Natural convective boundary-layer flow in a heat generating porous medium with a prescribed wall heat flux

The free convection boundary-layer flow on a vertical surface in a porous medium with local heat generation proportional to (T – T_∞)^p, where T is the local temperature and T_∞ is the ambient temperature, is considered when the surface is thermally insulated. The way in which the flow develops from the leading edge is seen to depend critically on the exponent p. For p ≤ 2 there is a boundary-layer flow for all x > 0, where x measures distance from the leading edge, with the internal heating having a significant effect at large x. For p ≥ 5 there is also a boundary-layer flow to large x but now the internal heating has an increasingly weaker effect as x increases. For 2 < p <  5 the boundary-layer solution breaks down at a finite x, with a singularity developing leading to thermal runaway at a finite distance along the surface.     

Natural convective boundary-layer flow in a heat generating porous medium with a prescribed wall heat flux

The free convection boundary-layer flow on a vertical surface in a porous medium with local heat generation proportional to (T – T_∞)^p, where T is the local temperature and T_∞ is the ambient temperature, is considered when the surface is thermally insulated. The way in which the flow develops from the leading edge is seen to depend critically on the exponent p. For p ≤ 2 there is a boundary-layer flow for all x > 0, where x measures distance from the leading edge, with the internal heating having a significant effect at large x. For p ≥ 5 there is also a boundary-layer flow to large x but now the internal heating has an increasingly weaker effect as x increases. For 2 < p <  5 the boundary-layer solution breaks down at a finite x, with a singularity developing leading to thermal runaway at a finite distance along the surface.     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

Regularization of a non-characteristic Cauchy problem for the heat equation

The non-characteristic Cauchy problem for the heat equation u_xx(x,t) = u₁(x,t), 0 ? x ? 1, ? ∞ < t < ∞, u(0,t) = φ(t), u_x(0, t) = ψ(t), ? ∞ < t < ∞ is regularizèd when approximate expressions for φ and ψ are given. Properties of the exact solution are used to obtain an explicit stability estimate.     

Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model

Flow and thermal field in nanofluid is analyzed using single phase thermal dispersion model proposed by Xuan and Roetzel [Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707]. The non-dimensional form of the transport equations involving the thermal dispersion effect is solved numerically using semi-explicit finite volume solver in a collocated grid. Heat transfer augmentation for copper–water nanofluid is estimated in a thermally driven two-dimensional cavity. The thermo-physical properties of nanofluid are calculated involving contributions due to the base fluid and nanoparticles. The flow and heat transfer process in the cavity is analyzed using different thermo-physical models for the nanofluid available in literature. The influence of controlling parameters on convective recirculation and heat transfer augmentation induced in buoyancy driven cavity is estimated in detail. The controlling parameters considered for this study are Grashof number (10³ < Gr < 10⁵), solid volume fraction (0 < ? < 0.2) and empirical shape factor (0.5 < n < 6). Simulations carried out with various thermo-physical models of the nanofluid show significant influence on thermal boundary layer thickness when the model incorporates the contribution of nanoparticles in the density as well as viscosity of nanofluid. Simulations incorporating the thermal dispersion model show increment in local thermal conductivity at locations with maximum velocity. The suspended particles increase the surface area and the heat transfer capacity of the fluid. As solid volume fraction increases, the effect is more pronounced. The average Nusselt number from the hot wall increases with the solid volume fraction. The boundary surface of nanoparticles and their chaotic movement greatly enhances the fluid heat conduction contribution. Considerable improvement in thermal conductivity is observed as a result of increase in the shape factor.     

