On Cesàro-Volterra Method in Orthotropic Saint-Venant Beam

The linearly elastic and orthotropic Saint-Venant beam model, with a spatially constant Poisson tensor and fiberwise homogeneous elastic moduli, is investigated by a coordinate-free approach. A careful reasoning reveals that the elastic strain, fulfilling the whole set of differential conditions of integrability and a differential condition imposed by equilibrium, is defined on the whole ambient space in which the beam is immersed. At this stage the shape of the beam cross-section is inessential and Cesàro-Volterra formula provides the general integral of the differential conditions of kinematic compatibility. The cross-section geometrical shape comes into play only when differential and boundary equilibrium conditions are imposed to evaluate the warping displacement field. The treatment of an orthotropic Saint-Venant beam is applied to investigate about the locations of the shear and twist centres. It is shown that the position of the shear centre can be expressed in terms of the sole cross-section twist warping. The advantage with respect to treatments in the literature is that the solution of a single Neumann-like problem is required.     

Study of the analytical solution to the heat transfer problem and surface temperature in a semi-infinite body with a constant heat flux at the surface and an initial temperature distribution

In many practical cases, one heats a semi-infinite solid with a constant heat flux source. For such an unsteady heat transfer problem, if the body has a uniform initial temperature, the analytical solution has been given by Carslaw and Jaeger. The surface temperature of the semi-infinite body follows the $\sqrt t $ -rule, that is, the surface temperature changes in proportion to square root of heating time. But if, instead of the uniform initial temperature, the body has a temperature distribution at the beginning of heating, the analytical solution has not yet been developed. Analytical solutions to the same problem with an exponential or a linear initial temperature distribution are obtained in this paper. It is shown, that in the case of a linear initial temperature distribution the surface temperature also changes according to $\sqrt t $ -rule Approximating the initial temperature distribution near the surface by its tangent at the surface, it is found that the surface temperature within a short time after the start of heating should also satisfy the $\sqrt t $ -rule, in spite of an arbitrary initial temperature distribution. The experimental data support this argument. Furthermore, the constant heat flux can be calculated after relationship between the surface temperature and heating time according to the equation derived in this paper, if the initial temperature distribution or its first-order derivative at the surface is known.     

Mikroskopische Untersuchung der Tropfenkondensation

The heat transfer during dropwise condensation can be calculated from the distribution function of the drop sizes and the growth function of single drops. Both functions are obtained from motion pictures with high magnification. The motion pictures are taken during condensation of steam of about 25 °C on vertical copper and brass surfaces. A simple approximation for the growth function is given which agrees with the exact solution ofUmur andGriffith [8] within the limits of about 1%.     

A New Insight into the Convective Boundary Condition

The steady mixed convection boundary layer flows over a vertical surface adjacent to a Darcy porous medium and subject respectively to (i) a prescribed constant wall temperature, (ii) a prescribed variable heat flux, $q_\mathrm{w} =q_0 x^{-1/2}$ q w = q 0 x ? 1 / 2 , and (iii) a convective boundary condition are compared to each other in this article. It is shown that, in the characteristic plane spanned by the dimensionless flow velocity at the wall ${f}^{\prime }(0)\equiv \lambda $ f ′ ( 0 ) ≡ λ and the dimensionless wall shear stress $f^{\prime \prime }(0)\equiv S$ f ′ ′ ( 0 ) ≡ S , every solution $(\lambda , S)$ ( λ , S ) of one of these three flow problems at the same time is also a solution of the other two ones. There also turns out that with respect to the governing mixed convection and surface heat transfer parameters $\varepsilon $ ε and $\gamma $ γ , every solution $(\lambda , S)$ ( λ , S ) of the flow problem (iii) is infinitely degenerate. Specifically, to the very same flow solution $(\lambda , S)$ ( λ , S ) there corresponds a whole continuous set of values of $\varepsilon $ ε and $\gamma $ γ which satisfy the equation $S=-\gamma (1+\varepsilon -\lambda )$ S = ? γ ( 1 + ε ? λ ) . For the temperature solutions, however, the infinite degeneracy of the velocity solutions becomes lifted. These and further outstanding features of the convective problem (iii) are discussed in the article in some detail.     

Geometric continuum mechanics

Geometric Continuum Mechanics ( GCM) is a new formulation of Continuum Mechanics ( CM) based on the requirement of Geometric Naturality ( GN). According to GN, in introducing basic notions, governing principles and constitutive relations, the sole geometric entities of space-time to be involved are the metric field and the motion along the trajectory. The additional requirement that the theory should be applicable to bodies of any dimensionality, leads to the formulation of the Geometric Paradigm ( GP) stating that push-pull transformations are the natural comparison tools for material fields. This basic rule implies that rates of material tensors are Lie-derivatives and not derivatives by parallel transport. The impact of the GP on the present state of affairs in CM is decisive in resolving questions still debated in literature and in clarifying theoretical and computational issues. As a consequence, the notion of Material Frame Indifference ( MFI) is corrected to the new Constitutive Frame Invariance ( CFI) and reasons are adduced for the rejection of chain decompositions of finite elasto-plastic strains. Geometrically consistent notions of Rate Elasticity ( RE) and Rate Elasto-Visco-Plasticity ( REVP) are formulated and consistent relevant computational methods are designed.     

Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number $10^{-5}\le \hbox {Da}\le 10^{-3}$ , porous layer height ratio $0\le d/L\le 1$ , thermal conductivity ratio $1\le k_{1,3}\le 20$ , and dimensionless time $0\le \tau \le 1000$ on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.     

Global Solution to the Incompressible Oldroyd-B Model in the Critical L p Framework: the Case of the Non-Small Coupling Parameter

This paper is dedicated to the global well-posedness issue of the incompressible Oldroyd-B model in the whole space \({\mathbb{R}^d}\) with \({d \geqq 2}\) . It is shown that this set of equations admits a unique global solution in a certain critical L ^p -type Besov space provided that the initial data, but not necessarily the coupling parameter, is small enough. As a consequence, even through the coupling effect between the equations of velocity u and the symmetric tensor of constrains τ is not small, one may construct the unique global solution to the Oldroyd-B model for a class of large highly oscillating initial velocity. The proof relies on the estimates of the linearized systems of (u, τ) and \({(u, \mathbb{P}{\rm div}\tau)}\) which may be of interest for future works. This result extends the work by Chemin and Masmoudi (SIAM J Math Anal 33:84–112, 2001) to the non-small coupling parameter case.     

A similarity solution for the flow and heat transfer over a moving permeable flat plate in an external free stream: case of strong injection

The previous work of Bachok et al. (Heat Mass Transf. 47:1643–1649, 2011) on the forced convection heat transfer on an isothermal moving surface in an external free stream is extended to the case when fluid injection through the surface, characterized by the parameter γ, is large. The asymptotic solution derived in this limit shows that the boundary layer has a double region structure, with an inviscid inner region of thickness O(γ) and an outer shear layer. Some further aspects of the original problem not treated in Bachok et al. (Heat Mass Transf. 47:1643–1649, 2011) are discussed as well as the analogous problem for a constant surface heat flux, where relatively small injection rates are seen to give rise to large increases in the surface temperature.     

A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation

The differential equation considered is \(y'' - xy = y|y|^\alpha \) . For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields similarity solutions to the well-known Korteweg-de Vries equation. Solutions are sought which satisfy the boundary conditions (1) y(∞)=0 (2) (1) $$y{\text{(}}\infty {\text{)}} = {\text{0}}$$ (2) $$y{\text{(}}x{\text{) \~( - }}\tfrac{{\text{1}}}{{\text{2}}}x{\text{)}}^{{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ as }}x \to - \infty $$ It is shown that there is a unique such solution, and that it is, in a certain sense, the boundary between solutions which exist on the whole real line and solutions which, while tending to zero at plus infinity, blow up at a finite x. More precisely, any solution satisfying (1) is asymptotic at plus infinity to some multiple kA i(x) of Airy's function. We show that there is a unique k^*(α) such that when k=k^*(α) the condition (2) is also satisfied. If 0 ^*, the solution exists for all x and tends to zero as x→-∞, while if k>k ^* then the solution blows up at a finite x. For the special case α=2 the differential equation is classical, having been studied by Painlevé around the turn of the century. In this case, using an integral equation derived by inverse scattering techniques by Ablowitz & Segur, we are able to show that k^*=1, confirming previous numerical estimates.     

Unsteady MHD double-diffusive convection boundary-layer flow past a radiate hot vertical surface in porous media in the presence of chemical reaction and heat sink

This paper presents an analytical study of the unsteady MHD free convective heat and mass transfer flow of a viscous, incompressible, gray, absorbing-emitting but non-scattering, optically-thick and electrically conducting fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot vertical plate with constant velocity. We employ a Darcian viscous flow model for the porous medium the Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The homogeneous chemical reaction of first order is accounted in mass diffusion equation. The governing equations are solved in closed form by Laplace-transform technique. A parametric study of all involved parameters is conducted and representative set of numerical results for the velocity, temperature, concentration, shear stress function $\frac{\partial u}{\partial y} \vert_{y=0}$ , temperature gradient $\frac{\partial \theta }{ \partial y}\vert_{y=0}$ , and concentration gradient $\frac{ \partial \phi }{\partial y}\vert_{y=0}$ is illustrated graphically and physical aspects of the problem are discussed.     

