Nanofluids are widely known to enhance the heat transfer rate resulting in a cooled system. In the present paper, we show mathematically that the nanofluids indeed cool the system as the nanoparticles volume fraction is increased. The key role is explained for a two-dimensional laminar free nanofluid jet and for a circular axisymmetric free nanofluid jet issuing into the same nanofluid medium. Exact nanofluid flow results are obtained and, integral flux relations of momentum and thermal layers concerning five most studied nanofluids, respectively Ag, Cu, CuO, Al O and TiO , are derived. A shape factor is defined controlling the momentum layer thickness. By means of another shape factor representing the thermal layer thickness, the relevant energy equation enables one to identify the regimes of nanoparticle size leading to a coolant jet, without a need to solve the energy equation fully. Two recently popular nanofluid models, resulting in the same conclusion, are examined on the considered free nanofluid jets. Additionally, an exact temperature field associated with the laminar two-dimensional free jet of nanoparticles is obtained offering explicit support to the current approach.
The homotopy analysis method (HAM), a general analytic technique for non-linear problems, is applied to analyze the solute dispersion process in non-Newtonian Carreau-Yasuda and Carreau fluids flow in a straight tube with the effect of wall absorption/reaction. Unlike the other analytical methods such as perturbation method, the HAM provides a simple way to get the convergent series solution. Any assumptions of small or high physical quantities are not required for the HAM. It provides us a great freedom to choose the so-called convergence control parameter which is used to guarantee the convergence of series solution. The convergent series solution is obtained by choosing the optimal value of convergence control parameter for which the series converges fastest. The optimal value of convergence control parameter is obtained by minimizing the square residual which also provides the convergence region for the series solution. The previous analytical studies on solute dispersion fail to justify the convergence of the series solution, whereas in this investigation, our results are convergent and valid for all physical parameters. In addition, present results are validated by the numerical and some existing results. This study explains elaborately about the advantage of the homotopy analysis method over perturbation and eigenfunction expansion methods for nonlinear problems. Owing to the great potential and flexibility of the homotopy analysis method, it is a more suitable analytical approach to solve the different types of non-linear problems in science and engineering. Besides, this study helps to gain some knowledge on the transportation process of drugs in the blood flow.
Computational Fluid Dynamics (CFD) simulations were performed to study the effect of the size of the column on mixing in an oscillating column. Volume of Fluid (VOF) method was employed to track the interface, and tracer simulations were carried out to estimate the mixing time. The fluid was assumed to be water, and the ratio of the height of the water in the column and the diameter of the column was equal to 1. The amplitude of the applied oscillation was kept constant at 0.25 cm, and the frequency was varied from 1 Hz to 20 Hz. Two cylindrical columns were considered in this work: one with a radius twice the benchmark geometry and the other with a radius half of it. The benchmark geometry was the cylindrical column with radius = 0.05 m, which was utilized in our previous work (Bale et al., 2017). It was observed that the mixing within an oscillating column was nonlinear with respect to the applied oscillation. The mixing time per unit volume was much smaller for the larger column, with similar power per unit mass applied to both the columns. The optimal condition to operate the two columns under consideration were reported. The numerical simulation results were corroborated by the stability chart determined theoretically by solving a series of Mathieu equation, contour plots through the column, and the snapshots of the free surface.
The buoyancy-driven fluid flow and heat transfer in a square cavity with partially active side walls filled with Cu–water nanofluid is investigated numerically. The active parts of the left and the right side walls of the cavity are maintained at temperatures and , respectively, with . The enclosure’s top and bottom walls as well as the inactive parts of its side walls are kept insulated. The governing equations in the two-dimensional space are discretized using the control volume method. A proper upwinding scheme is employed to obtain stabilized solutions. Using the developed code, a parametric study is undertaken, and the effects of the Rayleigh number, the locations of the active parts of the side walls, and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the cavity are investigated. It is observed from the results that the average Nusselt number increases with increasing both the Rayleigh number and the volume fraction of the nanoparticles. Moreover, the maximum average Nusselt number for the high and the low Rayleigh numbers occur for the bottom–middle and the middle–middle locations of the thermally active parts, respectively.
Direct numerical simulations of irregular unidirectional nonlinear wave evolution are performed within the framework of the Korteweg–de Vries equation for bimodal wave spectra model cases. The additional wave system co-existence effect on the evolution of the wave statistical characteristics and spectral shapes, and also on the attained equilibrium state is studied. The concerned problem describes, for example, the interaction between wind waves and swell in shallow seas. It is next demonstrated that a low-frequency spectral component yields more asymmetric waves with more extreme statistical properties.
