Similarity solution for a spherical shock wave in a non‐ideal gas under the influence of gravitational field and monochromatic radiation with increasing energy

The propagation of a spherical shock wave in a non‐ideal gas with or without gravitational effects is investigated under the action of monochromatic radiation. Similarity solutions are obtained for adiabatic flow between the shock and the piston. The numerical solutions are obtained using the Runge‐Kutta method of the fourth order. The density of the gas is assumed to be constant. The total energy of the shock wave is non‐constant and varies with time. The effects of change in values of non‐idealness parameter, gravitational parameter, shock Mach number, radiation parameter, and adiabatic exponent of the gas on shock strength and flow variables are worked out in detail. It is investigated that the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and, therefore, the distance between the inner contact surface and the shock surface is reduced. A comparison is also made between the solutions in the cases of the gravitating and the non‐gravitating media. It is manifested that the gravitational parameter and the radiation parameter have in general opposite behaviour on the flow variables and the shock strength.     

Potential flow of fluid from an elevated,two-dimensional source

When fluid is pumped from an elevated source it flows downward and then outward once it hits the base. In this paper we consider a simple two dimensional model of flow from a single line source elevated above a horizontal base and consider its downward flow into a spreading layer on the bottom. A hodograph solution and linear solutions are obtained for high flow rates and full nonlinear solutions are obtained over a range of parameter space. It is found that there is a minimum flow rate beneath which no steady solutions exist. Overhanging surfaces are found for a range of parameter values. This flow serves as a model for a two-dimensional water fountain, or approximates a similar flow in a density stratified environment.     

Spherical shock wave generated by a moving piston in mixture of a non-ideal gas and small solid particles under a gravitational field

Self-similar solutions are obtained for one-dimensional isothermal and adiabatic unsteady flows behind a strong spherical shock wave propagating in a dusty gas. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of a non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-conditions are maintained and variable energy input is continuously supplied by the piston. The medium is under the influence of the gravitational field due to a heavy nucleus at the origin (Roche model). The effects of an increase in the mass concentration of solid particles, the ratio of the density of the solid particles to the initial density of the gas, the gravitational parameter and the parameter of non-idealness of the gas in the mixture, are investigated. It is shown that due to presence of gravitational field the compressibility of the medium at any point in the flow-field behind the shock decreases and all other flow-variables and the shock strength increase. A comparison has also been made between the isothermal and adiabatic flows. It is investigated that the singularity in the density and compressibility distributions near the piston in the case of adiabatic flow are removed when the flow is isothermal.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

Self-similar strong shocks in non-uniform atmosphere

The propagation of strong shocks in an atmosphere of variable density at rest is studied. The energy gain of the flow enveloped by the shock is assumed to be time-dependent. Analytical and numerical solutions of the similarity flows behind such shocks are obtained.     

UNSTEADY NATURAL CONVECTIVE BOUNDARY LAYER FLOW AND HEAT TRANSFER OF FRACTIONAL SECOND-GRADE NANOFLUIDS WITH DIFFERENT PARTICLE SHAPES

The present study is concerned with unsteady natural convective boundary layer flow and heat transfer of fractional second-grade nanofuids for different particle shapes. Nonlinear boundary layer governing equations are formulated with time fractional derivatives in the momentum equation. The governing boundary layer equations of continuity, momentum and energy are reduced by dimensionless variable. Numerical solutions of the momentum and energy equations are obtained by the finite difference method combined with L1-algorithm. The quantites of physical interest are graphically presented and discussed in details. It is found that particle shape, fractional derivative parameter and the Grashof number have profound influences on the the flow and heat transfer.     

Evolution of weak discontinuities in a non-ideal radiating gas

A method of wavefront analysis is used to study the formation of shock waves in a two-dimensional steady supersonic flow of a non-ideal radiating gas past plane and axisymmetric bodies. The gas is taken to be sufficiently hot for the effect of thermal radiation to be significant, which is, of course, treated by the optically thin approximation to the radiative transfer equation. Transport equations, which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront, is derived. The influence of the parameter of the non-idealness, upstream flow Mach number in the presence of thermal radiation on the behavior of the wavefront are examined.     

Dense underflow into a lake or reservoir—Sub-critical flow.

Steady two-dimensional flow of a dense stream down a slight embankment into a lake or a reservoir is considered. The inflowing water is separated from the ambient lake water by a density interface. This work follows on from earlier work in which the flows down a steep incline with a relatively high flow rate were considered. Here, the flow is slow and the entry angle is small, resulting in waves on the interface. The fluid is assumed to be of finite depth and the incoming channel makes an angle α to the horizontal. Limiting flows are found when the fluid separates at a stagnation point or alternatively when the waves reach maximum steepness. The regions in parameter space where such solutions are obtained are delineated for different flow conditions.     

