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In this paper we propose and analyze a novel stream formulation of the virtual element method (VEM) for the solution of the Stokes problem. The new formulation hinges upon the introduction of a suitable stream function space (characterizing the divergence free subspace of discrete velocities) and it is equivalent to the velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beirão da Veiga et al., J. Comput. Phys., 228 (2009), pp. 7215–7232] (up to a suitable reformulation into the VEM framework). Both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem. Several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one.

Purpose The purpose of this paper is to investigate analytically the steady general three-dimensional stagnation-point flow of an aqueous titania-copper hybrid nanofluid past a circular cylinder that has a sinusoidal radius variation. Design/methodology/approach First, the analytic modeling of hybrid nanofluid is presented, and using appropriate similarity variables, the governing equations are transformed into nonlinear ordinary differential equations in the dimensionless stream function, which is solved by the well-known function bvp4c from MATLAB. Findings The current solution demonstrates good agreement with those of the previously published studies in the special cases of regular fluid and nanofluids. Graphical results are presented to investigate the influences of the titania and copper nanoparticle volume fractions and also the nodal/saddle indicative parameter on flow and heat transfer characteristics. Here, the thermal characteristics of hybrid nanofluid are found to be higher in comparison to the base fluid and fluid containing single nanoparticles. An important point to note is that the developed model can be used with great confidence to study the flow and heat transfer of hybrid nanofluids. Originality/value Analytic modeling of hybrid nanofluid is the important originality of present study. Hybrid nanofluids are potential fluids that offer better heat transfer performance and thermophysical properties than convectional heat transfer fluids (oil, water and ethylene glycol) and nanofluids with single nanoparticles. In this investigation, titania (TiO2, 50 nm), copper (Cu, 20 nm) and the hybrid of these two are separately dispersed into the water as the base fluid and analyzed.

Generalised two-dimensional (2-D) fluid dynamics is characterised by a relationship between a scalar field $q$ , called generalised vorticity, and the stream function $\\psi$ ,namely $q = (-\\nabla ^2)^{\\frac {\\alpha }{2}} \\psi$ . We study the transition of cascades in generalised 2-D turbulence by systematically varying the parameter $\\alpha$ and investigating its influential role in determining the directionality (inverse, forward or bidirectional) of these cascades. We derive upper bounds for the dimensionless dissipation rates of generalised energy $E_G$ and enstrophy $\\Omega _G$ as the Reynolds number tends to infinity. These findings corroborate numerical simulations, illustrating the inverse cascade of $E_G$ and forward cascade of $\\Omega _G$ for $\\alpha \\gt 0$ , contrasting with the reverse behaviour for $\\alpha \\lt 0$ . The dependence of dissipation rates on system parameters reinforces these observed transitions, substantiated by spectral fluxes and energy spectra, which hint at Kolmogorov-like scalings at large scales but discrepancies at smaller scales between numerical and theoretical estimates. These discrepancies are possibly due to non-local transfers, which dominate the dynamics as we go from positive to negative values of $\\alpha$ . Intriguingly, the forward cascade of $E_G$ for $\\alpha \\lt 0$ reveals similarities to three-dimensional turbulence, notably the emergence of vortex filaments within a 2-D framework, marking a unique feature of this generalised model.

We consider a slowly condensing droplet levitating near the surface of an evaporating layer, and develop a mathematical model to describe diffusion, heat transfer and fluid flow in the system. The method of separation of variables in bipolar coordinates is used to obtain the series expansions for temperature, vapour concentration and the Stokes stream function. This framework allows us to determine temperature profiles and condensation rates at the surface of the droplet, and to calculate the upward force that allows the droplet to levitate. Somewhat counter-intuitively, condensation is found to be the strongest near the bottom of the droplet, which faces the hot liquid layer. The experimentally observed deviations from the classical law predicting the square of the radius to grow linearly in time are explained by the model. A spatially non-uniform phase change rate results in a contribution to the force not considered in previous studies, and comparable to droplet weight and the upward force calculated from the Stokes drag law. The levitation conditions are formulated accordingly, resulting in the prediction of levitation height as a function of droplet size without any fitting parameters. A simple criterion is formulated to define the parameter ranges in which levitation is possible. The results are in good agreement with the experimental data except that the model tends to slightly underpredict the levitation height.

