The Horton–Rogers–Lapwood Problem Revisited: The Effect of Pressure Work

The problem of the onset of convective roll instabilities in a horizontal porous layer with isothermal boundaries at unequal temperatures, well known as the Horton–Rogers–Lapwood problem, is revisited including the effect of pressure work and viscous dissipation in the local energy balance. A linear stability analysis of rolls disturbances is performed. The analysis shows that, while the contribution of viscous dissipation is ineffective, the contribution of the pressure work may be important. The condition of marginal stability is investigated by adopting two solution procedures: method of weighted residuals and explicit Runge–Kutta method. The pressure work term in the energy balance yields an increase of the value of the Darcy–Rayleigh number at marginal stability. In other words, the effect of pressure work is a stabilizing one. Furthermore, while the critical value of the Darcy– Rayleigh number may be considerably affected by the pressure work contribution, the critical value of the wave number is affected only in rather extreme cases, i.e. for very high values of the Gebhart number. A nonlinear stability analysis is also performed pointing out that the joint effects of viscous dissipation and pressure work result in a reduction of the excess Nusselt number due to convection, when the Darcy–Rayleigh number is replaced by the superadiabatic Darcy–Rayleigh number.     

The R-γ transition prediction model

The R_t turbulence closure (Goldberg 2003 Goldberg, U. 2003. “Turbulence Closure with a Topography-parameter-free Single Equation Model.” IJCFD 17 (1): 27–38.[Web of Science ®] , [Google Scholar]) is coupled with an intermittency transport equation, γ, to enable prediction of laminar-to-turbulent flow by-pass transition. The model is not correlation-based and is completely topography-parameter-free, thus ready for use in parallelized Computational Fluid Dynamics (CFD) solvers based on unstructured book-keeping. Several examples compare the R-γ model's performance with experimental data and with predictions by the Langtry–Menter γ-Re_θ transition closure (2009) Langtry, R.B., and F.R. Menter. 2009. “Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes.” AIAA J 47 (12): 2894–2906.[Crossref] , [Google Scholar]. Like the latter, the R-γ model is very sensitive to freestream turbulence levels, limiting its utility for engineering purposes.     

The discrete Babuška-Brezzi condition

Summary In the treatment of saddle point problems with finite elements a discrete Babuka-Brezzi condition is encountered. We demonstrate what this discrete Babuka-Brezzi condition entails and explain the mechanical background of this somewhat abstract inf sup condition.

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Die diskrete Babuka-Brezzi-Bedingung

Übersicht Behandelt man Sattelpunktprobleme mit Finiten Elementen, dann stößt man auf eine diskrete Babuka-Brezzi Bedingung. Wir zeigen, welche Rolle diese diskrete Babuka-Brezzi-Bedingung bei der numerischen Behandlung spielt und verdeutlichen den mechanischen Hintergrund dieser etwas abstrakten inf sup Bedingung.

     

The torsion problem—A new twist

An electrical analogy for the solution of Poisson's equation which is both quick and cheap is described. Accuracy to within a few percent may be obtained. Tests giving comparisons with calculated values for typical cross sections are described.     

The Generalized Proudman�CJohnson Equation Revisited

We demonstrate the existence of solutions to the inviscid generalized Proudman–Johnson equation for parameters a lying in the open interval (−5,−1) which develop singularities in finite time; moreover, we show that there are solutions which exist for all times if a = −1. Finally, a simple blow-up criterion for solutions arising from a special class of initial data is given.     

The Continuous Coagulation-Fragmentation¶Equations with Diffusion

Existence of global weak solutions to the continuous coagulation-fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition. Our approach relies on weak and strong compactness methods in L ¹ in the spirit of the DiPerna-Lions theory for the Boltzmann equation. Under the detailed balance condition the large-time behaviour is also studied.     

