bathymetry of the water region is not uniform. This motivated us to study the problem of water wave scattering by a $ \boldsymbol \sqcap $ -shaped floating breakwater (which is a combination of thick and thin parts) in the presence of a rectangular submarine trench. For analytical solution of the corresponding boundary value problem, the geometric symmetry of the problem simplifies the mathematical difficulty considerably. For this, the domain of definition is divided into smaller subdomains and by suitable matching conditions at the various boundaries of the subdomains, the corresponding boundary value problem is formulated in terms of two Fredholm-type integral equations. The multi-term Galerkin approximation technique, involving Chebyshev polynomials and ultraspherical Gegenbauer polynomials as basis functions, is employed to solve the integral equations. It is important to note here that the choice of basis functions depends on the types of singularities at the corner of the trench and at the submerged sharp edge of the breakwater. The solutions of the integral equations are then employed to evaluate reflection and transmission coefficients. Graphical illustrations show that the length, width of the trench and the width of the breakwater have definite impact on the reflected wave, transmitted wave and the force on the breakwater. It is observed that the presence of trench produces oscillations in the reflection and transmission coefficients. When the width of the trench is equal to the width of the breakwater, reflection and transmission coefficients show smooth uniform oscillations. However, when the width of the trench and the width of the breakwater are unequal, nonuniform, nonsmooth oscillations in the reflection and transmission coefficients are observed. It is interesting to note that for a particular width of the breakwater, the reflected wave of certain wavelength shows resonating behaviour. Also, with the increase of submerged length of the side wall of the $\large \boldsymbol \sqcap $ -shaped breakwater, the horizontal force on the breakwater enhances and the vertical force diminishes. Additionally, it is observed that when the breakwater is present in a water region with a trench, the horizontal force is more than the vertical force. However, when the breakwater is present in uniformly deep water without a trench, the vertical force is more than the horizontal force.
Abstract The present study characterizes the impact of brash ice present at the topmost surface of an ocean on a submerged stationary prolate spheroidal body (both axisymmetric and nonaxisymmetric) within the framework of linear water wave theory. The brash ice cover is assumed to behave as an inertial surface composed of uniformly distributed, noninteracting particles in the presence of surface tension. Taking advantage of the body’s symmetry, we adopt a prolate spheroidal coordinate system. The solution is constructed through the method of image singularities, using Havelock’s theorem to transform the fundamental Green’s function into a prolate spheroidal coordinate system. The diffraction potentials are expanded in terms of the multipoles of the Green’s function using spherical harmonics. The zero-velocity condition, along with suitable orthogonality relations, is used to derive the final linear system. The hydrodynamic forces (such as surge, heave and sway) and moment acting on the spheroid are numerically evaluated, and analysed for different values of the brash ice parameter, coefficients of surface tension, depths of submergence, eccentricity of the body and wave-heading angles. We observe reduction in the magnitudes of forces due to the presence of brash ice and further reduction has been noticed as the layer of brash ice gets heavier. The yaw moment is found to attain its maximum value at a specific angle of wave incidence.
We investigate the dynamics arising from an idealized model of a spherical active particle immersed in a cubic Poiseuille field inspired by fluid flow through an equilateral triangular duct. Starting from a general Hamiltonian formulation, we describe the equations of motion, analyse equilibrium points and their stability and classify trajectories based on their initial position. Motion of an active particle within the Poiseuille flow of an equilateral triangular duct is an interesting case to examine given its symmetry group and a velocity field described by a cubic polynomial. In addition to trajectory types previously identified in other duct geometries, including central and vertical swinging, tumbling, off-centred trapping and wandering, we observe some exotic orbits within the triangular geometry. We also examine the chaotic behaviour by using Poincar & eacute; maps and Lyapunov exponents over a range of parameter values and initial conditions. This work enhances the broader understanding of idealized microswimmer motion via a case where the fluid flow has a straightforward closed-form description.
