Prandtl Type Research Articles

Global existence and decay estimates of solutions for the compressible Prandtl type equations with small analytic data

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G=defu+y2〈t〉φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ, which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data.

Global well-posedness of 2D Hyperbolic perturbation of the Navier–Stokes system in a thin strip

In this paper, we study a hyperbolic version of the Navier–Stokes equations, obtained by using the approximation by relaxation of the Euler system, evolving in a thin strip domain. The formal limit of these equations is a hyperbolic Prandtl type equation, our goal is to prove the existence and uniqueness of a global solution to these equations for analytic initial data in the tangential variable, under a uniform smallness assumption. Then we justify the limit from the anisotropic hyperbolic Navier–Stokes system to the hydrostatic hyperbolic Navier–Stokes system with small analytic data.

Some effective solutions for Prandtl's type integro‐differential equation

Prandtl's type integro‐differential equations with different coefficients are investigated. Using the methods of the theory of analytic functions and integral transformations, the singular integro‐differential equation is reduced to boundary value problems of the theory of analytic functions. Effective solutions of this equation and the corresponding asymptotic estimates are obtained.

On the stability of shear flows of Prandtl type for the steady Navier-Stokes equations

In this paper, we study the stability of shear flows of Prandtl type as $$\left({U\left({y/\sqrt \nu} \right),0} \right)$$ for the steady Navier-Stokes equations under a natural spectral assumption on the linearized NS operator. The key ingredient is to solve the Orr-Sommerfeld equation. For this, we develop a direct energy method combined with the compactness method, which may be of independent interest.

Global hydrostatic approximation of the hyperbolic Navier-Stokes system with small Gevrey class 2 data

We investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations, which is obtained by using the Cattaneo type law instead of the Fourier law, evolving in a thin strip ℝ × (0, ε). The formal limit of these equations is a hyperbolic Prandtl type equation. We first prove the global existence of solutions to these equations under a uniform smallness assumption on the data in the Gevrey class 2. Then we justify the limit globally-in-time from the anisotropic hyperbolic Navier-Stokes system to the hyperbolic Prandtl system with such Gevrey class 2 data. Compared with Paicu et al. (2020) for the hydrostatic approximation of the 2-D classical Navier-Stokes system with analytic data, here the initial data belongs to the Gevrey class 2, which is very sophisticated even for the well-posedness of the classical Prandtl system (see Dietert and Gérard-Varet (2019) and Wang et al. (2021)); furthermore, the estimate of the pressure term in the hyperbolic Prandtl system give rise to additional difficulties.

Optimal convergence rate of the vanishing shear viscosity limit for compressible Navier-Stokes equations with cylindrical symmetry

We consider the initial boundary value problem for the isentropic compressible Navier-Stokes equations with cylindrical symmetry. The existence of boundary layers is well-known when the shear viscosity vanishes. In this paper, we derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate of the vanishing shear viscosity limit without any smallness assumption on the initial and boundary data.

Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces

In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the monotonicity condition on the velocity field. Moreover, we show that if the tangential magnetic field of shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Gevrey function space provided that the initial velocity shear flow is non-degenerately critical at the same point.

On the Boundary Layer in the Case of Viscous Fluid Flowing Around Confuser with the Ladyzhenskaya Rheological Law

We consider the Prandtl type boundary layer equations for a fluid with the nonlinear Ladyzhenskaya rheological equation. In a nonstandard case of a fluid flow through a confuser (a tube of variable cross-section), we study the behavior of the boundary layer and derive an asymptotic estimate for the solution.

Mechanistic modelling of cyclic voltage-capacity response for lithium-ion batteries

One of the challenging tasks related to lithium-ion batteries (LIBs) remains a comprehensive approach for battery behaviour modelling. An approach is presented that enables modelling the voltage-capacity response of LIBs that are subjected to variable temperature and current load histories. A detailed presentation of the developed macro-scale phenomenological model embedding the mechanistic properties of the Prandtl type hysteresis operator and the concept of the force-voltage analogy is made. The necessary input data preparation for the model calibration is also presented. Accuracy of the model is confirmed with experimental observations for both nested current load history at two different temperatures and for arbitrary current load history. The same measured data is used to calibrate and to simulate response of the first order Thevenin equivalent circuit topology in order to amply compare the obtained results.