The Strongest Structures with Macrostiffness Constraints∗

An example of a truss structure with limited global stiffness, measured by some parameter k, is discussed. The optimal structural design problem of maximizing the external load intensity, with axial stresses in the rods of the truss remaining within an allowable range, is well known and leads to a statically determinate solution with uniformly loaded elements. However, if one takes into account the additional constraint on stiffness k, the problem is open and the solution leads, in general, to a redundant truss. The paper presents this problem and proposes its application to optimal design of supporting structures     

The Forced Response of the Lamped Dynamic Systems

This paper deals with the forced harmonie responses of the discrete or continuous dampedsystems to harmonic excitation,where the viscous damping matrix cannot be diagonalized.Theexplicit expressions of the response solutions are given.Hence,with these expressions,some generaland analytical study of some phenomena in vibration is made.For example,the“fixed amplitudepoint”phenomena in the single damped systems have been demonstrated generally.The conditions,under which all forces that exert on the system and possess common phase,which will excite com-mon phase responses,are also discussed.The solutions deduced here only involve the inverse matrices of lower order.Thus,in thenumerical computation for digital computers,the method is more simple,economical,and accuratethan others.The method described here can be used in the analysis of unbalance responses of the rotorsystems.     

The Logarithmic Tail of Néel Walls

  We study the multiscale problem of a parametrized planar 180° rotation of magnetization states in a thin ferromagnetic film. In an appropriate scaling and when the film thickness is comparable to the Bloch line width, the underlying variational principle has the form

The quaternion beam model for hard-magnetic flexible cantilevers

The recently developed hard-magnetic soft(HMS) materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields, e.g., soft robotics, flexible electronics,and biomedicine. Theoretical investigations on large deformations of HMS structures are significant foundations of their applications. This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar ...     

The “Butterfly Effect” in a Porous Slab

It is shown that the governing equation for the stream function of the Darcy free convection boundary layer flows past a vertical surface is invariant under arbitrary translations of the transverse coordinate y. The consequences of this basic symmetry property on the solutions corresponding to a prescribed surface temperature distribution T _w (x) are investigated. It is found that starting with a “primary solution” which describes the temperature boundary layer on an impermeable surface, infinitely many “translated solutions” can be generated which form a continuous group, the “translation group” of the given primary solution. The elements of this group describe free convection boundary layer flows from permeable counterparts of the original surface with a transformed temperature distribution [(T)\tilde]_w ( x ){\tilde {T}_w \left( x \right)}, when simultaneously a suitable lateral suction/injection of the fluid is applied. It turns out in this way that several exact solutions discovered during the latter few decades are in fact not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.     

The moiré method applied to three-dimensional elastic problems

The purpose of this paper is to develop the technique of moiré for three-dimensional strain and stress analysis. The classical problem of contact of a sphere with a semi-infinite space is studied, and experimental results are compared with those of Hertz's solution. Results show that the method is accurate and has a high sensitivity. The results prove also that the strain distribution in a very small region with a steep gradient can be detected quite precisely.Paper was presented at the 1964 SESA Spring Meeting held in Salt Lake City, Utah, on May 6–8.     

The Gel’fand Problem for the Biharmonic Operator

We study stable and finite Morse index solutions of the equation ${\Delta^2 u = {e}^{u}}$ . If the equation is posed in ${\mathbb{R}^N}$ , we classify radial stable solutions. We then construct nonradial stable solutions and we prove that, unlike the corresponding second order problem, no Liouville-type theorem holds, unless additional information is available on the asymptotics of solutions at infinity. Thanks to this analysis, we prove that stable solutions of the equation on a smoothly bounded domain (supplemented with Navier boundary conditions) are smooth if and only if ${N \leqq 12}$ . We find an upper bound for the Hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded Morse index, provided they are controlled in a suitable Morrey norm.     

On The Reissner Theory of Bending of Flastic Plates

The Reissner equations of elastic plates are rederived on the bases of the incomplete generalizedvariational principle of Complementary energy.The Stress function to is naturally obtained from thevariational Calculation in the form of Lagrange multiplier.The stucture of solutions of the Reissnerequations is thus defined.On the bases of these discussions,a simplified theory has been put forward,in which the equations of equilibrium involving the shearing influence can be reduced into a fourthorder differential equation similar to those of the Classical theory of plates.     

The First National Conference on Chemical and Biochemical Engineering

Recognizing the mounting interest in the union between chemical engineering and biochemical processing, Academician Prof. Pingkai Ouyang and Prof. Nanping Xu played pioneering roles in organizing the First National Conference on Chemical and Biochemical Engineering under the auspices of the Nanjing University of Technology and the Chemical Industry and Engineering Society of China,