Effect of Conduction in Bottom Wall on B��nard Convection in a Porous Enclosure with Localized Heating and Lateral Cooling

Darcy-Bénard convection in a square porous enclosure with a localized heating from below and lateral cooling is studied numerically in the present paper. A finite-thickness bottom wall is locally heated, the top wall is kept at a lower temperature than the bottom wall temperature, and the lateral walls are cooled. The finite difference method has been used to solve the dimensionless governing equations. The analysis in the undergoing numerical investigation is performed in the following ranges of the associated dimensionless groups: the heat source length?? ${0.2\leq H \leq 0.9}$ , the wall thickness?? ${0.05\leq D \leq 0.4}$ , the thermal conductivity ratio?? ${0.8\leq K_{\rm r} \leq 9.8}$ , and the Biot number?? ${0.1\leq Bi \leq 1.1}$ . It is observed that the heat transfer rate could increase with increasing heat source lengths, thermal conductivity ratio, and cooling intensity. There exists a critical wall thickness for a high wall conductivity below which the increasing wall thickness increases the heat transfer rate and above which the increasing wall thickness decreases the heat transfer rate.     

Effects of pressure work and viscous dissipation on Graetz problem for gas flows in parallel-plate channels

The analytical solutions are obtained for the Graetz problem with pressure work and viscous dissipation in the thermal entrance region of the parallel-plate channels for two basic boundary conditions of uniform wall temperature and uniform wall heat flux involving fully developed laminar gas flows. The asymptotic Nusselt number is found to be zero instead of the conventionally accepted value of 7.54 for the uniform wall temperature case and (140/17)/ [1+(27/17) PrEc] for uniform wall heat flux case. The effects of pressure work and viscous dissipation contribute significantly to the asymptotic results for heat transfer and cannot be neglected under any circumstances in the case of uniform wall temperature. Sample results are presented to illustrate the effects of pressure work and viscous dissipation on heat transfer characteristics in the thermal entrance region.     

A Comment on the Flow and Heat Transfer Past a Permeable Stretching/Shrinking Surface in a Porous Medium: Brinkman Model

An analytical solution is presented for the boundary-layer flow and heat transfer over a permeable stretching/shrinking surface embedded in a porous medium using the Brinkman model. The problem is seen to be characterized by the Prandtl number $Pr$ , a mass flux parameter $s$ , with $s>0$ for suction, $s=0$ for an impermeable surface, and $s<0$ for blowing, a viscosity ratio parameter $M$ , the porous medium parameter $\Lambda $ and a wall velocity parameter $\lambda $ . The analytical solution identifies critical values which agree with those previously determined numerically (Bachok et al. Proceedings of the fifth International Conference on Applications of Porous Media, 2013) and shows that these critical values, and the consequent dual solutions, can arise only when there is suction through the wall, $s>0$ .     

Busemann Functions for the N-Body Problem

Following ideas in Maderna and Venturelli (Arch Ration Mech Anal 194:283–313, 2009), we prove that the Busemann function of the parabolic homotetic motion for a minimal central coniguration of the N-body problem is a viscosity solution of the Hamilton–Jacobi equation and that its calibrating curves are asymptotic to the homotetic motion.     

Mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux

A model is developed for the study of mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux. The model combined natural convection dominated and forced convection dominated film condensation, including effects of pressure gradient and interfacial vapor shear drag has been investigated and solved numerically. The separation angle of the condensate film layer, φ _s is also obtained for various pressure gradient parameters, P ^* and their corresponding dimensionless Grashof?'s parameters, Gr ^*. Besides, the effect of P ^* on the dimensionless mean heat transfer, will remain almost uniform with increasing P ^* until for various corresponding available values of Gr ^*. Meanwhile, the dimensionless mean heat transfer, is increasing significantly with Gr ^* for its corresponding available values of P ^*. For pure natural-convection film condensation, is obtained.     

Linear Stability of a Developing Thermal Front Induced by a Constant Heat Flux

A developing thermal front is set up by suddenly imposing a constant heat flux on the lower horizontal boundary of a semi-infinite fluid-saturated porous domain. The critical time for the onset of convection is determined using two main forms of analysis. The first of these is an approximate method which is effectively a frozen-time model while the second implements a set of parabolic simulations of monochromatic disturbances placed in the boundary layer at an early time. Results from the two approaches are compared and it is found that instability only occurs when the nondimensional disturbance wavenumber, $k$ k , is less than unity. The neutral curve for the primary mode possesses a vertical asymptote at $k=1$ k = 1 in wavenumber/time parameter space which is in contrast to the more usual teardrop shape which occurs when the surface is subject to a constant temperature. Asymptotic analyses are performed for the frozen-time model which yield excellent predictions for both branches of the neutral curve and the locus of the maximum growth rate curve at late times.