A naive approach to the two‐dimensional boundary layer equation in fluid flow

Viscoelastic fluids can be difficult to model due to the wide range of different physical behaviors that polymer melts can exhibit. One such feature is the viscous elastic boundary layer. We address the particular problem of a viscoelastic shear-dependent fluid flowing past a corner and investigate how the properties of the boundary layer change for a White-Metzner fluid. The boundary layer equations are derived and the upstream layer is matched with the far-field flow. It was found that if the fluid is sufficiently shear thinning then the viscoelastic boundary layer formulation fails due to the inertial forces becoming dominant. The depth of the boundary layer is controlled by the shear-thinning parameters. These effects are not a feature of other shear-thinning models, such as the Phan-Thien-Tanner model. This study provides insight in the different effects of some commonly used viscoelastic models in corner flows in the upstream boundary layer, the downstream boundary layer is not addressed.

This paper studies the boundary layer flow of unsteady incompressible and viscous fluid passing over a porous sphere with forced convection. The non-magnetic viscous fluid was induced by magnetic field of the magnetic porous sphere and it becomes Magnetohydrodynamics (MHD) viscous fluid. The dimensional governing equations are developed from continuity equation, momentum equation, and energy equation. Thus, the governing equations are transformed into boundary layer equations. The boundary layer equations are further converted into non-similar boundary layer and solved numerically by using the Keller-Box method. The outcomes of this research are the effect of magnetic parameter, convection parameter, Prandtl number, porosity parameter, and permeability parameter towards velocity and temperature profiles. When magnetic parameter, convection parameter, and porosity parameter increase and permeability parameter decreases, the velocity profiles increase while the temperature profiles decrease. As an addition, the increment of Prandtl number does not affect the velocity profiles significantly.

Nitrogen oxides have been shown to catalyze the destruction of stratospheric ozone. Previous estimates of nitric oxide emissions from high‐altitude aircraft have taken into account engine emissions only. This study was undertaken to determine whether nitric oxide production in the hot viscous boundary layer surrounding the skin of a hypersonic aircraft can significantly increase estimates of total NO emissions. The described model approximates the viscous boundary layer as a series of finite stream tubes with time‐dependent and temperature‐dependent chemical kinetics and mass flow rates. Nitric oxide equilibrium mole fraction for air peaks at ≅ 3700 K. At hypersonic speeds the viscous effects near the aircraft skin will induce these very high temperatures. Along the skin surface of a 60 m craft, the NO mole fraction can be locally as high as 0.044. Nitric oxide output from this thin boundary layer becomes significant when integrated over the entire trailing edge of the aircraft. This model predicts that nitric oxide production in the boundary layer should be taken into account at speeds above Mach 8. Above Mach 8, boundary layer volume and temperature increase rapidly. At stratospheric speeds approaching Mach 16, the nitric oxide production in the boundary layer increases to the point where it roughly equals the nitric oxide output from the engines. Above Mach 16, the boundary layer produces the majority of the NO emissions. This additional source of nitric oxide will play an ever‐increasing role as aircraft fly at higher speeds and altitudes. Significant errors in total NO emission estimates will result if the viscous boundary layer is not taken into account at speeds above Mach 8.

The famous Toms effect (1948) consists of a substantial increase of the critical Reynolds number when a small amount of soluble polymer is introduced into water. The most noticeable influence of polymer additives is manifested in the boundary layer near solid surfaces. The task includes the ratio of two characteristic length scales, one of which is the Prandtl scale, and the other is defined as the square root of the normalized coefficient of relaxation viscosity (Frolovskaya and Pukhnachev, 2018) and does not depend on the characteristics of the motion. In the limit case, when the ratio of these two scales tends to zero, the equations of the boundary layer are exactly integrated. One of the goals of the present paper is group analysis of the boundary layer equations in two mathematical models of the flow of aqueous polymer solutions: the second grade fluid (Rivlin and Ericksen, 1955) and the Pavlovskii model (1971). The equations of the plane non-stationary boundary layer in the Pavlovskii model are studied in more details. The equations contain an arbitrary function depending on the longitudinal coordinate and time. This function sets the pressure gradient of the external flow. The problem of group classification with respect to this function is analyzed. All functions for which there is an extension of the kernels of admitted Lie groups are found. Among the invariant solutions of the new model of the boundary layer, a special place is taken by the solution of the stationary problem of flow around a rectilinear plate.