The inward flow between two parallel, co-rotating disks undergoes a thorough examination by analytical, experimental and numerical means. The analytical approach utilizes the asymptotical truncated series solution provided by Batista (2011) and extends it by a correction for an arbitrary mean tangential velocity at the rotor inlet. Taylor series expansions of the analytical results provide an estimate for the orders of magnitude of velocity components and the polynomial order of their profile shapes. The common assumption of parabolic velocity distributions is only appropriate in the radial direction. In parallel, a unique test rig provides the experimental counterpart of the velocity profiles inside a rotor gap, that is suitably narrow for turbomachinery applications. The optical flow measurements are based on a novel calibration technique and volumetric particle tracking evaluation. Both laminar and turbulent operating conditions are examined. Finally, numerical studies using commercial CFD software provide insight into the flow field inside the test rig rotor where experimental methods fall short and provide an additional means to investigate the effects of the approximations made in the derivation of the analytical results. The velocity distributions acquired by analytical, numerical and experimental means agree well, the asymptotical nature of the analytical solution by Batista (2011) can be observed. The comparison of experimental and numerical results of a turbulent case suggests that the Shear Stress Transport turbulence model reproduces turbulent flow inside the rotor gap appropriately.
The energy and momentum transfer between the atmosphere and the ocean has typically been studied for conditions where the waves have almost or already reached a local equilibrium with a uniform wind. The purpose of this work is to investigate the early stages of the generation of waves under non-stationary wind conditions and to describe the momentum and energy exchange at the air–water interface for non-equilibrium wind conditions. Some experiments with a characteristic wind acceleration were conducted in a large wind-wave facility at the Institut Pythéas (Marseille-France). Momentum fluxes were estimated through hot-wire anemometry and, the free surface displacement was measured along the wave tank by resistance and capacitance wire probes. Wind speed and water elevation measurements were acquired at a high sampling rate. During the experiments, the wind speed was increased with a constant acceleration over time, reaching a constant maximum intensity of 13 ms . Under accelerated wind conditions, the degree of wave field development associated with a certain value of wind speed depends on the wind acceleration. Accordingly, once the rough flow regime is established, the drag coefficient values associated with a certain wind speed also vary depending on wind acceleration. It was observed that higher wind speed is needed to reach a rough flow regime as the wind acceleration increases. Also, the momentum transfer is reduced as wind acceleration increases. Under the rough flow regime, a less developed wave field induces a higher increase of drag coefficient with wind speed.
The phenomena of concentration and cavitation and the formation of delta shock waves and vacuum states in vanishing pressure limits of solutions to the generalized Chaplygin Euler equations of compressible fluid flow are analyzed. It is proved that, as the pressure vanishes, any two-shock-wave Riemann solution of the generalized Chaplygin Euler equations of compressible fluid flow tends to a delta-shock solution to the transport equations, and the intermediate density between them tends to a weighted -measure that forms a delta shock wave; any two-rarefaction-wave solution is shown to tend to two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution of the transport equations. Moreover, some numerical simulations completely coinciding with the theoretical analysis are presented.
This paper considers the steady translational motion of two collinear rigid spheres in an incompressible couple stress fluid. The two spheres are assumed to be of different sizes and are moving with two different speeds along a line axis connecting their centers. First, the general solution for the steady motion of an incompressible couple stress fluid past an axially symmetric particle is obtained analytically in the form of an infinite series. Second, the obtained solution is employed to construct the general solution for the steady motion of a couple stress fluid flow past two translating spheres using the principle of superposition. The boundary collocation technique is then used to satisfy the imposed boundary conditions on the surfaces of the two spheres. The normalized drag force acting on each of the two spherical particles is evaluated and presented numerically through tables and graphs. The tabulated results show that the convergence of the numerical results is achieved rapidly. In addition, it is observed that the increase in the couple stress viscosity parameter increases the values of the normalized drag force on each of the spherical particles.
The normal impingement of the rotational stagnation-point flow on a surface executing bi-axial stretching is studied. This problem is governed by two parameters, the dimensionless stretching along the -axis, and the dimensionless stretching along the -axis. Consideration is given to both stretching and shrinking surfaces. At fixed turning points are found at for which no solutions exist for and dual solutions exist for all . Shear stress results are presented in graphical form as are sample similarity velocity profiles. The boundary for the existence of solutions is presented over a range of the governing parameters. Shrinking in both directions occurs only over the relatively small region and . An investigation of the zero stress values of reveals a bifurcation in solutions for at that leads to triple solutions for This bifurcation is responsible for a discontinuity that appears in the lower branch of the curve at this value of .
