Analysis of flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of disks contracting, rotation and heat transfer on the viscous fluid between heated contracting rotating disks. By introducing the Von Kármán type similarity transformations through which we reduced the highly nonlinear partial differential equation to a system of ordinary differential equations. This system of differential equations with appropriate boundary conditions is responsible for the flow behavior between large but finite coaxial rotating and heated disks. It is important to note that the lower disk is rotating with angular velocity Ω while the upper one with SΩ, the disks are also contracting and the temperatures of the upper and lower disks are T₁ and T₀, respectively. The agents which driven the flow are the contraction and also the rotation of the disks. On the other hand the velocity components and especially radial component of velocity strongly influence the temperature distribution inside the flow regime. The basic equations which govern the flow are the Navier Stokes equations with well known continuity equation for incompressible flow. The final system of ordinary differential equations is then solved numerically with given boundary conditions. In addition, the effect of physical parameters, the Reynolds number (Re), the wall contraction ratio (γ) and the rotation ratio (S) on the velocity and pressure gradient, as well as, the effect of Prandtl number (Pr) on temperature distribution are also observed.     

Dirac-bracket structure in multidimensional mode conversion

The intersection of two (2n − 1)-dimensional dispersion manifolds D_a and D_b in the 2n-dimensional ray phase space P yields a (2n − 2)-dimensional conversion manifold M≡D_a∩D_b that naturally possesses a Dirac-bracket structure that is inherited from the canonical Poisson bracket on ray phase space. The canonical symplectic two-form Ω ≡ Ω_∥ + Ω_⊥, defined on the 2n-dimensional tangent plane T_z0P≡T_z0M⊕(T_z0M)_⊥, can thus be decomposed into the Dirac two-form Ω_∥ on the (2n − 2)-dimensional tangent plane T_z0M at a conversion point z₀∈M, and the symplectic two-form Ω_⊥ on its orthogonal 2-dimensional complement (T_z0M)_⊥. These two symplectic two-forms are introduced in our analysis of multidimensional mode conversion, where their respective geometrical roles are defined. We note that since the Dirac-bracket structure Ω_∥ vanishes identically when n = 1, it represents a new structure in multidimensional (n > 1) mode conversion theory.     

Existence and uniqueness of similarity solutions of imbibition equation

The existence and uniqueness of the similarity profiles for the imbibition phenomenon which may arise due to the difference in the wetting abilities of the two immiscible fluids involved in the displacement process through porous media is discussed. By assuming the validity of the Darcy’s law, a mathematical model has been described and it is found that the investigated flow system is governed by a nonlinear diffusivity type equation. The existence and uniqueness of its similarity solutions have been proved by considering the bounds on the saturation coefficient,N(S _w) which is regarded as positive and piecewise continuously differentiable.     

Estimation of large domain Al foam permeability by Finite Difference methods

Classical methods to calculate permeability of porous media have been proposed mainly for high density (e.g. granular) materials. These methods present shortcomings in high porosity, i.e. high permeability media (e.g. metallic foams). While for dense materials permeability seems to be a function of bulk properties and occupancy averaged over the volume, for highly porous materials these parameters fail to predict it. Several authors have attacked the problem by solving the Navier-Stokes equations for the pressure and velocity of a liquid flowing through a small domain (Ω_s) of aluminium foam and by comparing the numerical results with experimental values (prediction error approx. 9%). In this article, we present calculations for much larger domains (Ω_L) using the Finite Difference (FD) method, solving also for the pressure and velocity of a viscous liquid flowing through the Packed Spheres scenario. The ratio Vol(Ω_L)/Vol(Ω_s) is around 10³. The comparison of our results with the Packed Spheres example yields a prediction error of 5% for the intrinsic permeability. Additionally, numerical permeability calculations have been performed for Al foam samples. Our geometric modelling of the porous domain stems from 3D X-ray tomography, yielding voxel information, which is particularly appropriate for FD. Ongoing work concerns the reduction in computing times of the FD method, consideration of other materials and fluids, and comparison with experimental data. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantitative unique continuation for the semilinear heat equation in a convex domain

In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation t_∂u−△u=g(u), with the homogeneous Dirichlet boundary condition, over Ω×(0,T_∗). Ω is a bounded, convex open subset of Rd, with a smooth boundary for the subset. The function g:R→R satisfies certain conditions. We establish some observation estimates for (u−v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω×{T}, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0,T]. At least two results can be derived from these estimates: (i) if ‖(u−v)(⋅,T)_‖L²(ω)=δ, then ‖(u−v)(⋅,T)_‖L²(Ω)?Cδα where constants C>0 and α∈(0,1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω×{T}, then they coincide over Ω×[0,Tm). Tm indicates the maximum number such that these two solutions exist on [0,Tm).     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.     

