Viscous Case Research Articles

Piezo-viscous micropolar fluid flow between two parallel annular disks

A theoretical analysis of the piezo-viscous dependence of micropolar fluid flow between parallel annular disks has been conducted. It is assumed that viscosity changes exponentially with pressure. Pressure is analytically approximated using a small perturbation method with the viscosity coefficient as the perturbation parameter. Squeeze film time, pressure, and load carrying capacity are calculated and graphically demonstrated across various viscosity–pressure characteristics. The study examines how an increase in cross-viscosity correlates with notable growth in pressure, response time, and bearing load. According to the results, micropolar fluid offers an increase in the load-bearing capacity and therefore lengthens the response time to prevent the contact of parallel annular disks. These observations are useful for bearings, seals, and lubrication systems, where controlling viscosity and pressure is crucial for smooth operation and productivity. As the limiting case of micro-polar fluid when the coupling number tends to zero or the couple stress parameter approaches infinity, without the annular ring, the present derivation agrees well with the Newtonian viscous case of parallel planar squeezing disks described by Bird et al. [Transport Phenomena (John Wiley & Sons, 2006)].

Computational analysis of radiative flow of power law fluid with heat generation effects: Galerkin finite element simulations

This research aims to presents the free convective flow power law fluid due to vertical cone. The investigation for observing the heat transfer phenomenon is accounted to heat generation and radiative effects. The assumption of variable viscosity is taken into account. A vertical cone induced the flow. The dimensionless variables are followed to attains the simplified form. The numerical computations are performed with help of famous finite element method (FEM). The FEM algorithm is supported with applications of quadratic Lagrange polynomials. The results are confirmed with peak accuracy. The physical aspect of problem is presented in view of shear thickening, shear thinning and viscous material case. A comparative thermal reflection in absence and presence of heat generation have been endorsed. Moreover, the insight of various parameters on Nusselt number is also presented.

The adjoint Rayleigh and Orr-Sommerfeld equations: Green function and eigenmodes

The Rayleigh and Orr-Sommerfeld equations are ODEs which arise from the linearized Euler and Navier-Stokes equation around a shear flow. In this paper, we consider the adjoints of the Rayleigh and Orr-Sommerfeld equations on [0,∞) with respect to the complex L2 product. In the viscous case, we consider a family of viscosity-dependent Navier boundary conditions, which in the limit corresponds to the no-slip condition. We rigorously establish existence and asymptotic properties of their eigenvalues, eigenmodes and Green functions away from critical layers. The adjoint operators are useful because they also allow us to deduce properties about the kernels and images of the original operators.

Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

The stability characteristics of a thin film of viscoelastic (Walters’ B′ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number kc which include the viscoelastic parameter Γ, angle of inclination of the plane θ, Reynolds number Re and Weber number We. The weakly nonlinear stability analysis that is based on the second Landau constant J2, reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is 3(1+3Γ)Re−3cotθ−4ReWek2 = 0. It is found that all the four distinct flow zones of this problem exist in Re-k-, θ-k- and Γ-k-plane after the critical value of Rec,θc and Γc, respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of Γ irrespective of the values of Re and θ, as for example, a solution of polyisobutylene in cetane, compared with the viscous (Γ=0) film flow case. Finally, we scrutinize the effect of Γ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