Series solutions of the dynamics of capillary flows in a vertical circular tube are obtained by the homotopy analysis method (HAM). The model proposed by Maggi and Alonso-Marroquin (2012) is considered but with assumptions of constant contact angle and negligible air. In the case of capillary flows oscillating around the equilibrium height, the series solution of the penetration distance of the meniscus has the basis and . By contrast, in the case of capillary flows rising monotonically to the equilibrium height, the series solution of the penetration distance of the meniscus can have either the basis , if , or the basis . The computed velocity of the capillary flow is found to be larger than that of the experimental results. This discrepancy should be mainly caused by the neglect of both the variation of contact angle and the expulsion of air out of the capillary. Besides, the velocity of the capillary flow at the initial stage is determined by the Bond number and the viscosity effect becomes significant in the latter stage, which is consistent with the results of previous literature.
Planar stagnation-point flow normally impinging a rotating plate is studied. Symmetries show that along a circle of radius , the radial velocity can be computed from results in one quadrant and the angular velocity can be computed from results in two quadrants. The results of this study for various values of the dimensionless angular velocity are compared with axisymmetric Homann stagnation-point flow on a rotating plate first studied by Hannah.
An influence of a spatial temperature modulation of the heat release/consumption at the interface on nonlinear convective flows in the 47v2 silicone oil–water system, has been investigated. Periodic boundary conditions on the lateral walls, corresponding to an infinite two-layer system, have been considered. Transitions between the flows with different spatial structures, have been studied. It is shown that the spatial modulation can change the sequence of bifurcations and lead to the appearance of new oscillatory regimes. Specifically, pulsating traveling waves changing the direction of propagation, as well as regimes corresponding to period doubling, period-four and period-eight bifurcations, have been obtained.
The Fourier–Kochin representation of the oscillatory part of the flow pressure at the hull surface of a ship that travels at a constant speed in calm water of large depth – and the related Fourier–Kochin representations of the wave drag, hydrodynamic lift and pitch-moment experienced by the ship – are considered within the framework of the potential flow theory based on the Green function that satisfies the Kelvin–Michell linear boundary condition at the free surface. The convergence of the Fourier integrals in these Fourier–Kochin representations of the flow pressure, the wave drag, the lift and the pitch-moment are studied via a parametric numerical analysis, based on Hogner’s approximate theory, for two families of 120 simple ship models at Froude numbers within the range The analysis shows that the cutoff wavelength associated with negligible short waves is significantly influenced by the Froude number and the hull shape, and yields an analytical relation that explicitly determines in terms of and three major hull-shape parameters: beam/length ratio, draft/length ratio, and (nondimensional) bow or midship lengths.
A two-dimensional incompressible smoothed particle hydrodynamics scheme is used to simulate the interaction of micron-sized particles suspended in quiescent medium. A uniform electric field is applied to the particles, causing them to approach one another due to dielectrophoretic forces and form a chain. Both fluid and particles are assumed to be polarizable and non-conductive where the permittivity of the fluid is assumed to be lower than that of the particles. The numerical scheme is validated by comparing its predictions for simpler case of a pair of particles with results available in the literature. The effects of initial orientation, Reynolds number and differences in particle permittivity on the chaining behavior are studied afterwards. The results show that the particles may follow a convergent or divergent–convergent trajectory, depending on the initial orientation of the particle pair with the electric field. Increasing the field intensity in low-Reynolds regime expedites the chaining process without affecting the particle trajectory. However, the particles may diverge at larger Reynolds numbers. Assigning different permittivities to particles skews the chaining position toward the particle with lower permittivity. Simulating the process for multiple particles results in longer chains branching and encompassing the entire computational domain, much like those observed in experiments.
A Galerkin boundary cover method (GBCM) is developed for modeling of two-dimensional incompressible viscous fluid flows. In this approach, a combination of the finite cover system employed in the numerical manifold method with the Galerkin approximation of the boundary integral equations is presented. It is applicable to both interior and exterior domains due to its naturally variational formulations. In contrast to the domain-type approach, only the boundary data is required in the newly explained method, makes it suitable for the exterior problems. To increase the solution accuracy without refining the local mesh, different cover functions can be implemented on different covers; thereby, the p-adaptive computations can be carried out conveniently. Further, the boundary conditions are imposed accurately and the system matrices are both symmetric and positive definite. The given numerical examples demonstrate the high accuracy and convergence rates of the proposed methodology by using a few covers.