On marginal distributions and isomorphisms of stationary processes

LetT be an invertible ergodic aperiodic measure preserving transformation of a Lebesgue space, letA be a finite alphabet, and let π be a probability measure onA ⁿ which admits a mixing shift-invariant measureμ _π onΩ=A ^? such that the marginals of anyn successive coordinates are π and the entropyh(T) ofT is smaller than the entropy of the shift in (Ω,μ _π). Then there exists a shift invariant measure ν_π in Ω which also has marginals π and for whichT is isomorphic to the shift in (Ω, ν_π). This contains Krieger's finite generator theorem and strengthens the measure theoretic part of his approximation theorem for shift-invariant measures by showing that the preassigned marginal π can not only be achieved up to an ε>0 but exactly. Our result also contains an as yet unpublished theorem of Krieger, which says thatT can be embedded in an arbitrary mixing subshift of finite type, as long as the entropy of the subshift under the measure with maximal entropy exceeds that ofT. In the final section we show that the method can be extended to yield also exact marginals for the generator in the Jewett-Krieger theorem, i.e.T is shown to be isomorphic to a shift in (Ω, ν_π) where ν_π has exact marginals π and the shift is uniquely ergodic on the support of ν_π.     

An Abstract Characterization of De Morgan (2, 5)-Semigroup of 5-ary Functions

Let Ω be a set and T _Ω be the set of all 5-ary functions over Ω. In this paper, by considering five Mann’s compositions on T _Ω, we obtain a (2, 5)-semigroup and an abstract characterization of this algebra.     

Analysis of melting behavior of PCMs in a cavity subject to a non-uniform magnetic field using a moving grid technique

Melting flow and heat transfer of electrically conductive phase change materials subjecting to a non-uniform magnetic field are addressed in a square enclosure. The top and bottom walls of the cavity are adiabatic, and the sidewalls are isothermal at different temperatures. The temperature of the hot wall is higher than the fusion temperature of PCM (T_f), and the cold wall is at the fusion temperature or lower. At the initial time, the cavity is filled with a solid saturated PCM. In the vicinity to the hot wall, there is an external line-source magnet, inducing a magnetic field. The location of the magnetic source (Y₀) can be changed along the hot wall. The cavity domain is divided into two parts of the liquid domain and the solid domain. The moving grid method is utilized to track the phase change interface at the exact fusion temperature of T_f. The governing equations for continuity, flow and heat transfer associated with the Arbitrary Lagrangian–Eulerian (ALE) moving mesh technique are solved using the finite element method. The results are investigated for the melting behavior of PCM by the study of Hartmann number (0 ≤ Ha ≤ 50) and the location of the magnetic source (0 ≤ Y₀ ≤ 1). Outcomes show that the effect of the magnetic field on the melting behavior of PCM is negligible at the initial stages of the melting (Fo < 1.15). However, after the initial stages of the melting, the effect of the presence of a magnetic field becomes significant. Moreover, the location of the magnetic source induces a feeble effect on the melting front at the initial melting stages, but its effect on the shape of the melting front increases by the increase of the non-dimensional time. The location of the magnetic source also significantly affects the streamlines patterns. Changing the position of the magnetic source from the bottom of the cavity (Y₀ = 0.2) to the almost middle of the cavity (Y₀ = 0.6) would decrease the required non-dimensional time of full melting from Fo = 10.4 to Fo = 9.0.     

Optimal Control of the Obstacle for a Parabolic Variational Inequality

An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L²(0, T; H²(Ω) ∩ H¹₀(Ω)) with ψ_t ∈ L²(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Parametric study of liquid droplets impinging on porous surfaces

This work presents a numerical simulation of the fluid dynamics of a liquid droplet during impact/absorption onto a porous medium. The main focus of this paper is on a parametric study of the influence of the governing parameters upon the fluid flow characteristics. The problem is described in a non-dimensional form, and the influence of the main governing parameters is investigated, including their variation along the range of physical configurations of interest. This procedure revealed 7 main governing parameters: Reynolds number (Re), Darcy number (Da), porosity (ε), Froude number (Fr), Weber number (We), contact angle (θ) and the ratio between pore and particle diameter size in the porous substrate (α). The results indicate that the values of Da and Re are more related to the amount of momentum dissipation due to the drag of the solid matrix of the substrate, while the values of We, α and θ can be mainly related to capillary pressure.     

