High-Reynolds-number flow about a submerged circular cylinder induced by free-surface travelling waves on finite depth

We consider the fluid flow induced when free-surface travelling waves, on fluid whose depth is finite and uniform in its undisturbed state, pass over a submerged circular cylinder. The wave amplitude is assumed to be small, and a suitably defined Reynolds number large. Thus, the inviscid flow may be pursued by perturbation methods, as may viscous effects that are confined to thin boundary layers on the cylinder and bed beneath it. Particular attention is focused on the steady streaming motion, which induces a circulation about the cylinder. The consequences of this on bed scouring beneath the cylinder, when the bed is erodible, are considered.     

Nonlinear streaming due to the oscillatory stretching of a sheet in a viscous fluid

Summary An elastic sheet is stretched back and forth in a viscous fluid. The problem is governed by a nondimensional parameterS which represents the relative magnitude of frequency to stretching rate. The Navier-Stokes equations are solved by matched asymptotic expansions for largeS. Due to nonlinearity there exists boundary layers ofO(S ^–1/2). The unsteady oscillatory flow contains both basic and higher harmonic oscillations. The induced steady streamlines show a saddle like flow which is different from that of acoustic streaming.With 5 Figures     

Rotating compressible flow between non torsionally oscillating disks

A viscous compressible gas between two disks is initially in a state of rigid rotation. The initial density distribution depends on the distance from the axis of rotation. A three dimensional flow is created relative to a rotating frame by imposing small amplitude non torsional oscillations on the disks. The solution is obtained using the Laplace transform. The structure of the boundary layers formed on the disks due to interaction of viscous force, Coriolis force and compressibility is analysed for various ranges of values of the forcing frequency. The theory reveals interesting features in comparison with the non oscillatory case or the case of an incompressible fluid.     

The dynamic stability of a pipe conveying a pulsatile flow

This investigation concern the small displacements of a pipe conveying a pressurized flow whose velocity possesses a harmonic fluctuation about a mean value. A new derivation of the fluid forces is used to obtain the partial differential equation of motion. General equations applicable to any set of boundary conditions are specialized for the case of a simply supported pipe and the Galerkin method is utilized to find solutions. An analysis of the case of steady flow shows that the pipe exhibits the divergence type of instability which is predictable by static structural theory. In the presence of pulsatile flow the pipe has regions of dynamic instability whose extent increases with increased amounts of fluctuation. The results have great similarity to those for beams carrying pulsating end forces.     

Inertial Waves and Steady Flows in a Liquid Filled Librating Cylinder

The fluid flow in a non-uniformly rotating (librating) cylinder about a horizontal axis is experimentally studied. In the absence of librations the fluid performs a solid-body rotation together with the cavity. Librations lead to the appearance of steady zonal flow in the whole cylinder and the intensive steady toroidal flows near the cavity corners. If the frequency of librations is twice lower than the mean rotation rate the inertial waves are excited. The oscillating motion associated with the propagation of inertial wave in the fluid bulk leads to the appearance of an additional steady flow in the Stokes boundary layers on the cavity side wall. In this case the heavy particles of the visualizer are assembled on the side wall into ring structures. The patterns are determined by the structure of steady flow, which in turn depends on the number of reflections of inertial wave beams from the cavity side wall. For some frequencies, inertial waves experience spatial resonance, resulting in inertial modes, which are eigenmodes of the cavity geometry. The resonance of the inertial modes modifies the steady flow structure close to the boundary layer that is manifested in the direct rebuilding of patterns. It is shown that the intensity of zonal flow, as well as the intensity of steady flows excited by inertial waves, is proportional to the square of the amplitude of librations.     

Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid

Summary. This paper deals with some steady unidirectional flows of an Oldroyd 8-constant magnetohydrodynamic (MHD) fluid in bounded domains. The fluid is electrically conducting in the presence of a uniform magnetic field. Three nonlinear flows are produced by the motion of a boundary or by sudden application of a constant pressure gradient or by the motion of a boundary and pressure gradient. The governing nonlinear differential equations are solved analytically using homotopy analysis method (HAM). Expressions for the velocity distribution are given. It is noted that for steady flow the solutions are strongly dependent on the non–Newtonian and magnetic parameters. The MHD solutions for a Newtonian fluid, as well as those corresponding to the Oldroyd 3 and 6-constant fluids, a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. Finally, a physical interpretation of the results is given with the help of several graphs.     