Intrusive Gravity Currents

Intrusive gravity currents arise when a fluid of intermediate density intrudes into an ambient fluid. These intrusions may occur in both natural and human-made settings and may be the result of a sudden release of a fixed volume of fluid or the steady or time-dependent injection of such a fluid. In this article we analytically and numerically analyze intrusive gravity currents arising both from the sudden release of a fixed volume and the steady injection of fluid having a density that is intermediate between the densities of an upper layer bounded by a free surface and a heavier lower layer resting on a flat bottom. For the physical problems of interest we assume that the dynamics of the flow are dominated by a balance between inertial and buoyancy forces with viscous forces being negligible. The three-layer shallow-water equations used to model the two-dimensional flow regime include the effects of the surrounding fluid on the intrusive gravity current. These effects become more pronounced as the fraction of the total depth occupied by the intrusive current increases. To obtain some analytical information concerning the factors effecting bore formation we further reduce the complexity of our three-layer model by assuming small density differences among the different layers. This reduces the model equations from a 6×6 to a 4×4 system. The limit of applicability of this weakly stratified model for various ranges of density differences is examined numerically. Numerical results, in most instances, are obtained using MacCormack's method. It is found that the intrusive gravity current displays a wide range of flow behavior and that this behavior is a strong function of the fractional depth occupied by the release volume and any asymmetries in the density differences among the various layers. For example, in the initially symmetric sudden release problem it is found that an interior bore does not form when the fractional depth of the release volume is equal to or less than 50% of the total depth. The numerical simulations of fixed-volume releases of the intermediate layer for various density and initial depth ratios demonstrate that the intermediate layer quickly slumps from any isostatically uncompensated state to its Archimedean level thereby creating a wave of opposite sign ahead of the intrusion on the interface between the upper and lower layers. Similarity solutions are obtained for several cases that include both steady injection and sudden releases and these are in agreement with the numerical solutions of the shallow-water equations. The 4×4 weak stratification system is also subjected to a wavefront analysis to determine conditions for the initiation of leading-edge bores. These results also appear to be in agreement with numerical solutions of the shallow-water equations.     

Exponentially stable equilibria to an indefinite nonlinear Neumann problem in smooth domains

We prove existence of two nonconstant exponentially stable equilibria to the heat equation supplied with a nonlinear Neumann boundary condition in any smooth n-dimensional domain (n ≥ 2), independently of its geometry. The Neumann boundary condition reflects the fact that the flux on the boundary is proportional to the product of a prescribed bistable function of the density or concentration with an indefinite weight. Such solutions are obtained via variational methods, by minimizing the corresponding energy functional on suitable invariant sets to the semiflow generated by the parabolic problem. But this is possible only if the parameter in the boundary condition is sufficiently large, otherwise we prove using the Implicit Function Theorem the uniqueness of constant equilibrium solutions. The same theorem allows us to derive isolation and smooth dependence on the parameter for nonconstant exponentially stable equilibria found.     

Microorganisms swimming through radiative Sutterby nanofluid over stretchable cylinder: Hydrodynamic effect

In the present article, radiative Sutterby nanofluid flow over a stretchable cylinder is considered. The suspended swimming microorganisms have been deliberated in the fluid analysis. Different processes such as Brownian motion, thermophoresis, Joules heating, and viscous dissipation have been inspected in the presences of stratification parameters. The solutions for flow profiles have been obtained via optimal homotopy analysis method. Impacts of different physical involved variables on non-dimensional velocity, temperature, nanofluid concentration, and concentration of density of swimming microorganisms have been debated. Coefficient of skin friction, local Nusselt number, Sherwood number, and density of motile organisms have been calculated. The results reveal that Sutterby fluid parameter enhances the skin friction and has a reverse impact on the velocity, while an increase in stratification causes a declination in the flow boundary layers. The temperature of the flow is also seen to be boosted by the increment in Brownian motion parameter. Analysis of entropy generation shows that the concentration difference parameter maximizes the entropy and minimizes the dimensionless Bejan number.     

An energy flow model for high-frequency vibration analysis of two-dimensional panels in supersonic airflow

Many energy flow models have been proposed for high-frequency forced vibration analysis of structures. In this paper, a novel energy flow model is developed to predict the high-frequency vibration response of panels in supersonic airflow and quantify the effects of supersonic airflow on high-frequency forced vibration characteristics. The additional damping due to supersonic airflow is derived from the motion equation of a two-dimensional panel. The relationship between the wavenumber and the group velocity is introduced to describe the energy transmission property considering the effects of supersonic airflow. Then the energy density governing equation (i.e. energy flow model) is established and solved by the energy flow analysis (EFA) and the energy finite element method (EFEM). Finally, comparing the vibration responses obtained by the present energy flow model with the corresponding exact analytical solutions, the developed energy flow model is verified to be effective for high-frequency vibration analysis of panels in supersonic airflow. Furthermore, the numerical simulations indicate that supersonic airflow can affect the equivalent damping of the propagating elastic waves, and thus change the energy density distribution of the panel.     

