Inertia effects in circular squeeze film bearing using Herschel–Bulkley lubricants

Recent engineering trends in lubrication emphasize that in order to analyze the performance of bearings adequately, it is necessary to take into account the combined effects of fluid inertia forces and non-Newtonian characteristics of lubricants. In the present work, the effects of fluid inertia forces in the circular squeeze film bearing lubricated with Herschel–Bulkley fluids with constant squeeze motion have been investigated. Herschel–Bulkley fluids are characterized by an yield value which leads to the formation of a rigid core in the flow region. The shape and extent of the core formation along the radial direction is determined numerically for various values of Herschel–Bulkley number and power-law index. The bearing performances such as pressure distribution and load capacity for different values of Herschel–Bulkley number, Reynolds number, power-law index have been computed. The effects of fluid inertia and non-Newtonian characteristics on the bearing performances have been discussed.     

Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – A mathematical model

The pulsatile flow of blood through catheterized artery has been studied in this paper by modeling blood as Herschel–Bulkley fluid and the catheter and artery as rigid coaxial circular cylinders. The Herschel–Bulkley fluid has two parameters, the yield stress θ and the power index n. Perturbation method is used to solve the resulting quasi-steady nonlinear coupled implicit system of differential equations. The effects of catheterization and non-Newtonian nature of blood on yield plane locations, velocity, flow rate, wall shear stress and longitudinal impedance of the artery are discussed. The existence of two yield plane locations is investigated and their dependence on yield stress θ, amplitude A, and time t are analyzed. The width of the plug core region increases with increasing value of yield stress at any time. The velocity and flow rate decrease, whereas wall shear stress and longitudinal impedance increase for increasing value of yield stress with other parameters held fixed. On the other hand, the velocity, flow rate and wall shear stress decrease but resistance to flow increases as the catheter radius ratio (ratio of catheter radius to vessel radius) increases with other parameters fixed. The results for power law fluid, Newtonian fluid and Bingham fluid are obtained as special cases from this model.     

Peristaltic transport of non-Newtonian fluid in a diverging tube with different wave forms

The present study investigates the peristaltic transport of non-Newtonian fluid, modeled as power law and Bingham fluid, in a diverging tube with different wall wave forms: sinusoidal, multi-sinusoidal, triangular, trapezoidal and square waves. Fourier series is employed to get the expressions for temporal and spatial dependent wall shapes. Solutions for time average pressure rise — flow rate relationship are computed for different amplitude ratios, φ, power law indices, n, yield stresses, τ₀, and wave shapes. Results indicate that φ and n play a vital role in peristalsis. When φ of the sinusoidal wave is increased from 0.6 to 0.8, the maximum pressure rise, increased by a factor of 10. Increasing n from 0.6 to 1 increased the by a factor of 3. For Bingham fluid with φ=0.5, a 25% increase in is obtained when τ₀, is reduced from 1 (non-Newtonian) to 0 (Newtonian). Of all the wave shapes considered, obtained is maximum for the square wave and minimum for the triangular wave (4–15 times less depending on φ). Finally, pathlines of massless particles are traced to investigate the occurrence of reflux. It is observed that, even for zero flow rate, reflux occurs near the tube wall and the thickness and shape of the reflux region strongly depends on φ, n, and shape of the peristaltic waves.     

Local Lipschitz continuity of solutions to a problem in the calculus of variations

This article studies the problem of minimizing ∫_ΩF(Du)+G(x,u) over the functions uW^1,1(Ω) that assume given boundary values on ∂Ω. The function F and the domain Ω are assumed convex. In considering the same problem with G=0, and in the spirit of the classical Hilbert–Haar theory, Clarke has introduced a new type of hypothesis on the boundary function : the lower (or upper) bounded slope condition. This condition, which is less restrictive than the classical bounded slope condition of Hartman, Nirenberg and Stampacchia, is satisfied if is the restriction to ∂Ω of a convex (or concave) function. We show that for a class of problems in which G(x,u) is locally Lipschitz (but not necessarily convex) in u, the lower bounded slope condition implies the local Lipschitz regularity of solutions.     

Preliminary test estimators and phi-divergence measures in generalized linear models with binary data

We consider the problem of estimation of the parameters in Generalized Linear Models (GLM) with binary data when it is suspected that the parameter vector obeys some exact linear restrictions which are linearly independent with some degree of uncertainty. Based on minimum -divergence estimation (ME), we consider some estimators for the parameters of the GLM: Unrestricted ME, restricted ME, Preliminary ME, Shrinkage ME, Shrinkage preliminary ME, James–Stein ME, Positive-part of Stein-Rule ME and Modified preliminary ME. Asymptotic bias as well as risk with a quadratic loss function are studied under contiguous alternative hypotheses. Some discussion about dominance among the estimators studied is presented. Finally, a simulation study is carried out.     