Computational modelling of a non-viscous fluid flow in a multi-walled carbon nanotube modelled as a Timoshenko beam

In the design of nanotube-based fluidic devices, a critical issue is the effect of the induced vibrations in the nanotube arising from the fluid flow, since these vibrations can promote structural instabilities, such as buckling transitions. It is known that the induced resonant frequencies depend on the fluid flow velocity in a significant manner. We have studied, for the first time, the flow of a non-viscous fluid in stubby multi-walled carbon nanotubes, using the Timoshenko classical beam theory to model the nanotubes as a continuum structure. We have obtained the variations of the resonant frequencies with the fluid flow velocity under several experimentally interesting boundary conditions and aspect ratios of the nanotube. The main finding from our work is that, compared to an Euler-Bernoulli classical beam model of a nanotube, the Timoshenko beam predicts the loss of stability at lower fluid flow velocities.     

Hydrodynamic tweezers: impact of design geometry on flow and microparticle trapping

Here we explore the role of microfabricated device geometry on frequency-dependent low Reynolds number steady streaming flow and particle trapping behavior. In our system, flow and particle trapping is induced near an obstruction or cavity located in an otherwise rectilinear oscillating flow of frequency ω and amplitude s in a fluid of kinematic viscosity ν. This work expands prior studies to characterize nine distinct obstruction/cavity geometries. The imaged microeddy flows show that the device geometry affects the eddy number, shape, structure, and strength. Comparison of measured particle trap locations with the computed eddy flow structure shows that particles trap closer to the wall than the eddy core. Trapping strength and location are controlled by the geometry and the oscillation frequency. In most cases, the trapping behavior is linearly proportional to the Stokes layer thickness, δ(AC) ~ O((ν/ω)(1/2)). We show that steady streaming in microfluidic eddies can be a flexible and versatile method for noncontact microparticle trapping, and hence we call this class of devices "hydrodynamic tweezers".     

Analysis of Silver Nanoparticles in Engine Oil: Atangana–Baleanu Fractional Model

The present article aims to examine the heat and mass distribution in a free convection flow of electrically conducted, generalized Jeffrey nanofluid in a heated rotatory system. The flow analysis is considered in the presence of thermal radiation and the transverse magnetic field of strength B₀. The medium is porous accepting generalized Darcy’s law. The motion of the fluid is due to the cosine oscillations of the plate. Nanofluid has been formed by the uniform dispersing of the Silver nanoparticles in regular engine oil. The problem has been modeled in the form of classical partial differential equations and then generalized by replacing time derivative with Atangana–Baleanu (AB) time-fractional derivative. Upon taking the Laplace transform technique (LTT) and using physical boundary conditions, exact expressions have been obtained for momentum, energy, and concentration distributions. The impact of a number of parameters on fluid flow is shown graphically. The numerical tables have been computed for variation in the rate of heat and mass transfer with respect to rooted parameters. Finally, the classical solution is recovered by taking the fractional parameter approaching unity. It is worth noting that by adding silver nanoparticles in regular engine oil, its heat transfer rate increased by 14.59%, which will improve the life and workability of the engine.     

Energetic balance for the Rayleigh–Stokes problem of a Maxwell fluid

The energetic balance in the Rayleigh–Stokes problem for a Maxwell fluid is studied for several initial and/or boundary conditions. In the case of the first problem of Stokes, in comparison with the Newtonian fluid, the power of the wall shear stress and the dissipation increase while the boundary layer thickness decreases. A similar result is obtained for a series solution. In the case of the decay of the previous steady motion as well as for the Newtonian fluid, the power of the wall shear stress is null and the boundary layer thickness is the same. Finally, the second problem of Stokes is also considered.     