Standing waves and jets on a sessile, incompressible bubble

We show numerically that large amplitude, shape deformations imposed on a spherical-cap, incompressible, sessile gas bubble pinned on a rigid wall can produce a sharp, wall-directed jet. For such a bubble filled with a permanent gas, the temporal spectrum for surface-tension-driven, linearized perturbations has been studied recently in D. Ding and J. Bostwick [“Oscillations of a partially wetting bubble,” J. Fluid Mech. 945, A24 (2022)]. in the potential flow limit. We reformulate this as an initial-value problem analogous in spirit to classical derivations in the inviscid limit by Kelvin [“Oscillations of a liquid sphere,” Math. Phys. Papers 3, 384–386 (1890)], Rayleigh [“On the instability of jets,” Proc. London Math. Soc. s1-10, 4–13 (1878)] or by Prosperetti [“Viscous effects on small-amplitude surface waves,” Phys. Fluids 19, 195–203 (1976)] and Prosperetti [“Motion of two superposed viscous fluids,” Phys. Fluids 24, 1217–1223 (1981)] for the viscous case. The first test of linear theory is reported here by distorting the shape of the pinned, spherical cap employing eigenmodes obtained from linearized theory. These are employed as the initial shape distortion of the bubble in numerical simulations. It is seen that linearized predictions show good agreement with nonlinear simulations at small distortion amplitude producing standing waves, which oscillate at the predicted frequency. Beyond the linear regime as the shape distortions are made sufficiently large, we observe the formation of a dimple followed by a slender, wall-directed jet, analogous to similar jets observed in other geometries from collapsing wave troughs [Farsoiya et al., “Axisymmetric viscous interfacial oscillations–theory and simulations,” J. Fluid Mech. 826, 797–818 (2017) and Kayal et al., “Dimples, jets and self-similarity in nonlinear capillary waves,” J. Fluid Mech. 951, A26 (2022).] This jet can eject with an instantaneous velocity exceeding nearly 20 times that predicted by linear theory. By projecting the shape of the bubble surface around the time instant of jet ejection, into the eigenspectrum we show that the jet ejection coincides with the nonlinear spreading of energy into a large number of eigenmodes. We further demonstrate that the velocity-field associated with the dimple plays a crucial role in evolving it into a jet and without which, the jet does not form. It is also shown that evolving the bubble shape containing a dimple but zero initial velocity-field everywhere, via linear theory, does not produce the jet. These conclusions accompanied by first principles analysis provide insight into the experimental observations of Prabowo and Ohl [“Surface oscillation and jetting from surface attached acoustic driven bubbles,” Ultrason. Sonochem. 18, 431–435 (2011)], where similar jets were reported earlier, albeit via acoustic forcing. Our inferences also complement well-known results of Naude and Ellis [“On the mechanism of cavitation damage by nonhemispherical cavities collapsing in contact with a solid boundary,” J. Fluids Eng. 83, 648–656 (1961)] and Plesset and Chapman [“Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary,” J. Fluid Mech. 47, 283–290 (1971)] demonstrating that wall-directed jets can be generated from volume preserving, shape deformations of a pinned bubble.

Insight into acoustic wave-driven gas bubble dynamics in Williamson's fluid

This study's motivation on the behavior of a gas bubble and surrounding fluid during the collapse phase of an acoustic wave leads to the emission of light, which can be modelled using the Rayleigh-Plesset equation. The significant contribution to the major addition to the understanding of gas bubble dynamics in viscoelastic fluids. The results of the study have the potential to impact a wide range of applications, from sonar and underwater acoustic propagation to biomedical applications and industrial operations. Understanding the dynamics of acoustic wave-driven gas bubble oscillations in non-Newtonian fluids is crucial for various applications in medical imaging, therapy, and microfluidics. The aim of this study is to discover the conditions under which these bubbles can undergo sonoluminescence, the emission of light due to the collapse of bubbles under severe sonic pressure. The complex ordinary differential equation is solved numerically, and graphs are generated to evaluates various parameters such as velocity, radius, and pressure. A comparison with the viscous case is also presented to assess the influence of Newtonian and non-Newtonian fluids. A program designed by Mathematica software (13.0) “parametric NDSolve package” is used to solve the equations. The numerical solution supports the boundary conditions, indicating stability and reliability. Notably, the study identifies distinct fluctuations in the phase and stable characteristics associated with non-Newtonian effects. Interestingly, certain fluid parameters lead to discrete group modulation of radial excursions, revealing a novel and previously unknown phenomenon. It is concluded that Higher Reynolds numbers result in turbulent flow, which causes the bubble to enlarge and increase its volume. Furthermore the pressure gradient is affected by magnetic drag force, and as size of bubble increases, required pressure gradient to counteract the drag force also increases. As the magnetic field intensity increases, the velocity of the bubble decreases. Also, considering the implications of these results, the study explores their relevance to medical ultrasonography applications.