Flatness and null controllability of 1-D parabolic equations

We present a recent result on null controllability of one-dimensional linear parabolic equations with boundary control. The space-varying coefficients in the equation can be fairly irregular, in particular they can present discontinuities, degeneracies or singularities at some isolated points; the boundary conditions at both ends are of generalized Robin-Neumann type. Given any (fairly irregular) initial condition θ₀ and any final time T, we explicitly construct an open-loop control which steers the system from θ₀ at time 0 to the final state 0 at time T. This control is very regular (namely Gevrey of order s with 1 < s < 2); it is simply zero till some (arbitrary) intermediate time τ, so as to take advantage of the smoothing effect due to diffusion, and then given by a series from τ to the final time T. We illustrate the effectiveness of the approach on a nontrivial numerical example, namely a degenerate heat equation with control at the degenerate side. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions

We consider the fourth-order degenerate diffusion equation, in one space dimension. This equation, derived from a lubrication approximation, models the surface-tension-dominated motion of thin viscous films and spreading droplets [15]. The equation with f(h) = |h| also models a thin neck of fluid in the Hele-Shaw cell [10], [11], [23]. In such problems h(x,t) is the local thickness of the the film or neck. This paper considers the properties of weak solutions that are more relevant to the droplet problem than to Hele-Shaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets. We consider two problems: The first has initial data h₀ ≥ 0 and f(h) = |h|ⁿ, 0 < n < 3. We show that there exists a weak nonnegative solution for all time. Also, we show that this solution becomes a strong positive solution after some finite time T*, and asymptotically approaches its means as t → ∞. The weak solution is in the classical sense of distributions for 3/8 < n < 3 and in a weaker sense introduced in [1] for the remaining 0 < n ≤ 3/8. Furthermore, the solutions have high enough regularity to just include the unique source-type solutions [2] with zero slope at the edge of the support. They do not include any of the less regular solutions with positive slope at the edge of the support. Second, we consider strictly positive initial data h₀ ≥ m > 0 and f(h) = |h|ⁿ, 0 < n < ∞. For this problem we show that even if a finite-time singularity of the form h → 0 does occur, there exists a weak nonnegative solution for all time t. This weak solution becomes strong and positive again after some critical time T*. As in the first problem, we show that the solution approaches its mean as t → ∞. The main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity. We show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result. © 1996 John Wiley & Sons, Inc.     

Free convection boundary layers on a vertical surface in a heat-generating porous medium

The natural convection boundary-layer flow on a solid verticalsurface with heat generated within the boundary layer at a rateproportional to (T – T)^p (p 1) is considered. The surfaceis held at the ambient temperature T except near the leadingedge where it is held at a temperature above ambient. The behaviourof the flow as it develops from the leading edge is examinedand is seen to become independent of the initial heat input;however, it does depend strongly on the exponent p. For 1 p 2, the local heating eventually dominates at large distancesand there is a convective flow driven by this mechanism. Forp 4, the local heating does not have a significant effect,the fluid temperature remains relatively small throughout andthe heat transfer dies out through a wall jet flow. For 2 <p < 4, the local heating has a significant effect at relativelysmall distances, with a thermal runaway developing at a finitedistance along the surface.     

Distribution of Lattice Points on Hyperboloids

Consider the region Ω₀ on the hyperboloid 1 = b² + ac defined by the conditions

Maximal Probability Inequalities for Stochastic Processes

Let X_a,b be nonnegative random variables with the property that X_a,b ≦ X_a,c + X_c.b for all 0≦ a < c < b ≦ T, where T > 0 is fixed. We define M_a,b = sup {X_a,c: a < c ≦ h} and establish bounds for P[M_a,b ≧ λ] in terms of given bounds for P[X_a,b ≧ λ], where λ runs through some interval (0, λ_o), 0 < λ_o ≦ ∞ fixed. These bounds explicitly involve a nonnegative function g(a, b) assumed to be quasi-superadditive with an index Q, i.e., g(a, c) + g(c, b) ≦ Qg(a, b) for all 0≦ a < c < b < T, where 1 ≦ Q < 2 is fixed. Maximal inequalities obtained in this way can be applied to stochastic processes exhibiting long-range dependence. Among others, these applications may include certain self-similar processes such as fractional Brownian motion, stochastic processes occurring in linear time series models, etc.     