Oscillatory non-Newtonian flow in tubes of slowly varying radius

Summary The paper studies low Reynolds number flow of a non-Newtonian fluid in an axisymmetric tube of slowly varying radius which is subjected to an axial oscillatory pressure gradient. It is observed that the leading approximation is affected by the visco-elastic coefficient. In the higher approximation particular attention is centered around the steady streaming components for both small and large values of the frequency of oscillation. On the overall the combined effect of visco-elast c and cross viscosity parameter is to induce a radial pressure gradient. For the velocity components the Newtonian and non-Newtonian effects are of the same magnitude when the frequency is small; but when this frequency is large the non-Newtonian effects swamp the flow velociites. When the results are applied to a locally constricted tube, flow reversal is possible downstream of this constriction. The most striking feature however is the condition of zero velocity at a locally constricted tube for the steady streaming velocities—upstream of this constriction the velocity is positive while downstream it is negative. In pathophysiology thrombus formation in constrictions is believed to be caused by aggregation of platelets due to endothelium damage. The condition of zero steady streaming velocity at the constriction is another possible explaination of platelet accumulation and possible blood cloting.With 2 Figures     

Free convection from a torsionally oscillating vertical plane

Free convective flow of a viscous incompressible fluid from a uniformly heated vertical plane lamina, undergoing small-amplitude sinosoidal torsional oscillations, is investigated. The non-axisymmetric fluid motion consists of the primary Rosenblat flow and the secondary buoyancyinduced cross-flow. Numerical-analytical and asymptotic solutions of the energy equation, covering the whole range of the values of the Prandtl number of the fluid, are derived for their subsequent use in the analysis of the unsteady cross-flow. The mean cross-flow is found to dominate the steady components of the primary flow.     

Trapping of water waves by moored bodies

Certain types of floating bodies are known to support trapped modes, with oscillatory fluid motion near the body and no energy radiation in the far field. Previous work has considered either fixed bodies, where the boundary conditions are homogeneous, or bodies which are freely floating and moving without any exciting force. For a fixed body the existence of a trapped mode implies that there is no unique solution of the boundary-value problem for the velocity potential with a prescribed body motion. For a free body which supports a trapped mode, the solution of the coupled problem for the motions of the fluid and body does not have a unique solution. A more general case is considered here, of a body with a linear restoring force such as an elastic mooring. The limiting cases of a fixed and free body correspond to infinite or zero values of the corresponding spring constant. A variety of body shapes are found including cylinders in two dimensions and axisymmetric bodies in three dimensions, which illustrate this more general case of trapping and provide a connection between the fixed and free cases.     

Magnetohydrodynamic flow due to non-coaxial rotations of a porous oscillating disk and a fluid at infinity

An exact solution of the Navier-Stokes equations is constructed for the case of flow due to non-coaxial rotations of a porous disk and a fluid at infinity. The disk executes oscillations in its own plane and is non-conducting. The viscous fluid is incompressible and electrically conducting. Analytical solution is established by the method of Laplace transform. The velocity fields are obtained for the cases when the angular velocity is greater than, smaller than or equal to the frequency of oscillations. The structure of the steady and the unsteady velocity fields are investigated. The difficulty of the hydrodynamic steady solution associated with the case of resonant frequency is resolved in the present analysis.     

Influence of wall elasticity and poiseuille flow on peristaltic induced flow of a particle-fluid mixture

The effect of Poiseuille flow on peristaltic transport of a particle-fluid mixture has been investigated in a two-dimensional mathematical model for the case where the walls of the channel execute sinusoidal motion of small amplitude. The driving mechanism of the muscle is represented by assuming the channel walls to be elastic or viscoelastic. The fluid-particle interaction problem is investigated by considering equations of motion for both the fluid and particle phases with the deformable boundaries. The wall characteristics appear in their equations of motion, which are solved to represent boundary conditions of the fluid motion. Solutions for free pumping case and interaction with Poiseuille flow are obtained for small values of Poiseuille flow parameter in closed form as well as by a series method. The effect of the particulate phase is observed throughout the analysis both qualitatively and quantitatively.     

Steady two-layer flows over an obstacle

Nonlinear waves in a forced channel flow of two contiguous homogeneous fluids of different densities are considered. Each fluid layer is of finite depth. The forcing is due to an obstruction lying on the bottom. The study is restricted to steady flows. First a weakly nonlinear analysis is performed. At leading order the problem reduces to a forced Korteweg-de Vries equation, except near a critical value of the ratio of layer depths which leads to the vanishing of the nonlinear term. The weakly nonlinear results obtained by integrating the forced Korteweg-de Vries equation are validated by comparison with numerical results obtained by solving the full governing equations. The numerical method is based on boundary integral equation techniques. Although the problem of two-layer flows over an obstacle is a classical problem, several branches of solutions which have never been computed before are obtained.     