Energy transfer and third-order law in forced anisotropic magneto-hydrodynamic turbulence with hyper-viscosity

The third-order law links energy transfer rates in the inertial range of magneto- hydrodynamic (MHD) turbulence with third-order structure functions. Anisotropy, a typical property in the solar wind, challenges the applicability of the third-order law with the isotropic assumption. To shed light on the energy transfer process in the presence of anisotropy, we conducted direct numerical simulations of forced MHD turbulence with normal and hyper-viscosity under various strengths of the external magnetic field ( $B_0$ ), and calculated three forms of third-order structure function with or without averaging of the azimuthal or polar angles with respect to $B_0$ direction. Correspondingly, three estimated energy transfer rates were obtained. The result shows that the peak of normalized third-order structure function occurs at larger scales closer to the $B_0$ direction, and the maximum of longitudinal transfer rates shifts away from the $B_0$ direction at larger $B_0$ . Compared with normal viscous cases, hyper-viscous cases can attain better separated inertial range, thus facilitating the estimation of the energy cascade rates. We find that the widespread use of the isotropic form of the third-order law in estimating the energy transfer rates is questionable in some cases, especially when the anisotropy arising from the mean magnetic field is inevitable. In contrast, the direction-averaged third-order structure function properly accounts for the effect of anisotropy and predicts the energy transfer rates and inertial range accurately, even at very high $B_0$ . With limited statistics, the third-order structure function shows a stronger dependence on averaging of azimuthal angles than the time, especially for high $B_0$ cases. These findings provide insights into the anisotropic effect on the estimation of energy transfer rates.

Energy conservation for weak solutions of incompressible fluid equations: The Hölder case and connections with Onsager's conjecture

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way.Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one –in the viscous case– with Hölder type hypotheses, but without additional assumptions on the pressure.

On Inhibition of the Rayleigh-Taylor Instability by a Horizontal Magnetic Field in 2D Non-Resistive MHD Fluids: The Viscous Case

On Inhibition of the Rayleigh-Taylor Instability by a Horizontal Magnetic Field in 2D Non-Resistive MHD Fluids: The Viscous Case

A wave appropriate discontinuity sensor approach for compressible flows

In this work, we propose a novel selective discontinuity sensor approach for numerical simulations of the compressible Navier–Stokes equations. Since transformation to characteristic space is already a common approach to reduce high-frequency oscillations during interpolation to cell interfaces, we exploit the characteristic wave structure of the Euler equations to selectively treat the various waves that the equations comprise. The approach uses the Ducros shock sensing criterion to detect and limit oscillations due to shocks while applying a different criterion to detect and limit oscillations due to contact discontinuities. Furthermore, the method is general in the sense that it can be applied to any method that employs characteristic transformation and shock sensors. However, in the present work, we focus on the gradient-based reconstruction family of schemes. A series of inviscid and viscous test cases containing various types of discontinuities are carried out. The proposed method is shown to markedly reduce high-frequency oscillations that arise due to improper treatment of the various discontinuities; i.e., applying the Ducros shock sensor in a flow where a strong contact discontinuity is present. Moreover, the proposed method is shown to predict similar volume-averaged kinetic energy and enstrophy profiles for the Taylor–Green vortex simulation compared to the base Ducros sensor, indicating that it does not introduce unnecessary numerical dissipation when there are no contact discontinuities in the flow.

Asymptotic behaviors for the Jordan–Moore–Gibson–Thompson equation in the viscous case

In this paper, we study large-time behaviors for a fundamental model in nonlinear acoustics, precisely, the viscous Jordan–Moore–Gibson–Thompson (JMGT) equation in the whole space Rn. This model describes nonlinear acoustics in perfect gases under irrotational flow and equipping Cattaneo’s law of heat conduction. By employing refined WKB analysis and Fourier analysis, we derive first- and second-order asymptotic profiles of solution to the Moore–Gibson–Thompson (MGT) equation as t≫1, which illustrates novel optimal estimates for the solutions even subtracting its profiles. Concerning the nonlinear JMGT equation, via suggesting a new decomposition of nonlinear portion, we investigate the existence and large-time profiles of global (in time) small data Sobolev solutions with suitable regularity. These results help bridge a new connection between the JMGT equation and diffusion-waves as t≫1.