This work delves into the peristaltic rheology of two-wave sinusoidal cilia beating within a tubular pipe. Cilia movement drives the dynamic phenomenon of peristaltic fluid flow. Employing the traditional long-wavelength lubrication assumption, the flow equations are transformed into similarity form. The main objective is to take into account the true peristaltic-ciliary motion effects. We then derive analytical solutions for the radial and axial velocities of fluid particles within the tube. Notably, at this leading approximation level, the impacts of cilia beating are negligible, suggesting the motion is solely driven by peristaltic surface waves. However, analyzing the correction to the long-wavelength limit reveals the emergence of ciliated boundary effects through their largely eccentric elliptic paths. This correction enables us to extract expressions for the pressure gradient, stream function, axial and radial velocities, resultant pressure rise, and drag force, all based on the time-averaged mean flow rate across the pipe. Finally, we present a general discussion of fluid rheology due to cilia-assisted peristaltic motion, illustrated with informative graphical displays. It is shown that the drag force on the tube walls owing to the cilia beating waves in biology or biomedical applications necessitates addition of correction terms to the traditional long-wavelength adoption.

A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a ‘pure’ point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter $A$ in the hybrid solutions approaches the limits $A\\to 0,\\infty$. While $A\\to 0$ reproduces the input equilibrium, $A\\to \\infty$ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by $A$ as a ‘Liouville link’. In some cases, the emergent point vortex equilibrium as $A\\to \\infty$ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a ‘Liouville chain’. We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717) and by Krishnamurthy et al. (J. Fluid Mech., vol. 874, 2019, R1) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (Proc. R. Soc. A, vol. 476, 2020, 20200310) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the two-dimensional Euler equation.

Purpose The purpose of this study is to peruse natural convection in a CuO-water nanofluid-filled complex-shaped enclosure under the influence of a uniform magnetic field by using control volume finite element method. Design/methodology/approach Governing equations formulated in dimensionless stream function, vorticity and temperature variables using the single-phase nanofluid model with the Koo–Kleinstreuer–Li correlation for the effective dynamic viscosity and the effective thermal conductivity have been solved numerically by control volume finite element method. Findings Effects of various pertinent parameters such as Rayleigh number, Hartmann number, volume fraction of nanofluid and shape factor of nanoparticle on the convective heat transfer characteristics are analysed. It was observed that local and average heat transfer rates increase for higher value of Rayleigh number and lower value of Hartmann number. Among various nanoparticle shapes, platelets were found to be best in terms of heat transfer performance. The amount of average Nusselt number reductions was found to be different when nanofluids with different solid particle volume fractions were considered due to thermal and electrical conductivity enhancement of fluid with nanoparticle addition. Originality/value A comprehensive study of the natural convection in a CuO-water nanofluid-filled complex-shaped enclosure under the influence of a uniform magnetic field by using control volume finite element method is addressed.

We consider the dynamics of a two-dimensional incompressible perfect fluid on a Möbius strip embedded in $\\mathbb {R}^{3}$. The vorticity–stream function formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the Möbius strip and the distinction between forms and pseudo-forms this introduces lead to unusual properties: a boundary condition is provided by the conservation of circulation along the single boundary of the strip, and there is no integral conservation for the vorticity density or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the Möbius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability and decaying turbulence.

Purpose The purpose of this study is to investigate partial magnetic source (MS) effect on natural convection (NC) flow of a ferrofluid flow in a cavity with sinusoidally heated vertical walls. The combination of ferrohydrodynamics and magnetohydrodynamics due to the variable magnetic field (MF) and magnetite nanoparticles in one part of the cavity, and the classical NC in the other part of the cavity are concerned. Design/methodology/approach The dimensionless equations in stream function-vorticity form are numerically solved by radial basis functions (RBF) based collocation method. Findings A remarkable change in fluid flow and heat transfer is noted if the MS location is close to the left sinusoidally heated wall. In particular, the average Nusselt number is the smallest for the middle centered partial MF through the left wall at a large Hartmann number. Research limitations/implications RBF collocation approach is limited to small geometries due to the obtained solution globally in the entire domain of the problem. Practical implications If the partial restriction of the effect of MF is done in real life, it would be a control parameter at some required/requested areas of the concerned problem. Social implications This is a physical problem. Originality/value If the proposed idea of partial variable MF is able to be applied to a system in real life, it would be a good controller on fluid flow and heat transfer. RBF-based methods are also alternative numerical procedures to solve heat transfer and fluid flow problems.

This study investigates the behaviour of subtropical high-pressure systems and the Hadley cell, which affect the weather of South Africa, using the ERA-Interim database and ensemble of 14 global circulation models from Phase 6 of the Coupled Model Intercomparison Project (CMIP6). Mass stream function was used to represent the Hadley cell. To analyse the behaviour of the subtropical anticyclones, monthly sea level pressure, the 1018 hPa isobar and the maximum isobar in the study area were used. The seasonal variation of the anticyclones and Hadley circulation is consistent with rainfall over South Africa. During austral summer, a less intense, narrow mass stream function, South Atlantic Subtropical Anticyclone and Mascarene High are located more southwards, causing rainfall over the eastern parts of South Africa. During the austral winter, Hadley circulation, as well as the anticyclones, is stronger and located more northwards, causing rainfall over the southern and southwestern parts of South Africa.