Sobolev Stability of Prandtl Expansions for the Steady Navier–Stokes Equations

We show the $H^1$ stability of shear flows of Prandtl type: $U^\nu = (U_s(y/\sqrt{\nu}),0)$, in the steady two-dimensional Navier-Stokes equations, under the natural assumptions that $U_s(Y) > 0$ for $Y > 0$, $U_s(0) = 0$, and $U_s'(0) > 0$. Our result is in sharp contrast with the unsteady ones, in which at most Gevrey stability can be obtained, even under global monotonicity and concavity hypotheses. It provides the first positive answer to the inviscid limit problem in Sobolev regularity for a non-trivial class of steady Navier-Stokes flows with no-slip boundary condition.

Convergence of boundary layers for the Keller–Segel system with singular sensitivity in the half-plane

Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63]), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by ε≥0. Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with ε=0) plus the inner layer as ε→0, where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63], and open a new window in the future theoretical study of chemotaxis models.

On the steady Prandtl type equations with magnetic effects arising from 2D incompressible MHD equations in a half plane

In this paper, we study a steady Prandtl type boundary layer system of equations, which is derived from the two-dimensional incompressible magnetohydrodynamics equations without magnetic diffusion under the no-slip boundary condition. By the classical von Mises transformation, this Prandtl type system is reduced to a single second order quasilinear parabolic equation, in which the coefficient depends on the strength of magnetic field. For different ratios of the magnetic field and the velocity field, we prove the global existence of solutions to the boundary layer system and the non-existence of global smooth solutions, respectively.

Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows

We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^\nu(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrt{\nu}})\,, \, 0 \big )\, , \quad 0<\nu \ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrt{\nu}$ then $u^\nu$ is stable over some time interval $(0,T)$, $T$ independent of $\nu$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$). Moreover, in the case where $U^{BL}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr-Sommerfeld operator.

Bearing Capacity Factors for Isolated Surface Strip Footing Resting on Multi-layered Reinforced Soil Bed

In this paper, the ultimate bearing capacity of an isolated surface strip footing resting on a reinforced c–ϕ soil bed is determined using the upper-bound limit analysis. The soil is assumed to obey the Mohr–Coulomb failure criteria along with associated flow rule. A Prandtl type failure mechanism along with kinematically admissible velocity field is considered in this analysis. The variation of bearing capacity is studied with respect to the angle of internal friction of soil (ϕ) and the number of reinforcement layers (m). The results obtained from the upper bound limit analysis are compared with the finite element solution performed separately using PLAXIS 2D. The present theoretical observations are generally found in good agreement with those results available in literature.

Consistent equations for open-channel flows in the smooth turbulent regime with shearing effects

Consistent equations for turbulent open-channel flows on a smooth bottom are derived using a turbulence model of mixing length and an asymptotic expansion in two layers. A shallow-water scaling is used in an upper – or external – layer and a viscous scaling is used in a thin viscous – or internal – layer close to the bottom wall. A matching procedure is used to connect both expansions in an overlap domain. Depth-averaged equations are then obtained in the approximation of weakly sheared flows which is rigorously justified. We show that the Saint-Venant equations with a negligible deviation from a flat velocity profile and with a friction law are a consistent set of equations at a certain level of approximation. The obtained friction law is of the Kármán–Prandtl type and successfully compared to relevant experiments of the literature. At a higher precision level, a consistent three-equation model is obtained with the mathematical structure of the Euler equations of compressible fluids with relaxation source terms. This new set of equations includes shearing effects and adds corrective terms to the Saint-Venant model. At this level of approximation, energy and momentum resistances are clearly distinguished. Several applications of this new model that pertains to the hydraulics of open-channel flows are presented including the computation of backwater curves and the numerical resolution of the growing and breaking of roll waves.

Boundary layer problem and zero viscosity-diffusion limit of the incompressible magnetohydrodynamic system with no-slip boundary conditions