Nonintrusive Aerodynamic Shape Optimisation with a POD-DEIM Based Trust Region Method

This work presents a strategy to build reduced-order models suitable for aerodynamic shape optimisation, resulting in a multifidelity optimisation framework. A reduced-order model (ROM) based on a discrete empirical interpolation (DEIM) method is employed in lieu of computational fluid dynamics (CFD) solvers for fast, nonlinear, aerodynamic modelling. The DEIM builds a set of interpolation points that allows it to reconstruct the flow fields from sets of basis obtained by proper orthogonal decomposition of a matrix of snapshots. The aerodynamic reduced-order model is completed by introducing a nonlinear mapping function between surface deformation and the DEIM interpolation points. The optimisation problem is managed by a trust region algorithm linking the multiple-fidelity solvers, with each subproblem solved using a gradient-based algorithm. The design space is initially restricted; as the optimisation trajectory evolves, new samples enrich the ROM. The proposed methodology is evaluated using a series of transonic viscous test cases based on wing configurations. Results show that for cases with a moderate number of design variables, the approach proposed is competitive with state-of-the-art gradient-based methods; in addition, the use of trust region methodology mitigates the likelihood of the optimiser converging to, shallower, local minima.

Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface

Under consideration here are time-dependent three-dimensional stratified water flows of finite depth with a free surface and an interface (playing the role of an internal wave and separating two layers of constant and different densities). The main outcome of the paper is that, under the assumption that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other, we prove that bounded solutions to the three-dimensional equations are essentially two-dimensional. More precisely, the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. In addition, one of the horizontal components of the velocity field is constant throughout the flow and the vorticity also points in the direction orthogonal to the wave propagation direction. The rigid lid case is analyzed from the perspective of relaxed assumptions concerning the vorticity vectors. We also consider the viscous water wave problem case with normal and tangential stress conditions at the free surface and find that under similar conditions on the vorticity vectors, the conclusion about the two-dimensionality remains in place. Along the way, we also provide an explicit solution for the time-dependent three-dimensional inviscid water wave problem with an interface.

High-Precision Flow Field Simulation of Aerostatic Bearings Based on the Interior Penalty Discontinuous Galerkin Method

All the static performance, dynamic characteristics, and stability are strongly associated with the flow field inside the aerostatic bearings. Therefore, a high-precision numerical method is beneficial for the detailed description of the bearing flow field. To this end, a modified interior penalty discontinuous Galerkin method was introduced here. Actually, a lift operator was included to eliminate the so-called homogeneity tensor connecting the viscous term and variable gradient, which could improve the numerical feasibility. The accuracy of the above numerical method has been comprehensively validated through viscous cases, including Couette flow and shear-driven cavity flow. Then, the flow fields of three aerostatic bearings were simulated with different orifice geometries. As a result, the Mach number distributions and static pressure could be estimated together with the integration of the pressure acting upon the thrust surface. The acceleration within the orifice and air film could be detected, and the influence of the orifice geometry has been systematically discussed.

Bioconvective Homann flow of tangent hyperbolic nanofluid due to spiraling disk with convective and zero mas flux constraints

The phenomenon of bioconvection in nanomaterials presents novel applications in the biotechnology, biofuels, enzymes, biomedical engineering and energy systems. Current exploration explores the applications of bioconvection in Homann flow of nanofluid due to spiraling of disk theoretically. The generalized model namely tangent hyperbolic fluid is used to predicts the rheological and thermal impact. The stability of nanofluid is ensured with interaction of motile microorganisms. The Boungrino nanofluid model with thermophoresis and Brownian motion features is used to perform the analysis The thermal distribution of nanofluids is proceeded by utilizing the zero mass thermal constraints. The similarity variables are introduced in order to estimating the dimensionless formulation. The Keller Box method with higher efficiency is imposed with implicit finite difference numerical algorithm. The main observations reveal that with enhancing the radial velocity and azimuthal velocity decreases with increasing sparling angle. For highly viscous case, a decrement in the azimuthal velocity has been observed.

The role of time integration in energy conservation in Smoothed Particle Hydrodynamics fluid dynamics simulations

The choice of a time integration scheme is a crucial aspect of any transient fluid simulation, and Smoothed-Particle Hydrodynamics (SPH) is no exception. The influence of the time integration scheme on energy balance is here addressed. To do so, explicit expressions allowing to compute the deviations from the energy balance, induced by the time integration scheme, are provided. These expressions, computed a posteriori, are valid for different integration methods. Besides, a new formulation that improves energy conservation by enhancing stability, based on an implicit integration scheme, is proposed. Such formulation is tested with the simulation of a two-dimensional non-viscous impact of two jets, with no artificial dissipation terms. To the best of our knowledge, this is the first stable simulation of a non-dissipative system with a weakly-compressible SPH method. A viscous case, the Taylor–Green vortex, has also been simulated. Results show that an implicit time integration scheme also behaves better in a viscous context.

Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation

The one-dimensional Kardar-Parisi-Zhang (KPZ) equationis becoming an overarching paradigm for the scaling of nonequilibrium, spatially extended, classical and quantum systems with strong correlations. Recent analytical solutions have uncovered a rich structure regarding its scaling exponents and fluctuation statistics. However, the zero surface tension or zero viscosity case eludes such analytical solutions and has remained ill-understood. Using numerical simulations, we elucidate a well-defined universality class for this case that differs from that of the viscous case, featuring intrinsically anomalous kinetic roughening (despite previous expectations for systems with local interactions and time-dependent noise) and ballistic dynamics. The latter may be relevant to recent quantum spin chain experiments which measure KPZ and ballistic relaxation under different conditions. We identify the ensuing set of scaling exponents in previous discrete interface growth models related with isotropic percolation, and show it to describe the fluctuations of additional continuum systems related with the noisy Korteweg-de Vries equation. Along this process, we additionally elucidate the universality class of the related inviscid stochastic Burgers equation.

New Thought on Matsumura–Nishida Theory in the L_p–L_q Maximal Regularity Framework

This paper is devoted to proving the global well-posedness of initial-boundary value problem for Navier–Stokes equations describing the motion of viscous, compressible, barotropic fluid flows in a three dimensional exterior domain with non-slip boundary conditions. This was first proved by an excellent paper due to Matsumura and Nishida (Commun Math Phys 89:445–464, 1983). In [10], they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to L_2 in space-time, detailed statement of Matsumura and Nishida theorem is given in Theorem 1 of Sect. 1 of context. This requirement is essentially used to estimate the L_infty norm of necessary order of derivatives in order to enclose the iteration scheme with the help of Sobolev inequalities and also to treat the material derivatives of the mass density. On the other hand, this paper gives the global wellposedness of the same problem as in [10] in L_p (1 <p le 2) in time and L_2cap L_6 in space maximal regularity class, which is an improvement of the Matsumura and Nishida theory in [10] from the point of view of the minimal requirement of the regularity of solutions. In fact, after changing the material derivatives to time derivatives by Lagrange transformation, enough estimates obtained by combination of the maximal L_p (1 <p le 2) in time and L_2cap L_6 in space regularity and L_p–L_q decay estimate of the Stokes equations with non-slip conditions in the compressible viscous fluid flow case enable us to use the standard Banach’s fixed point argument. Moreover, one of the purposes of this paper is to present a framework to prove the L_p–L_q maximal regularity for parabolic-hyperbolic type equations with non-homogeneous boundary conditions and how to combine the maximal L_p–L_q regularity and L_p–L_q decay estimates of linearized equations to prove the global well-posedness of quasilinear problems in unbounded domains, which gives a new thought of proving the global well-posedness of initial-boundary value problems for systems of parabolic or parabolic-hyperbolic equations appearing in mathematical physics.

Asymptotic Stability of Solutions to a Hyperbolic-Elliptic Coupled System of Radiating Gas on the Half Line

This paper is concerned with the asymptotic stability of the solution to an initial-boundary value problem on the half line for a hyperbolic-elliptic coupled system of radiating gas, where the data on the boundary and at the far field state are defined as $u_-$ and $u_+$ satisfying $u_-<u_+$. For the scalar viscous conservation law case, it is known by the work of Liu, Matsumura, and Nishihara [SIAM J. Math. Anal., 29 (1998), pp. 293--308] that the solution tends toward a rarefaction wave or stationary solution or superposition of these two kinds of waves depending on the distribution of $u_\pm$. Motivated by their work, we prove the stability of the above three types of wave patterns for the hyperbolic-elliptic coupled system of radiating gas with small perturbation. A singular phase plane analysis method is introduced to show the existence and the precise asymptotic behavior of the stationary solution, especially for the degenerate case: $u_-<u_+=0$ such that the system has inevitable singularities. The stability of the rarefaction wave, the stationary solution, and their superposition is proved by applying the standard $L^2$-energy method.