A non-Newtonian fluid flow model for blood flow through a catheterized artery—Steady flow

In this paper the effects catheterization and non-Newtonian nature of blood in small arteries of diameter less than 100 μm, on velocity, flow resistance and wall shear stress are analyzed mathematically by modeling blood as a Herschel–Bulkley fluid with parameters n and θ and the artery and catheter by coaxial rigid circular cylinders. The influence of the catheter radius and the yield stress of the fluid on the yield plane locations, velocity distributions, flow rate, wall shear stress and frictional resistance are investigated assuming the flow to be steady. It is shown that the velocity decreases as the yield stress increases for given values of other parameters. The frictional resistance as well as the wall shear stress increases with increasing yield stress, whereas the frictional resistance increases and the wall shear stress decreases with increasing catheter radius ratio k (catheter radius to vessel radius). For the range of catheter radius ratio 0.3–0.6, in smaller arteries where blood is modeled by Herschel–Bulkley fluid with yield stress θ = 0.1, the resistance increases by a factor 3.98–21.12 for n = 0.95 and by a factor 4.35–25.09 for n = 1.05. When θ = 0.3, these factors are 7.47–124.6 when n = 0.95 and 8.97–247.76 when n = 1.05.     

Combinatorial congruences and -operators

The ψ-operator for (,Γ)-modules plays an important role in the study of Iwasawa theory via Fontaine's big rings. In this note, we prove several sharp estimates for the ψ-operator in the cyclotomic case. These estimates immediately imply a number of sharp p-adic combinatorial congruences, one of which extends the classical congruences of Fleck (1913) and Weisman [Some congruences for binomial coefficients, Michigan Math. J. 24 (1977) 141–151].     

Self-Dual Vortices in Chern–Simons Hydrodynamics

The classical theory of a nonrelativistic charged particle interacting with a U(1) gauge field is reformulated as the Schrödinger wave equation modified by the de Broglie–Bohm nonlinear quantum potential. The model is gauge equivalent to the standard Schrödinger equation with the Planck constant for the deformed strength of the quantum potential and to the pair of diffusion–antidiffusion equations for the strength . Specifying the gauge field as the Abelian Chern–Simons (CS) one in 2+1 dimensions interacting with the nonlinear Schrödinger (NLS) field (the Jackiw–Pi model), we represent the theory as a planar Madelung fluid, where the CS Gauss law has the simple physical meaning of creation of the local vorticity for the fluid flow. For the static flow when the velocity of the center-of-mass motion (the classical velocity) is equal to the quantum velocity (generated by the quantum potential velocity of the internal motion), the fluid admits an N-vortex solution. Applying a gauge transformation of the Auberson–Sabatier type to the phase of the vortex wave function, we show that deformation parameter , the CS coupling constant, and the quantum potential strength are quantized. We discuss reductions of the model to 1+1 dimensions leading to modified NLS and DNLS equations with resonance soliton interactions.     

On the similarity of dual operators

Let k be a field with an involution σ and a non-degenerate sesquilinear form, where V,W are n-dimensional k-spaces. Assume that ΛEnd(V) and Λ^*End(W) are dual operators. We show that if Λ and Λ^* are similar, then Λ^*=Λ^-1, where :V→W is Hermitian.     

Finite element analysis of an A– method for time-dependent Maxwell’s equations based on explicit-magnetic-field scheme

In this paper, error analysis of a finite element A– method for the time-dependent Maxwell’s equations is presented. An explicit-magnetic-field scheme is applied. Provided that the time-stepsize τ is sufficiently small, the proposed algorithm yields for finite time T an error of in the L²-norm for the electric field E and the magnetic field H, where h is the mesh size.     

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

We show that the L^p spatial–temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier–Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443–458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, submitted for publication]. For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier–Stokes solutions for general L^p space when the initial velocity is in , 1<rq<∞ (1<r<q=∞).     

Existence and Regularity of the Pressure for the Stochastic Navier–Stokes Equations

We prove, on one hand, that for a convenient body force with values in the distribution space (H ^-1(D)) ^d , where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V of the divergence free subspace V of (H ¹ ₀(D)) ^d , in general it is not possible to solve the stochastic Navier–Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier–Stokes equations could be meaningful for them.     

Existence and multiplicity results for some nonlinear problems with singular -Laplacian

Using Leray–Schauder degree theory we obtain various existence and multiplicity results for nonlinear boundary value problems

where l(u,u^′)=0 denotes the Dirichlet, periodic or Neumann boundary conditions on [0,T], is an increasing homeomorphism, (0)=0. The Dirichlet problem is always solvable. For Neumann or periodic boundary conditions, we obtain in particular existence conditions for nonlinearities which satisfy some sign conditions, upper and lower solutions theorems, Ambrosetti–Prodi type results. We prove Lazer–Solimini type results for singular nonlinearities and periodic boundary conditions.     