The behavior of glottal flow can, to a large extent, be characterized by development and separation of the boundary layer. The point of flow separation is known to vary during the phonatory cycle due to change in channel configuration. To take the movable nature of the separation point into account, the boundary-layer equation is solved numerically, and the values of the characteristic quantities are determined as well as the point of separation. Development of the boundary layer in general reduces the effective size of the channel, and, therefore, increases the core flow velocity, which, in turn provides the boundary condition of the boundary-layer equation. The interaction between the viscous (boundary layer) and inviscid (core flow) parts of the glottal flow is, therefore, strongly indicated. To apply this viscous-inviscid interaction, the expression of the core flow is derived for a two-dimensional flow field, and is solved jointly with the boundary-layer equation. Numerical results are shown to examine the effect of the Reynolds number and glottal configuration, with special emphasis on the comparison of flow models developed for one- and two-dimensional flow fields. Numerical results are also quantitatively compared with data obtained from flow measurement experiments.

Steady flow of a viscous fluid around a transversely oscillating, simply-supported beam is driven by forces represented in the nonlinear terms of the Navier-Stokes’ equation. This flow occurs inside and outside the viscous boundary layer. The flow-velocity amplitude is proportional to the square of the amplitude of the beam transverse velocity and thus it is essentially only observable during high velocity beam vibration. This flow, known as ‘edge streaming’, does not occur in an ideal fluid. In the flow the fluid is ‘drawn in’ at the edges of the beam near the vibrational antinodal points and ‘expelled’ at the nodal points. When a rigid plate is placed parallel to the long axis of the vibrating beam and normal to the vibration velocity, vortices of different orientation are generated depending on the separation of the plate and beam surfaces. When the separation is an order of magnitude greater than the boundary layer thickness, the vortices have the rolling characteristic discussed by Rayleigh [1]. But for small separation of the order of the boundary layer thickness, the vortical axes are perpendicular to that of the former, parallel to the vibration velocity. They have been experimentally observed [6]. Owing to the streaming motion, damping, as measured by the ratio of the dissipation of energy and the total vibration energy over a time period, has a nonlinear hard representation with respect to the vibration amplitude.

The present study is devoted to discrepancies between experimental and theoretical runup heights on an inclined plane, which have occasionally been reported in the literature. In a new study on solitary wave-runup on moderately steep slopes, in a wave tank with 20 cm water depth, detailed observations are made for the shoreline motion and velocity profiles during runup. The waves are not breaking during runup, but they do break during the subsequent draw-down. Both capillary effects and viscous boundary layers are detected. In the investigated cases the onshore flow is close to the transitional regime between laminar and turbulent boundary layers. The flow behaviour depends on the amplitude of the incident wave and the location on the beach. Stable laminar flow, fluctuations (Tollmien-Schlichting waves), and formation of vortices are all observed. Comparison with numerical simulations showed that the experimental runup heights were markedly smaller than predictions from inviscid theory. The observed and computed runup heights are discussed in the context of preexisting theory and experiments. Similar deviations are apparent there, but have often been overlooked or given improper physical explanations. Guided by the absence of turbulence and irregular flow features in parts of the experiments we apply laminar boundary layer theory to the inundation flow. Outer flows from potential flow models are inserted in a nonlinear, numerical boundary layer model. Even though the boundary layer model is invalid near the moving the shoreline, the computed velocity profiles are found to compare well with experiments elsewhere, until instabilities are observed in the measurements. Analytical, linear boundary layer solutions are also derived both for an idealized swash zone motion and a polynomial representation of the time dependence of the outer flow. Due to lacking experimental or theoretical descriptions of the contact point dynamics no two-way coupling of the boundary layer model and the inviscid runup models is attempted. Instead, the effect of the boundary layer on the maximum runup is estimated through integrated losses of onshore volume transport and found to be consistent with the differences between inviscid theory and experiments.

The inlet region of a plane-parallel pressure flow is the subject of this study. The patterns of variations in the hydrodynamic inlet region under unsteady plane-parallel pressure flow of a viscous fluid are examined in this research. Based on the boundary layer equation and flow characteristics, the boundary conditions of the problem were determined and a boundary value problem was formulated. The problem's boundary conditions were established and a boundary value problem was developed based on the boundary layer equation and flow characteristics. In order to find patterns of velocity change over time and over the length of the inlet region under general boundary conditions, a method for integrating the boundary value conditions was created. Solutions for scenarios with a constant and parabolic velocity distribution in the inlet region were derived from the general solutions. Regularities of pressure and velocity change were found along the entire hydrodynamic inlet region. Using computer analysis, graphs of velocity changes over time at various points along the entire length of the inlet region are constructed. The patterns of velocity distribution along the entire length of the inlet region depending on time can be seen using graphs. This allows one to estimate the length of the hydrodynamic inlet region and calculate the fluid flow velocity at any point in this region. The findings enable revealing the essence of the processes running in an hydropneumatic automation system's transition sections. Based on the revealed regularities of the hydrodynamic parameters of viscous incompressible liquid during unsteady flows, it is possible to correctly design of the automatic systems' channels of regulating units ensuring their smooth and accurate operation