We describe the results of a direct numerical simulation inspired by laboratory experiments (Carstensen et al., 2012), which showed the formation of turbulent spots in an oscillatory boundary layer over a rough wall. Even though, differently from the experiments, the wall we consider is characterized by a regular roughness, turbulent spots with similar characteristics are observed. The numerical results provide information on transition to turbulence and on the early stages of formation of the turbulent spots. In particular, the formation of low-speed streaks and the subsequent generation of turbulent spots are described. The speed of the extreme points of the spots is obtained from the numerical results. Moreover, the effects of the wall roughness on the speed of the turbulent spots are discussed.
In this paper, we study a plane-parallel flow of the homogeneous inviscid incompressible fluid. The close attention is paid to the free boundary deformation under the forces of self-gravitation and fluid rotation. The problem is studied numerically, i.e., we propose the numerical algorithm for the free boundary calculation. The algorithm is based on the boundary element method. Algorithm applications include calculation of thin cumulative jets. The second point of interest is numerical results validation. The computational accuracy is controlled through the conservation laws. In this paper, we provide the derivation of conservation laws of energy and angular momentum for the case of rotating flows. In addition, we analytically investigate dynamics of two-dimensional near-equilibrium shapes and check the consistency of the analytical and numerical results.
In this paper, a droplet slipping along a filament under gravity is numerically simulated with a two-phase lattice Boltzmann method. In the simulations, a droplet encompasses the bottom end of the filament as an initial state. Here the contact angle hysteresis for non-ideal surface, prescribed by advancing angle and receding angle , is taken into account. Besides, the dimensionless parameters: Ohnesorge number ( ), Bond number ( ), and the initial wetting length ( ) are considered. The contact angle hysteresis tends to prevent the motion of contact lines, but gravity would drive the droplet to slip accompanied by its surface’s deformation. Accordingly, four dynamic modes for droplet moving on wetting filament and two for non-wetting filament have been identified for the first time. In addition, three maps of modes distribution depending on , , , and are shown clearly. It is indicated that both the contact angle hysteresis and the sharp point at the bottom end of filament, are able to result in the contact line pinning, which has decisive effect on the motion modes of droplet. It is also found that the breakup of droplet during the slipping process can only occur for wetting filament, but for non-wetting filament, the droplet either keeps suspended at the end of filament, or detaches from the filament completely. It is more likely for droplet to fall down from the filament at smaller . Adopting larger , the droplet tends to be hung at the end of wetting filament, but oppositely, for the non-wetting filament, the whole droplet is inclined to depart from filament totally. These results could be beneficial to the understanding of droplet’s dynamical behavior on filament.
In this paper, numerical study of emulsion of drops in a channel under uniform electric field is investigated. The electric field is created by imposing an electric potential difference across the channel walls. The leaky dielectric model developed by Taylor is used to compute the electric force. The electric conductivity ratio and electric permittivity ratio are used to study oblate and prolate drops The interaction of drop-to-wall and drop-to-drop are investigated by changing the dimensionless numbers. Oblate drops and prolate drops with , are always attracted to the walls, but prolate drops with , are attracted by or repelled from the walls depending on the initial distance and electric properties. Drop-to-drop interaction depends on the initial configuration of drops. Drop pairs that are placed at an angle relative to the direction of the electric field, are always attracted towards each other. If they have an initial separation distance in the direction of the electric field, they are attracted to the walls or towards each other depending on their separation distance. Prolate drops with , obtain a different configuration in their final state. They may stick together or have a small separation distance, depending on their electric properties. Increasing the electric capillary number, changes the vertical and horizontal separation distance of drop pairs at steady state configuration. It also decreases the time required to reach steady state configuration. The behavior of emulsion of drops is studied by considering oblate drops, prolate drops with and prolate drops with . All drops form fiber chains in the direction of the electric field that will break in stronger Poiseuille flow. Oblate drops either settle on the walls or migrate to the channel center line, but prolate drops form chains of fibers that sometimes extend between the walls. Presence of drops will change the well-known parabolic velocity profile especially for prolate drops that are suspended in the channel. The channel flow rate is also studied for different emulsion of drops and electric field. At a constant electric field and pressure gradient, the channel flow rate for oblate emulsion is higher than prolate ones. As the electric field strength increases, the channel flow rate will also increase for oblate emulsions.