Navier–Stokes equations with nonhomogeneous boundary conditions in a convex bi-dimensional domain

We prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a convex and bounded region, ΩR², with nonhomogeneous Dirichlet boundary conditions on ∂Ω. These results are also extended to flow domain surrounding an obstacle.     

On the continuation of solutions for elliptic equations in two variables

We consider nonlinear elliptic differential equations of second order in two variables

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. Supposing analyticity of F, we prove analyticity of the real solution z=z(x,y) in the open set Ω. Furthermore, we show that z may be continued as a real analytic solution for F=0 across the real analytic boundary arc Γ∂Ω, if z satisfies one of the boundary conditions z= or z_n=ψ(x,y,z,z_t) on Γ with real analytic functions and ψ, respectively (z_n denotes the derivative of z w.r.t. the outer normal n on Γ and z_t its derivative w.r.t. the tangent). The proof is based on ideas of H. Lewy combined with a uniformization method. Studying quasilinear equations, we get somewhat better results concerning the initial regularity of the given solution and a little more insight.     

A mathematical model for nonlinear analysis of flow pulses utilizing an integral technique

An existing one-dimensional mathematical model, for the arterial tree was extended to include the effects of radial variation of axial fluid velocity by the application of an integral technique. The resulting formulation reduced to a system of characteristics equations similar, in form to the equations for the onedimensional model and the computer program was modified to accommodate the integral formulation. The need for a kinematic boundary condition on the axial component of wall velocity was demonstrated. Results were obtained for a variety of velocity profiles. It was found that the slope of the front and back of the waves as well as the wave, amplitude are sensitive to changes in the velocity profile and the axial component of wall velocity. The velocity of the waves is also effected but not significantly.

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Zusammenfassung Die eindimensionale Theorie von Anliker et al. (ZAMP22, 217 (1971)) wird in dieser Arbeit dahingehend erweitert, dass der Einfluss des Geschwindigkeitsprofiles mitberücksichtigt wird. Die Navier-Stokes-Gleichungen und die Kontinuitätsgleichung werden mit Hilfe einer Integraltechnik auf ähnliche, für die Rechnung mit dem Computer geeignete Gleichungen zurückgeführt, wie sie von Anliker et al. verwendet wurden. Die charakteristischen Grössen des Geschwindigkeitsprofiles sowie die Geschwindigkeit der Gefässwand gehen als Parameter in die Theorie ein, so dass parametrische Studien durchgeführt werden können.

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Nomenclature a local internal radius of the vessel - a _f , A _f constants in the cosine profile - b defined by equation (22) - local normal and tangential unit vectors (see Figure 1) - f(r/a), g (z,t) defined by equation (9) - f _R friction factor - local mass flux into the vessel - p local pressure - p _c capillary pressure - p _o pressure at the terminal end - r, z radial and axial coordinates - S local cross sectional area - t time - flow velocity at the wall interface - u, v, w radial, circumferential and axial components of flow velocity - u _w , v _w , w _w radial, circumferential and axial components of flow velocity at the wall interface - wall velocity at the interface - U _w , W _w radial and axial components of wall velocity at the interface - W mass average flow velocity defined by equation (13) - w _o maximum flow velocity - ₀, ₁, ₂, ₃ parameters defined by equation (10) - ₄ W _w /w _w - _A , _B , _C parameters defined by equation (18) - outflow parameter - wave length - _L Lagrangian multiplier - viscosity coefficient for the fluid - density of the fluid - kinematic viscosity - ()' nondimensional quantity of order one - ()₊, ()_– values of () associated with roots of equation (23) This analysis was initiated during the authors appointment as a NASA-ASEE Summer Faculty Fellow to the Stanford-Ames Program and completed through the facilities of the Computer Science Center at the University of Maryland.     

A characterization of convex calibrable sets in with respect to anisotropic norms

A set is called “calibrable” if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the “-calibrability” of bounded convex sets in with respect to a norm (called anisotropy in the sequel) by the anisotropic mean -curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body C satisfying a -ball condition contains a convex -calibrable set K such that, for any V[|K|,|C|], the subset of C of volume V which minimizes the -perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set.     

Conditional Empirical Processes Defined by Nonstationary Absolutely Regular Sequences

K. I. Yoshihara (1990,Comput. Math. Appl.19, No. 1, 149–158) proved the weak invariance of the conditional nearest neighbor regression function estimator called the conditional empirical process based on-mixing observations. In this paper, we extend the result for nonstationary and absolutely regular random variables which have applications for Markov processes, for which the initial measure is not necessary, the invariant measure.