On well-posedness of the initial boundary-value problem for the derivative nonlinear Schrödinger equation

This paper deals with an initial boundary-value problem for the generalized derivative nonlinear Schrödinger equation. The cases of zero Dirichlet and generalized periodic boundary conditions are considered. The global existence of a solution inL _∞(0,∞;H _b ¹) is proved. The uniqueness inL _∞(0,T;H _b ¹)∩{u: ?u/?x εL _∞(Ω×(0,T))} is also established.     

Irreducible collineation groups with two orbits forming an oval

Let G be a collineation group of a finite projective plane π of odd order fixing an oval Ω. We investigate the case in which G has even order, has two orbits Ω₀ and Ω₁ on Ω, and the action of G on Ω₀ is primitive. We show that if G is irreducible, then π has a G-invariant desarguesian subplane π₀ and Ω₀ is a conic of π₀.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

Asymptotically Nonexpansive Mappings in Modular Function Spaces

In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying a Δ₂-type condition, C a convex, ρ-bounded, ρ-a.e. compact subset of L_ρ, and T: C → C a ρ-asymptotically nonexpansive mapping, then T has a fixed point. In particular, any asymptotically nonexpansive self-map defined on a convex subset of L¹(Ω, μ) which is compact for the topology of local convergence in measure has a fixed point.     

A lower bound on the Kobayashi metric near a point of finite type in ℂ n

Let Ω be a bounded domain in ? ⁿ andbΩ smooth pseudoconvex near z₀ ∈bΩ of finite type. Then there are constantsc>0 and ε′>0 such that the Kobayashi metric,K _Ω(z; X), satisfiesK _Ω(z; X)≥c|X|δ(z)^?t′ for allX ∈T _z ^1,0 ? ⁿ in a neighborhood ofz ₀. Here δ(z) denotes the distance fromz tobΩ. As an application, we prove the Hölder continuity of proper holomorphic maps onto pseudoconvex domains.     

MHD free convective flow through porous medium in the presence of hall current, radiation and thermal diffusion

An unsteady free convective flow through porous media of viscous, incompressible, electrically conducting fluid through a vertical porous channel with thermal radiation is studied. A magnetic field of uniform strength is applied perpendicular to the vertical channel. The magnetic Reynolds number is assumed very small so that the induced magnetic field effect is negligible. The injection and suction velocity at both plates is constant and is given by v ₀. The pressure gradient in the channel varies periodically with time along the axis of the channel. The temperature difference of the plates is high enough to induce the radiative heat. Taking Hall current and Soret effect into account, equations of motion, energy, and concentration are solved. The effects of the various parameters, entering into the problem, on velocity, temperature and concentration field are shown graphically.     

Group method analysis of combined heat and mass transfer by MHD non-Darcy non-Newtonian natural convection adjacent to horizontal cylinder in a saturated porous medium

The group theoretic method is applied for solving problem of combined magneto-hydrodynamic heat and mass transfer of non-Darcy natural convection about an impermeable horizontal cylinder in a non-Newtonian power law fluid embedded in porous medium under coupled thermal and mass diffusion, inertia resistance, magnetic field, thermal radiation effects. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The ordinary differential equations are solved numerically for the velocity using shooting method. The effects of magnetic parameter M, Ergun number Er, power law (viscosity) index n, buoyancy ratio N, radiation parameter R_d, Prandtl number Pr and Lewis number Le on the velocity, temperature fields within the boundary layer, heat and mass transfer are presented graphically and discussed.     

Compact operators and Toeplitz algebras on multiply-connected domains

If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C^?-subalgebra of τ generated by Toeplitz operators with symbols in H^∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M₁ of trivial Gleason parts.     

Small into-isomorphism from L∞ (A, μ) into L∞(B, υ)

In this paper we shall assert that if T is an isomorphism of L_∞(Ω₁, A, μ) into L_∞(Ω₂, B, υ) satisfying the condition ‖T‖·‖T ^?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions.