The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Stochastic equations in infinite dimensions

Physics of shock waves and high-temperature hydrodynamic phenomena - Ya.B. Zeldovich and Yu.P. Raizer (translated from the Russian and then edited by Wallace D. Hayes and Ronald F. Probstein); Dover Publications, New York, 2002, 944 pp., $34.

* The only book describing applications of nonlinear fiber optics * Two new chapters on the latest developments: highly nonlinear fibers and quantum applications* Coverage of biomedical applications* Problems provided at the end of each chapterThe development of new highly nonlinear fibers - referred to as microstructured fibers, holey fibers and photonic crystal fibers - is the next generation technology for all-optical signal processing and biomedical applications. This new edition has been thoroughly updated to incorporate these key technology developments.The book presents sound coverage of the fundamentals of lightwave technology, along with material on pulse compression techniques and rare-earth-doped fiber amplifiers and lasers. The extensively revised chapters include information on fiber-optic communication systems and the ultrafast signal processing techniques that make use of nonlinear phenomena in optical fibers.New material focuses on the applications of highly nonlinear fibers in areas ranging from wavelength laser tuning and nonlinear spectroscopy to biomedical imaging and frequency metrology. Technologies such as quantum cryptography, quantum computing, and quantum communications are also covered in a new chapter.This book will be an ideal reference for: RD scientists involved with research on fiber amplifiers and lasers; graduate students and researchers working in the fields of optical communications and quantum information. * The only book on how to develop nonlinear fiber optic applications* Two new chapters on the latest developments; Highly Nonlinear Fibers and Quantum Applications* Coverage of biomedical applications

Applications of nonlinear fiber optics

The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously.

/pdf/vanishing-of-vacuum-states-and-blow-up-phenomena-of-the-w4fs3n6gcj.pdf

Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

The current article aims to present a numerical analysis of MHD Williamson nanofluid flow maintained to flow through porous medium bounded by a non-linearly stretching flat surface. The second law of thermodynamics was applied to analyze the fluid flow, heat and mass transport as well as the aspects of entropy generation using Buongiorno model. Thermophoresis and Brownian diffusion is considered which appears due to the concentration and random motion of nanoparticles in base fluid, respectively. Uniform magnetic effect is induced but the assumption of tiny magnetic Reynolds number results in zero magnetic induction. The governing equations (PDEs) are transformed into ordinary differential equations (ODEs) using appropriately adjusted transformations. The numerical method is used for solving the so-formulated highly nonlinear problem. The graphical presentation of results highlights that the heat flux receives enhancement for augmented Brownian diffusion. The Bejan number is found to be increasing with a larger Weissenberg number. The tabulated results for skin-friction, Nusselt number and Sherwood number are given. A decent agreement is noted in the results when compared with previously published literature on Williamson nanofluids.

https://www.mdpi.com/1099-4300/22/1/18/pdf

Entropy Generation and Consequences of Binary Chemical Reaction on MHD Darcy–Forchheimer Williamson Nanofluid Flow Over Non-Linearly Stretching Surface

We investigate the global strong solutions for a system of equations related to the incompressible viscoelastic fluids of Oldroyd--B type with the initial data close to a stable equilibrium. We obtain the existence and uniqueness of the global solution in a functional setting invariant by the scaling of the associated equations, where the initial velocity has the same critical regularity index as the incompressible Navier--Stokes equations, and one more derivative is needed for the deformation tensor. Like the classical incompressible Navier--Stokes equations, one may construct the unique global solution for a class of large highly oscillating initial velocity. Our result also implies that the deformation tensor $F$ has the same regularity as the density of the compressible Navier--Stokes equations.

Global Existence of Strong Solution for Equations Related to the Incompressible Viscoelastic Fluids in the Critical $L^p$ Framework

We consider the long-time behavior and optimal decay rates of global strong solution to three-dimensional isentropic compressible Navier–Stokes (CNS) system in the present paper. When the regular initial data also belong to some Sobolev space with l⩾4 and s∈[0, 1], we show that the global solution to the CNS system converges to the equilibrium state at a faster decay rate in time. In particular, the density and momentum converge to the equilibrium state at the rates (1 + t)−3/4−s/2 in the L2-norm or (1 + t)−3/2−s/2 in the L∞-norm, respectively, which are shown to be optimal for the CNS system. Copyright © 2010 John Wiley & Sons, Ltd.

Large time behavior of isentropic compressible Navier–Stokes system in ℝ 3

This paper aims to analyze the features of MHD Darcy-Forchheimer nanofluid flow over a nonlinear stretching sheet. A viscous incompressible nanofluid saturates the porous medium via Darcy-Forchheimer relation. Heat and mass transfer is analyzed through Brownian motion factor and Thermophoresis. A non-uniform magnetic field is induced to enhance the electric conductivity of the nanofluid. The model emphasis on small magnetic Reynolds number and boundary layers formulation. The governing Partial differential Equations are converted into nonlinear Ordinary differential equations using similarity transformations and thereafter solved utilizing homotopy analysis method. Graphs are sketched for different values of various fluid parameters. Drag force, heat flux and mass flux are interpreted numerically. The porosity factor results in rising the skin friction due to high resistance offered by porous medium. The heat and mass flux are reduced for a stronger porosity factor, which is an important finding for industrial applications of nanofluids. The temperature profile shows augments for increasing values of both the Brownian diffusion and Thermophoresis.

Magnetohydrodynamic Darcy–Forchheimer nanofluid flow over a nonlinear stretching sheet

In this paper, we consider the existence and asymptotic behavior of the global weak solutions to a two-dimensional (2D) viscous liquid-gas two-phase flow model. The analysis is based on several key a priori estimates, which are obtained by the ideas of studying the single-phase Navier–Stokes equations. This can be viewed to be a generalization of the results in [S. Evje and K. H. Karlsen, J. Differential Equations, 245 (2008), pp. 2660–2703] from one dimension to two dimensions.

Ting Zhang

Papers

Entropy Generation and Consequences of Binary Chemical Reaction on MHD Darcy–Forchheimer Williamson Nanofluid Flow Over Non-Linearly Stretching Surface

Global Existence of Strong Solution for Equations Related to the Incompressible Viscoelastic Fluids in the Critical $L^p$ Framework

Large time behavior of isentropic compressible Navier–Stokes system in ℝ 3

Magnetohydrodynamic Darcy–Forchheimer nanofluid flow over a nonlinear stretching sheet

Existence and Asymptotic Behavior of Global Weak Solutions to a 2D Viscous Liquid-Gas Two-Phase Flow Model