On a similarity solution in the theory of unsteady marginal separation

The present paper deals with initial value problems associated with the high Reynolds number asymptotic theory of unsteady, marginally separated boundary layer flows. In particular, the subsonic planar flow case is treated. Special emphasis is placed on solutions which blow up within finite time. As is well-known, steady solutions of the underlying equations only exist up to a critical value of the crucial parameter which controls the conditions leading to localized boundary layer separation. Our numerical analysis shows that any blow-up solution finally approaches a unique structure, entirely independently of the choice of initial data, sub- or super-critical flow conditions, and, if present at all, the type of forcing. Further support for the existence of a self-similar, unique blow-up structure is gained from asymptotic analysis.     

A finite element model for simulating acoustic streaming in cystic breast lesions with experimental validation

Streaming detection is an ultrasonic technique that can be used to distinguish fluid-filled lesions, or cysts, from solid lesions. With this technique, high intensity ultrasound pulses are used to induce acoustic streaming in cyst fluid, and this motion is detected using Doppler flow estimation methods. Results from a pilot clinical study were recently published in which acoustic streaming was successfully induced and detected in 14 of 15 simple breast cysts and four of 14 sonographically indeterminate breast lesions in vivo. In the study, the detected velocities were found to vary considerably among cysts and for different pulsing regimes. A finite element model of streaming detection is presented. This model is utilized to investigate methods of increasing induced acoustic streaming velocity while minimizing patient exposure to high intensity ultrasound during streaming detection. Parameters studied include intensity, frequency, acoustic beam shape, cyst-diameter, cyst fluid protein concentration, and cyst fluid viscosity. The model, which provides both transient and steady-state solutions, is shown to predict trends in streaming velocity accurately. Experimental results from studies investigating the potential for nonlinear streaming enhancement in cysts are also provided.     

Axisymmetric flow of two fluids in a pulsating pipe

The motion of a viscous thread surrounded by an annular viscous layer inside a pulsating cylindrical pipe whose radius is a periodic function of time is investigated. At zero Reynolds number, a stagnation-point-type solution may be written down in closed form. A Floquet linear stability analysis for Stokes flow reveals the pulsations either decrease or increase the growth rate of longwave disturbances depending on the initial radius of the thread. For a moderate-sized initial thread radius, increasing the amplitude of the pulsations decreases the critical wavenumber for instability to below the classical Rayleigh threshold. Increasing the viscosity contrast, so that the fluid in the annular layer becomes more viscous than the fluid in the thread, tends to decrease the growth rate of disturbances. In the second part of the paper, the basic stagnation-point-type flow at arbitrary Reynolds number is computed using a numerical method on the assumption that the interface is a circular cylinder at all times. During the motion, either the thread radius tends to increase and the thickness of the annular layer decreases, or else the thread tends to thin and the thickness of the annular layer increases, depending upon the initial conditions and the parameter values. For a judicious choice of initial condition, a time-periodic exact solution of the Navier–Stokes equations is identified.     

Separating inviscid flows

Summary The separating flow of an inviscid fluid is not only a limit solution of the steady separating, laminar fluid flow at high values of the Reynolds number but it is also part of its structure (Smith [1], [2]). This work aims at reexamining the separating flow of inviscid fluid past a bluff body which is fixed in an otherwise uniform stream of fluid. For the purpose of this paper we will assume that the bluff body is a circular cylinder but the theory is applicable to bodies of any shape. It is further assumed that the fluid is in steady two-dimensional motion and is inviscid and of constant density. The flow structure is assumed to consist of a separated flow region, caviting flows in which there exists a free surface on which the pressure is constant, and a wake. A twin spiral vortex model is used in order to determine the shape of the free streamline. Based on the free streamline theory the problem reduces to solving a mixed boundary value problem and a Hilbert solution for the inverse problem in the auxiliary plane is obtained. When we consider the flow in the physical plane the problem is transformed into a direct problem in which the geometry of the solid body is given in advance. We assume that the separation is smooth and thus the curvature of the free streamline at the point of free detachment be equal to that of the body surface. A numerical method for solving the two-dimensional potential flows past arbitrarily shaped curved bluff bodies is developed.When the cavitation number is zero the angle of separation is approximately 55° and the computed results predict that the position of the separation point will move backward as the cavition number increases. The relationships between the drag coefficient, and the width and length of the cavity is determined and is found to be in very good agreement with the predictions of Smith [1].