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Fourth-Order Ordinary Differential Equations
Published in Saad A. Ragab, Hassan E. Fayed, Introduction to Finite Element Analysis for Engineers, 2018
Saad A. Ragab, Hassan E. Fayed
This chapter presents a finite‐element model for fourth‐order ordinary differential equations. In solid mechanics, a typical equation is the Euler‐Bernoulli beam equation that governs deflection of beams. The beam equation is then combined with the bar equation of Chapter 2 to analyze structures such as plane frames and trusses. Principles of virtual displacements and minimum total potential energy are frequently used to derive finite‐element models in structural and solid mechanics. Starting with the equations of equilibrium of a continuum and boundary conditions, we derive the principle of virtual displacements, which is mathematically identical to the weak form of the problem. We then present the principle of minimum total potential energy and demonstrate its use for deriving finite‐element models for bars and beams. In fluid mechanics, a well‐known fourth‐order equation is the Orr‐Sommerfeld equation which is encountered in hydrodynamic stability analysis of shear flows. However, it is a differential eigenvalue problem and will be presented in Chapter 7.
Heat and mass transfer in silicon-based nanostructures
Published in Klaus D. Sattler, Silicon Nanomaterials Sourcebook, 2017
One of the limitations of this model is that it does not define the valid region pertaining to a spatial area. When there is no certain constraints, the model can cause misconception for defining the effective area of the boiling. On a confined boiling surface, the hydrodynamic stability would be maintained by balancing the up-flowing vapor columns and the down-flowing liquid (Zuber 1959; Lienhard and Dhir 1973). Based on the hydrodynamic instability model described in Figure 11.25, CHF occurs when the hydrodynamic stability is lost, inducing a large retarding force on the down-flowing liquid (Lu et al., 2011). Therefore, the impacts of hemi-wicking can be confined considering the hydrodynamic stability by adopting the Rayleigh–Taylor interfacial stability wavelength at the effective length scale. The Rayleigh–Taylor interfacial stability wavelength λRT is given by 2π(σ/(g(ρl-ρv)))1/2 in the most critical cases (Lamb 1959; Liter 2000). The wavelength simply indicates the characteristic length of an unstable interface between two fluids of different densities (i.e., up-flowing vapor and down-flowing liquid). Therefore, the amount of heat dissipated by hemi-wicking can be revised using the concept of the valid heating area for hydrodynamic stability as a criterion (Kim et al., 2014a):
Ocean Hydrodynamics
Published in Victor Raizer, Optical Remote Sensing of Ocean Hydrodynamics, 2019
Stability is an important aspect of any dynamical system. Helmholtz established the science of hydrodynamic stability in 1868. Since then, hydrodynamic stability has become the main tool in analysis of small perturbations in sheared flow including turbulence as well (Section 2.6).Hydrodynamic instabilities have been studied extensively; see books (Zakharov 1968; Chandrasekhar 1981; Riahi 1996; Moiseev et al. 1999; Schmid and Henningson 2001; Drazin 2002; Drazin and Reid 2004; Rahman 2005; Manneville 2010; Charru 2011; Sengupta 2012; Yaglom and Frisch 2012; Gaissinski and Rovenski 2018). One of the first experiments demonstrating instability of surface waves was conducted by Su and Green (1984).
Direct numerical simulations of second-order Stokes wave driven smooth-walled oscillatory channel: investigation of net current formation
Published in Journal of Turbulence, 2019
C. E. Ozdemir, X. Yu, S. Sororian, L. Zhu, M. Tyagi, S. Haddadian, D. Oliveira, C. N. Turnipseed, H. Kefelegn
As it is closely related to the findings that will be presented shortly in this section, it is helpful to describe the flow regimes in Stokes flows, i.e. , over smooth wall. With increasing , Stokes flow reaches the following states: (i) laminar () [20,21], (ii) disturbed laminar () [21,22], (iii) self-sustaining transitional flow () [23], (iv) intermittently turbulent () [20,22,24], and (v) fully turbulent () [24]. While there is absolute hydrodynamic stability in the laminar regime, there occur growth of infinitesimal velocity disturbance and formation of vortex rollers in the disturbed laminar regime. However, the aforementioned growth of velocity fluctuations never becomes nonlinear. In self-sustaining transitional regime, velocity fluctuations saturate and start growing nonlinearly but decay without establishing developed turbulence. With further increase in , the Stokes flow becomes intermittently turbulent where developed turbulence occurs twice a cycle: one during the forward flow and the other during the reverse flow. In fully turbulent flow, developed turbulence is maintained throughout the cycle.
Magnetorotational instability in Taylor-Couette flows between cylinders with finite electrical conductivity
Published in Geophysical & Astrophysical Fluid Dynamics, 2018
G. Rüdiger, M. Schultz, F. Stefani, R. Hollerbach
Taylor-Couette flows with stationary inner cylinder are clearly the most basic models for super-rotating fluids. There is a long history to probe their hydrodynamic stability or instability. Interestingly enough, it has been shown by numerical simulations for very high Reynolds numbers that a hydrodynamic instability does exist (Deguchi 2017). In a previous paper we have shown that under the influence of current-free azimuthal fields such flows become unstable even for much lower Reynolds numbers of order . The results, however, are only valid for narrow gaps between the cylinders, with the consequence that the solutions for perfectly conducting and insulating cylinders differ significantly. Only for perfectly conducting cylinders is small enough for realistic experiments (Rüdiger et al.2018b).
Mathieu stability of offshore triceratops under postulated failure
Published in Ships and Offshore Structures, 2018
Srinivasan Chandrasekaran, P. A. Kiran
Mathieu stability charts are also useful in assessing tether instability under dynamic tether tension variations (McLachlan 1947). Stability charts represent conditional solution for Mathieu's equation due to the fact that a general solution to the said equation cannot be obtained. This is due to the fact that Mathieu parameters are case-specific. Patel and Park (1991) investigated Mathieu stability of tethers considering them as simply supported columns under constant axial tension. Excluding nonlinear damping, governing equation is formulated using Galerkin's technique, which resulted in Mathieu equation. Mathieu stability charts are extended to large parameters using perturbation method and Runge–Kutta method as compliant structures exhibit large range of values for Mathieu functions (Ince 1925; Goldstein 1929). Safe operability of compliant structures under wave loads is assessed using Mathieu stability under exceeding of load amplitudes (Koo et al. 2004; Chandarsekaran et al. 2006; Banik & Datta 2008). Studies show successful examinations of deep-draft platforms like spar for Mathieu stability (Haslum & Faltinsen 1999; Rho et al. 2002, 2003). While deep-draft platform enables effective hydrodynamic stability to spar, instability can occur when pitch natural period becomes closer to twice of that of heave period (Simos & Pesce 1997; Zhang et al. 2002).