The viscous boundary layer on a finite flat plate in a stream which reverses its direction once (at t = 0) is analysed using an improved version of the approximate method described earlier (Pedley 1975). Long before reversal (t < −t1), the flow at a point on the plate will be quasi-steady; long after reversal (t > t2), the flow will again be quasi-steady, but with the leading edge at the other end of the plate. In between (−t1 < t < t2) the flow is governed approximately by the diffusion equation, and we choose a simple solution of that equation which ensures that the displacement thickness of the boundary layer remains constant at t = −t1. The results of the theory, in the form of the wall shear rate at a point as a function of time, are given both for a uniformly decelerating stream, and for a sinusoidally oscillating stream which reverses its direction twice every cycle. The theory is further modified to cover streams which do not reverse, but for which the quasi-steady solution breaks down because the velocity becomes very small. The analysis is also applied to predict the wall shear rate at the entrance to a straight pipe when the core velocity varies with time as in a dog's aorta. The results show positive and negative peak values of shear very much larger than the mean. They suggest that, if wall shear is implicated in the generation of atherosclerosis because it alters the permeability of the wall to large molecules, then an appropriate index of wall shear at a point is more likely to be the r.m.s. value than the mean.

The study of heat transfer along a wavy surface with a prescribed heat flux proportional to (1+x 2) m (wherem is constant) is investigated. The governing equations are first cast into a non-dimensional form by using suitable boundary layer variables that substract out the effects of the wavy surface from the boundary conditions. The boundary layer equations are then solved numerically by the implicit finite difference method. The numerical results are presented for various values of exponentm, amplitudea and Prandtl number Pr. It was found that for amplitudea=0.2 a point of separation appears atx=2.8 that makes the solution restricted at this amplitude. In addition the wavelength of the local Nusselt number and surface temperature variation was equal to the wavy surface. The wavelength of the average Nuselt number was half of the wavy surface.

In this paper we have analyzed the steady-state process leading to ignition of a combustible mixture of hydrogen, oxygen and nitrogen by a hot flat plate in a boundary layer flow. For plate temperatures larger than the crossover temperature dictated by the competition between the chain branching reaction H + 02 → OH + O and the chain breaking reaction H + 02 + M → H02 + M, the ignition event corresponds to a typical chain branching explosion with negligible heat release, in a first approximation. The boundary layer equations are solved using the fact that the activation energy of the chain branching reaction H + 02 → OH + O is relatively large, employing the reduced kinetic mechanism appropriate for this regime. The equations reduce to a single integro-differential equation for the concentration of atomic hydrogen. The ignition condition can be assumed to be reached when one of the shuffle reactions reaches partial equilibrium. On the other hand, for low plate temperatures, the ignition event is characterized by a thermal runaway. The governing equations reduce to a single one-parameter integro-differential equation, to be solved numerically.

The interaction of supernova shocks and interstellar clouds is an important astrophysical phenomenon since it can result in stellar and planetary formation. Our experiments attempt to simulate this mass-loading as it occurs when a shock passes through interstellar clouds. We drive a strong shock using the Omega laser (∼5 kJ) into a foam-filled cylinder with an embedded Al sphere (diameter D = 120µm) simulating an interstellar cloud. The density ratio between Al and foam is ∼9. We have previously reported on the interaction between shock and cloud, the ensuing Kelvin-Helmholtz and Widnall instabilities, and the rapid stripping of all mass from the cloud. We now present a theory that explains the rapid mass-stripping. The theory combines (1) the integral momentum equations for a viscous boundary layer, (2) the equations for a potential flow past a sphere, (3) Spalding’s law of the wall for turbulent boundary layers, and (4) the skin friction coefficient for a turbulent boundary layer on a flat plate. The theory gives as its final result the mass stripped from a sphere in a turbulent high Reynolds number flow, and it agrees very well with our experimental observations.

Numerical simulation of the boundary-layer flow by using the boundary-layer equations and an imposed edge condition inevitably encounters a difficulty when the flow tends to separate from the surface. For avoiding this numerical difficulty and ensuring a successful computation over the recirculating region, an interacting boundary-layer edge condition may be employed to substitute that imposed one. This interacting boundary-layer edge condition is based on the pressure distribution imposed on the body surface by its outer inviscid flow and also includes a correction that properly takes the boundary-layer's own effects into account. Consequently, the boundary-layer equations and their interacting edge condition are mutually coupled and the full linearization of both of them is essential to obtain an efficient computational procedure. Therefore this study focuses on the full linearization of the two-dimensional turbulent boundary-layer equations and their interacting edge condition; some strategies associate with it are also discussed.

Dynamics in the boundary layer of a flat particle

Air–Sea Gas Exchange