Effect of Hall current on transient hydromagnetic Couette–Poiseuille flow of a viscoelastic fluid with heat transfer

The unsteady Couette–Poiseuille flow of an electrically conducting incompressible non-Newtonian viscoelastic fluid between two parallel horizontal non-conducting porous plates is studied with heat transfer considering the Hall effect. A sudden uniform and constant pressure gradient, an external uniform magnetic field that is perpendicular to the plates and uniform suction and injection through the surface of the plates are applied. The two plates are kept at different but constant temperatures while the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained using finite difference approximations. The effect of the Hall term, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions is examined.     

Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape.     

Waves produced by sources distributed on an expanding disk

Explicit solutions of a nonhomogeneous wave equation are constructed. The solutions obtained describe waves generated by sources distributed on a disk that expands with velocities less than, equal to, or greater than the velocity of the wavefront. The structure and the directional transmission of waves are discussed. It is shown that the scalar solutions obtained can be applied to electromagnetic waves. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 38–47.     

Hall effect on unsteady MHD Couette flow and heat transfer of a Bingham fluid with suction and injection

The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid bounded by two parallel non-conducting porous plates is studied with heat transfer considering the Hall effect. An external uniform magnetic field is applied perpendicular to the plates and the fluid motion is subjected to a uniform suction and injection. The lower plate is stationary and the upper plate moves with a constant velocity and the two plates are kept at different but constant temperatures. Numerical solutions are obtained for the governing momentum and energy equations taking the Joule and viscous dissipations into consideration. The effect of the Hall term, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions are studied.     

Hamiltonian Structure and a Variational Principle for Grounded Abyssal Flow on a Sloping Bottom in a Mid‐Latitude β‐Plane

Observations, numerical simulations, and theoretical scaling arguments suggest that in mid‐latitudes, away from the high‐latitude source regions and the equator, the meridional transport of abyssal water masses along a continental slope correspond to geostrophic flows that are gravity or density driven and topographically steered. These dynamics are examined using a nonlinear reduced‐gravity geostrophic model that describes grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. It is shown that this model possesses a noncanonical Hamiltonian formulation. General nonlinear steady solutions to the model can be obtained for arbitrary bottom topography. These solutions correspond to nonparallel shear flows that flow across the planetary vorticity gradient. If the in‐flow current along the poleward boundary is strictly equatorward, then no shock can form in the solution in the mid‐latitude domain. It is also shown that the steady solutions satisfy the first‐order necessary conditions for an extremal to a suitably constrained potential energy functional. Sufficient conditions for the definiteness of the second variation of the constrained energy functional are examined. The theory is illustrated with a nonlinear steady solution corresponding to an abyssal flow with upslope and down slope groundings in the height field.     

Global stability and wavefronts in a cooperation model with state-dependent time delay

This article deals with a diffusive cooperative model with state-dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds. For the cooperative DDE system, the positivity and boundedness of solutions are firstly given. Using the comparison principle of the state-dependent delay equations obtained, the stability criterion of model is analyzed both from local and global points of view. When the diffusion is properly introduced, the existence of traveling waves is obtained by constructing a pair of upper–lower solutions and Schauder's fixed point theorem. Calculating the minimum wave speed shows that the wave is slowed down by the state-dependent delay. Finally, the traveling wavefront solutions for large wave speed are also discussed, and the fronts appear to be all monotone, regardless of the state dependent time delay. This is an interesting property, since many findings are frequently reported that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.     

A global meshless collocation particular solution method (integrated Radial Basis Function) for two-dimensional Stokes flow problems

A global version of the Method of Approximate Particular Solutions (MAPS) is developed to solve two-dimensional Stokes flow problems in bounded domains. The velocity components and the pressure are approximated by a linear superposition of particular solutions of the non-homogeneous Stokes system of equations with a Multiquadric Radial Basis Function as forcing term. Although, the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative since the particular solutions exactly satisfy the mass conservation equation. The present scheme is validated by comparing the obtained numerical result with the analytical solution of two boundary value problems constructed from the Stokeson exterior fundamental solution, i.e. regular everywhere except at infinity. For these two cases, convergence of the method and the influence of the value of the Multiquadric’s shape parameter on the numerical results are studied by computing the relative Root Mean Square (RMS) error for several homogeneous distributions of collocation points and values of the shape parameter. From this analysis is observed that the proposed MAPS results are stable and accurate for a wide range of shape parameter values. In addition, the lid-driven cavity and backward-facing step flow problems are solved and the obtained results compared with the solutions found with more conventional numerical schemes, showing good agreement between them.