Tangential Boundary Research Articles

Micro/nano thermal boundary layer equations with slip creep jump boundary conditions

At the micro- and nanoscale, the standard continuity boundary conditions at fluid-solid interfaces of classical transport phenomena do not apply and must be replaced by boundary conditions that allow discontinuities. In this study, the classical thermal laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition for tangential velocity and continuous temperature boundary conditions replaced by non-linear slip-creep-jump boundary conditions. These boundary conditions contain an arbitrary index parameter, denoted by n > 0, which appears in the coefficients of the coupled ordinary differential equations to be solved. As an independent check on the numerical procedure, the case of a boundary layer formed in a convergent channel with a sink, which corresponds to n = 1/2, is solved analytically for various values of the Prandtl number and zero Brinkham number. Other values of n for n > 1/2 which correspond to the thermal boundary layer formed in the flow past a wedge are solved numerically for various values of the Prandtl and Brinkham number and constant coefficients appearing in the non-linear slip-creep-jump boundary conditions. It is found that for 1/2 < n < 2, solutions may be found for all values of the constant coefficients, while for n ≥ 2 the constant coefficient for the creep term must be set to zero.

Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations

We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H 3 / 2 + ε , 3 / 4 + ε / 2 ( Q ) , with 0 < ε . In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t = 0 , is achieved by choosing a time varying control operator in a neighbourhood of t = 0 .

Nonlinear Static Response and Free Vibration Analysis of Doubly Curved Cross-Ply Panels

In this paper, the nonlinear static and free vibration analyses of doubly curved cross-ply laminated panels subjected to thermomechanical loading are examined. The shell theory adopted in the present case is an extension of Reissner’s shallow shell simplifications that accounts for parabolic distribution of the transverse shear strains through thickness of the shell and tangential stress-free boundary conditions. A multiterm Galerkin’s method is adopted to solve the governing nonlinear partial differential equations. The static equilibrium paths are traced using the Newton-Raphson method in conjunction with the Riks approach to overcome the limit points. The free vibration frequencies about a static equilibrium state of a deformed panel are reported by solving the linear eigenvalue problem. Analytical results are presented for symmetric [0/90/0] and antisymmetric [0/90] cross-ply laminated doubly curved panels that illustrate the influence of geometric properties, in-plane edge boundary conditions, latera...

Analysis of Direct Three-Dimensional Parabolic Panel Methods

Adherence boundary conditions for time dependent partial differential equations, via Chorin algorithm, can be reduced to a parabolic problem with Robin–Fourier boundary conditions in the three-dimensional context. In the spirit of panel methods, one establishes an integral formulation whose key point is the estimation of the potential density, introducing a kind of panel method for tangential kinematic boundary conditions. This paper discusses explicit estimations of this density in the general case of an arbitrarily shaped three-dimensional body, which leads to a fast numerical scheme. An error analysis is also provided, involving body smoothness, the Hölder exponent of the density, and whether the body presents torsion or not.

Comparison of boundary conditions for predicting the collection efficiency of cyclones

Gas-particle flow in three cyclones was numerically modeled using the Eulerian–Lagrangian approach. A tangential lift-off boundary condition was developed and included in the CFD simulations of cyclone performance via a user-defined function. Particles were considered as detached from the surface if they met the criterion of tangential lift-off; otherwise they were captured by the surface. Prediction of the collection efficiency in cyclones was conducted using the three different boundary conditions: (1) bottom trap only; (2) cone and bottom trap; and (3) the tangential lift-off, which is a combination of trap and reflect. Comparison with experimental data in literature indicates that the tangential lift-off boundary condition yields more accurate predictions than other boundary conditions.

On the harmonic response of plates with the shear deformation effect using the elastodynamic solution in the boundary element method

The harmonic response of plates is studied by using the frequency domain solution in conjunction with the direct boundary integral equation (DBIE) for Mindlin's model. The shear deformation and rotatory inertia effects in that model can be switched off in the presented formulation and a DBIE for the classical bending model is then obtained, which includes an additional degree of freedom for the tangential boundary rotation. The frequency responses were obtained to map the plate behavior and they were calculated both with and without the effects of the shear deformation and the rotatory inertia. The analyses were finished when the first natural frequency was identified. The results were compared to available solutions in the literature, employing three-dimensional elasticity theory, the Mindlin and the classical bending models.

Weighted estimates for tangential boundary behaviour

Let be a space of homogeneous type (here is a quasimetric and a measure). A function of modulus of continuity kind gives rise to approach regions at the boundary of , , where for a point , These are 'tangential' regions if . Weighted -estimates are proved for the corresponding maximal functions of integral operators. Applications of these estimates to potentials in and to multipliers of homogeneous expansions of holomorphic functions in the Hardy classes in the unit ball of are presented.