The purpose of this paper is to study the boundary layer problem and zero viscosity-diffusion limit of the initial boundary value problem for the incompressible viscous and diffusive magnetohydrodynamic (MHD) system with (no-slip characteristic) Dirichlet boundary conditions and to prove that the corresponding Prandtl's type boundary layer are stable with respect to small viscosity-diffusion coefficients. The main difficulty here comes from Dirichlet boundary conditions for the velocity and magnetic field. Firstly, we identify a non-trivial class of initial data for which we can establish the uniform stability of the Prandtl's type boundary layers and prove rigorously the solution of incompressible viscous-diffusion MHD system converges strongly to the sum of the solution to the ideal MHD system and the approximating solution to Prandtl's type boundary layer equation by using the elaborate energy methods and the special structure of the solution to ideal MHD system with the initial data we identify here, which yields that there exists the cancellation between the boundary layer of the velocity and that of the magnetic field. Secondly, for general initial data, we obtain zero viscosity-diffusion limit of the incompressible viscous and diffusive MHD system with the different horizontal and vertical viscosities and magnetic diffusions, when they go to zero with the different speeds, and, we prove rigorously the convergence of the incompressible viscous and diffusion MHD system to the ideal MHD system or the anisotropic MHD system by constructing the exact boundary layers and using the elaborate energy methods. We also mention that these results obtained here should be the first rigorous ones on the stability of Prandtl's type boundary layer for the incompressible viscous and diffusion MHD system with no-slip characteristic boundary condition.

Dissipated energy-based fatigue lifetime calculation under multiaxial plastic thermo-mechanical loading

The paper concerns the metallic material lifetime calculation under arbitrary multiaxial thermo-mechanical fatigue loading. An energy-based damage operator approach ([Formula: see text]) is introduced. In such an approach, fatigue lifetime is continuously computed by the Prandtl type damage operator. The dissipated energy of plastic deformation is applied as a damage parameter that is obtained at any moment by the recently developed multiaxial energy operator approach. Strain-life curves are transformed into energy-cycle damage curves in order to associate the damage parameter with the fatigue lifetime. The present approach is validated on turbocharger housing. A satisfactory evaluation of the fatigue lifetime is obtained even though visco-plasticity and mean stress correction are not addressed yet.

Review of Models Introduced to Estimate the Distribution of Longitudinal Velocity in Open Channels based on Navier-stokes Equations

Determination of the velocity profile in open channels (particularly in narrow channels) with turbulent flows has always been the subject of attention and studies of researchers. This is due to the fact that knowing the velocity distribution it is possible to calculate the distributions of shear stress and the discharge in the channels. However, due to the complexity of conditions governing the flow, which are the consequences of the non-isotropic state of turbulence and the existence of Prandtl type 2 secondary flows; it is difficult to introduce a model that would be capable of properly defining the velocity distribution in narrow open channels. Many researchers have tried to modify the famous logarithmic law, which responds well in the case of wide open channels, to suit the narrow channels as well, thereby allowing the modified model to describe the negative gradient of velocity beneath the free surface of water (known as dip phenomenon). However, in this context, they have been forced to set limitations on their models, which have in turn limited their applications. For this reason by using the laws of mathematics and simplifying the Navier-Stokes equations, some researchers have presented models that are capable of calculating the velocity profile in narrow channels in a satisfactory manner with the least constraint for application. Two models that are derived from Navier-Stokes are introduced in this study and their results are compared with experimental data.

Energy dissipation under multiaxial thermomechanical fatigue loading

The paper presents an approach to the energy dissipation calculation under arbitrary multiaxial thermomechanical fatigue (TMF) loading. In such an approach the total area of plastic hysteresis loops is taken as a measure of dissipated energy. The calculation is based on the concept of the developed temperature dependent Prandtl type operator. Energy dissipation is associated to irreversible dislocation movements represented by slider shifts of three independent operators. The dissipated energy is then obtained continuously at any time by collecting dissipated energy increments of each operator. It is shown that the multiaxial operator approach gives us the same total energy of plastic deformation as compared to the classical integration approach. Furthermore, the presented approach enables to automatically split the obtained dissipated energy between the “true” dissipated energy and the elastically “stored” energy. In order to satisfy the request for a minimum number of dedicated material tests, the approach assumes fixed principal directions. Therefore, the proportional as well as non-proportional loading conditions are addressed in the same manner, which is currently the main deficiency of the approach.

Boundary layers for the upper convected Maxwell fluid

We discuss boundary layers arising in the high Weissenberg number limit of viscoelastic flows and the scalings that arise in analyzing them. Two quite distinct mechanisms for the formation of viscoelastic boundary layers exist: One mechanism is analogous to that of the Prandtl boundary layer and is linked to the enforcement of the no-slip boundary condition. The other mechanism has nothing to do with the no-slip condition, but is linked to long memory which manifests itself in different stresses near the wall versus some distance from the wall. In certain situations, these stress boundary layers may be embedded within Prandtl type boundary layers.