On the spectrum of the Dirichlet Laplacian in a narrow strip

We consider the Dirichlet Laplacian Δ_∈ in a family of bounded domains {−a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ_∈.     

Spectral invariance for pseudodifferential operators on weighted function spaces

The spectrum of each symmetric ψ DO of the symbol class S⁰ _{1, γ}, 0≤γ<1, acting on B³ _p,q(w(x)) and F³ _p,q(w(x)), is independent of the choice ofs ∈ℝ, 0<p≤∞ (p<∞ in the F-case), 0<q≤∞ and the weight w(x)∈W.     

Homogenization of a non‐linear degenerate parabolic problem in a highly heterogeneous periodic medium

We consider the homogenization of a time‐dependent heat transfer problem in a highly heterogeneous periodic medium made of two connected components having comparable heat capacities and conductivities, separated by a third material with thickness of the same order ε as the basic periodicity cell but having a much lower conductivity such that the resulting interstitial heat flow is scaled by a factor λ tending to zero with a rate λ=λ(ε). The heat flux vectors a_j, j=1,2,3 are non‐linear, monotone functions of the temperature gradient. The heat capacities c_j(x) are positive, but may vanish at some subsets such that the problem can be degenerate (parabolic–elliptic). We show that the critical value of the problem is δ=lim_ε→0ε^p/λ and identify the homogenized problem depending on whether δ is zero, strictly positive finite or infinite. Copyright © 2005 John Wiley & Sons, Ltd.     

On the equivalence of measures on loop space

Let K be a simply-connected compact Lie Group equipped with an Ad _K -invariant inner product on the Lie Algebra ?, of K. Given this data, there is a well known left invariant “H ¹-Riemannian structure” on L(K) (the infinite dimensional group of continuous based loops in K), as well as a heat kernel ν_T(k ₀, ·) associated with the Laplace-Beltrami operator on L(K). Here T > 0, k ₀∈L(K), and ν _T (k ₀, ·) is a certain probability measure on L(K). In this paper we show that ν₁(e,·) is equivalent to Pinned Wiener Measure on K on ? _s0 ≡<x _t : t∈ [0, s ₀]> (the σ-algebra generated by truncated loops up to “time”s ₀). Recevied: 9 September 1999 / Revised version: 13 March 2000 / Published online: 22 November 2000     

On the chromatic number of the general Kneser-graph

Given integers k, n, 2 < k < n, let us define a graph with vertex set V = {F ?{1, 2, …, n}: ∩F = k}, and (F, F') is an edge if |F ∩ F′| ≤ 1. We show that for n > n₀(k) the chromatic number of this graph is (k - 1)() + rs, where n = (k - 1)s + r, 0 ≤ r < k - 1.     

Locally Complemented Subspaces and ℒ︁p-Spaces for 0 < p < 1

We develop a theory of ??_p-spaces for 0 < p < 1, basing our definition on the concept of a locally complemented subspace of a quasi-BANACH space. Among the topics we consider are the existence of basis in ??_p-spaces, and lifting and extension properties for operators. We also give a simple construction of uncountably many separable ??_p-spaces of the form ??_p(X) where X is not a ??_p-space. We also give some applications of our theory to the spaces H_p, 0 < p < 1.     

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the L^p functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the L^p norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p ₀ = 2+4π ² /β ² ω ₂ , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p ₀ and p > p ₀ . We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.     

Weak type estimates for some maximal operators on the weighted Hardy spaces

Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spacesH _ω ^p (0 <p < 1, ω ∈A ₁) (0<p<1, ω∞A₁); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means onH _ω ^p .     

On the jackson theorem for periodic functions in spaces with integral metric

We consider the approximation of periodic functions by trigonometric polynomials in metric (not normed) spaces that are generalizations of the spaces L _p , 0 < p < 1, and L ₀. In particular, we prove the multidimensional Jackson theorem in L _p (T ^m ), 0 < p < 1.