Lattice-Boltzmann method for simulating spherical bubbles with no tangential stress boundary conditions

A lattice-Boltzmann boundary rule has been developed to recover the slip boundary condition at a liquid-gas interface. This rule enables one to use a single-component lattice-Boltzmann model to simulate bubbly flows where bubbles are nearly spherical and coalescence is prohibited. Numerical tests showed this method to be robust and accurate in simulating both steady and unsteady flows around spherical bubbles in the Reynolds number range 0<Re<30.

Border detection for medical imaging

Methods and systems for detecting a border in a medical image are provided. A boundary is chosen as a connected curve whose tangent is substantially perpendicular to the gradient of the image everywhere along the curve. As an alternative to a tangent, a normal or other border direction may be used. At a given point within the image, the tangent to the boundary and the image gradient direction are orthogonal. Using an initial boundary detection, the boundary associated with the minimum cost or associated with the closest boundary where the boundary tangent and the image gradient directional are orthogonal for locations along the boundary is identified. By refining an initial border location to minimize divergence from the boundary tangent being orthogonal to the image gradient direction or by identifying a border based on the orthogonal relationship, accurate border detection may be provided in ultrasound images as well as other medical images.

Tangential boundary stabilization of Navier-Stokes equations

Introduction Main results Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): $d=3$ Boundary feedback uniform stabilization of the linearized system (3.1.4) via an optimal control problem and corresponding Riccati theory. Case $d=3$ Theorem 2.3(i): Well-posedness of the Navier-Stokes equations with Riccati-based boundary feedback control. Case $d=3$ Theorem 2.3(ii): Local uniform stability of the Navier-Stokes equations with Riccati-based boundary feedback control A PDE-interpretation of the abstract results in Sections 5 and 6 Appendix A. Technical material complementing Section 3.1 Appendix B. Boundary feedback stabilization with arbitrarily small support of the linearized system (3.1.4a) at the $(H^{\tfrac{3}{2}-\epsilon}(\Omega))^d \cap H$-level, with I.C. $y^0\in (H^{\frac{1}{2}-\epsilon}(\Omega))^d \cap H$. Cases $d=2,3$. Theorem 2.5 for $d=2$ Appendix C. Equivalence between unstable and stable versions of the optimal control problem of Section 4 Appendix D. Proof that $FS(\cdot) \in \mathcal {L}(W L^2(0,\infty (L^2(\Gamma))^d)$ Bibliography.

Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers

The present paper seeks to continue the analysis in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on tangential boundary stabilization of Navier–Stokes equations, d = 2 , 3 , as deduced from well-posedness and stability properties of the corresponding linearized equations. It intends to complement [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on two levels: (i) by casting the Riccati-based results of Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] for d = 2 , 3 in an abstract setting, thus extracting the key relevant features, so that the resulting framework may be applicable also to other stabilizing boundary feedback operators, as well as to other parabolic-like equations of fluid dynamics; (ii) by including, in the case d = 2 this time, also the low-level gain counterpart of the results in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] with both Riccati-based and spectral-based (tangential) feedback controllers. This way, new local boundary stabilization results of Navier–Stokes equations are obtained over [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear].

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C 2 , α -estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.

Tangential residual as error estimator in the boundary element method

In this paper a new error estimator based on tangential derivative Boundary Integral Equation residuals for 2D Laplace and Helmholtz equations is shown. The direct problem for general mixed boundary conditions is solved using standard and hypersingular boundary integral equations. The exact solution is broken down into two parts: the approximated solution and the error function. Based on theoretical considerations, it is shown that tangential derivative Boundary Integral Equation residuals closely correlate to the errors in the tangential derivative of the solution. A similar relationship is shown for nodal sensitivities and tangential derivative errors. Numerical examples show that the tangential Boundary Integral Equation residual is a better error estimator than nodal sensitivity, because of the accuracy of the predictions and the lesser computational effort.

Navier–Stokes revisited

A revision of Newton's law of viscosity appearing in the role of the deviatoric stress tensor in the Navier–Stokes equation is proposed for the case of compressible fluids, both gaseous and liquid. Explicitly, it is hypothesized that the velocity v appearing in the velocity gradient term ∇ v in Newton's rheological law be changed from the fluid's mass-based velocity v m , the latter being the velocity appearing in the continuity equation, to the fluid's volume velocity v v , the latter being a stand-in for the fluid's volume current (volume flux density n v ). A similar v m → v v alteration is proposed for the velocity v appearing in the no-slip tangential velocity boundary condition at solid surfaces. These proposed revisions are based upon both experiment and theory, including re-interpretation of the following three items: (i) experimental “near-continuum” thermophoretic and other low Reynolds number phoretic data for the movement of suspended particles in fluids under the influence of mass density gradients ∇ ρ , caused either by temperature gradients in single-component fluids undergoing heat transfer or by species concentration gradients in inhomogeneous two-component mixtures undergoing mass transfer; (ii) the hierarchical re-ordering of the Burnett terms appearing in the Chapman–Enskog gas-kinetic theory perturbation expansion of the viscous stress tensor from one of being based upon small Knudsen numbers to one of being based upon small Mach numbers; (iii) Maxwell's (1879) ubiquitous v m -based “thermal creep” or “thermal stress” slip boundary condition used in nonisothermal gas-kinetic theory models, recast in the form of a v v -based no-slip condition. The v v vs. v m dichotomy in the case of compressible fluids is shown to lead to a fundamental distinction between the fluid's tracer velocity as recorded by monitoring the spatio-temporal trajectory of a small non-Brownian particle deliberately introduced into the fluid, and the fluid's “optical” or “colorimetric” velocity as monitored, for example, by the introduction of a dye into the fluid or by some photochromic- or fluorescence-based scheme in circumstances where the individual fluid molecules are themselves responsive to being probed by light. Explicitly, it is argued that the fluid's tracer velocity, representing a strictly continuum nonmolecular notion, is v v , whereas its colorimetric velocity, which measures the mean velocity of the molecules of which the fluid is composed, is v m .

Divergence and flutter instability of elastic specially orthotropic plates subject to follower forces

Divergence and flutter instabilities of a rectangular, specially orthotropic plate (simply supported on a pair of opposite sides) subject to follower forces are computationally analyzed. The effects of the tangential follower parameter ( η ) , aspect ratio ( λ ) , boundary condition (on the side opposite the loaded edge), and material orthotropy on the magnitude of the critical load and the mode (divergence or flutter) of instability are detailed. For the three boundary conditions considered (built in, simply supported, and free free—for the side opposite the loaded edge) and given material orthotropy, the λ – η plane is divided into regions corresponding to different governing instabilities. Both the number and size of these regions depend strongly upon the boundary condition considered as does the governing instability. For a given boundary condition the configuration of the stability regions is independent of material orthotropy. Material orthotropy affects only the buckling loads corresponding to the instabilities and the size of the regions.

Free boundary and mixed boundary value problems for quasilinear elliptic equations: tangential oblique derivative and degenerate Dirichlet boundary problems

We present Hölder estimates and Hölder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector β does not vanish at any point on the boundary. We also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation.

Single-patch surfaces for tool shape design and finite element analysis of hot extrusion

In this paper, a new mathematical representation of single-patch surfaces is reported. With this representation, complex tool shapes of metal forming can be described with a single surface patch which is prescribed by the boundary of the die at its entry and exit. For non-linearly converging die-surfaces, the die shapes can be effectively altered by changing the coefficients of the tangential boundary conditions. Using the obtained die-surfaces, the finite element analysis of the extrusion of hot metals from a round billet to a hexagon workpiece was carried out. A comparison of the extrusion forces and the grain sizes for the formed parts was made for the linearly and non-linearly converging dies. It was found that the non-linearly converging dies could reduce the deformation resistance remarkably, and produce more uniform grains than those using the linearly converging dies.

THE MAIN INEQUALITY OF 3D VECTOR ANALYSIS

This paper proves some simple inequalities for Sobolev vector fields on nice bounded three-dimensional regions, subject to homogeneous mixed normal and tangential boundary data. The fields just have divergence and curl in L2. For the limit cases of prescribed zero normal, respectively zero tangential, data on the whole boundary, the inequalities were proved by Friedrichs who called the result the main inequality of vector analysis. For this mixed case, the optimal constants in the inequality are described, together with the fields for which equality holds. The detailed results depend on a special orthogonal decomposition and the analysis of associated eigenvalue problems.

A Global Solution to a Two-dimensional Riemann Problem Involving Shocks as Free Boundaries

We present a global solution to a Riemann problem for the pressure gradient system of equations. The Riemann problem has initially two shock waves and two contact discontinuities. The angle between the two shock waves is set initially to be close to 180 degrees. The solution has a shock wave that is usually regarded as a free boundary in the self-similar variable plane. Our main contribution in methodology is handling the tangential oblique derivative boundary values.

Prediction of friction factors and heat transfer coefficients for turbulent flow in corrugated tubes combined with twisted tape inserts. Part 2: heat transfer coefficients

A simple mathematical model following the suggestion of Smithberg and Landis has been created to predict the heat transfer coefficients for the case of a fully developed turbulent flow in a spirally corrugated tube combined with a twisted tape insert. The heat transfer can be predicted from the combined effects of the axial and the tangential boundary layer flows coupled with an additional “vortex mixing” effect near the wall through the solution of the corresponding momentun and energy transfer equations. The “wall roughness” has an effect simultaneously on the axial velocity, secondary fluid motion and the resulting swirl mixing. The model reflects the influence of the “wall roughness” and the twisted tape on the thermal resistances of the helicoidal core flow, twisting boundary layer flow and the viscous sublayer near the wall. The calculated heat transfer coefficients have been compared to 544 experimental points obtained from 57 tubes tested. Four hundred thirty-eight points (80.5%) have a relative difference of less than ±15% and 106 points (19.5%) have a relative difference between ±(15–20)%.