On the effects of uniform high suction on the steady hydromagnetic flow due to a rotating disc

Summary The effect of uniform high suction on the steady flow of conducting viscous liquid due to a rotating disc, in the presence of a transverse magnetic field is considered. Series solutions for velocity components are obtained in descending powers of the suction parameter (a). The solutions, thus obtained, are valid for small values of m (= B ₀ ² /). Tables for displacement thickness, momentum thickness, and velocity components are given for different values of a and m.     

Strong cylindrical shocks in a rotating gas

Similarity solutions describing the flow behind a diverging strong cylindrical shock wave, advancing into a nonuniform gas having solid body rotation, are studied. The effects of the angular velocity variation on the shock velocity are shown graphically. It is found that an increase in the initial angular velocity leads to a decrease in the shock velocity.Nomenclature c ₀ sound velocity in unperturbed state - c sound velocity in unperturbed state at the axis of symmetry - D nondimensional density in unperturbed state - E energy release per unit length - f nondimensional radial velocity in perturbed state - g nondimensional pressure in perturbed state - h nondimensional density in perturbed state - k nondimensional azimuthal velocity in perturbed state - M an integral - N another integral - P nondimensional pressure in unperturbed state - p pressure in perturbed state - p ₀ pressure in unperturbed state - p pressure at the axis in unperturbed state - p ₁ pressure immediately behind the shock front - R shock front radius - r radial coordinate - R ₀ a characteristic length parameter - t time coordinate - U shock front velocity - u particle velocity (radial) in perturbed state - u ₀ particle velocity (radial) in unperturbed state - u ₁ particle velocity (radial) immediately behind the shock front - v particle velocity (azimuthal) in perturbed state - v ₀ particle velocity (azimuthal) in unperturbed state - v ₁ particle velocity (azimuthal) immediately behind the shock - w nondimensional azimuthal velocity in unperturbed state - x a nondimensional independent variable - z axial coordinate of cylindrical coordinates - Z a nondimensional independent variable - ₀ angular velocity in unperturbed state - ₁ angular velocity immediately behind the shock - density in perturbed state - ₀ density in unperturbed state - ₁ density immediately behind the shock - density at r=0 in unperturbed state - adiabatic index of the gas - ₀ R ₀ ² ₀ ² /(c)²     

The influence of boundary layer growth on shock tube test times

Summary A theoretical and experimental investigation of the limitation on shock tube test times which is caused by the development of laminar and turbulent boundary layers behind the incident shock is presented. Two theoretical methods of predicting the test time have been developed. In the first a linearised solution of the unsteady one-dimensional conservation equations is obtained which describes the variations in the average flow properties external to the boundary layer. The boundary layer growth behind the shock is related to the actual extent of the hot flow and not, as in previous unsteady analyses, to its ideal extent. This new unsteady analysis is consequently not restricted to regions close to the diaphragm. Shock tube test times are determined from calculations of the perturbed shock and interface trajectories. In the second method a constant velocity shock is assumed and test times are determined by approximately satisfying only the condition of mass continuity between the shock and the interface. A critical comparison is made between this and previous theories which assume a constant velocity shock. Test times predicted by the constant shock speed theory are generally in agreement with those predicted by the unsteady theory, although the latter predicts a transient maximum test time in excess of the final asymptotic value. Shock tube test times have also been measured over a wide range of operating conditions and these measurements, supplemented by those reported elsewhere, are compared with the predictions of the theories; good agreement is generally obtained. Finally, a simple method of estimating shock tube test times is outlined, based on self similar solutions of the constant shock speed analysis.Nomenclature a speed of sound - A, B, C constants defined in section 5.3 - D shock tube diameter - K =/q ^m, boundary layer growth constant, see Appendices A and B - l hot flow length - m constant, =1/2 or 4/5 for laminar or turbulent boundary layers, respectively - M ₀ initial shock Mach number at the diaphragm - M _s shock Mach number at station x _s - M ₂ =(U ₀–u ₂)/a ₂, hot flow Mach number relative to the shock front - N = ₂ a ₂/ ₃ a ₃, the ratio of acoustic impedances across the interface - P pressure - P* =P _e–P ₂, perturbation pressure - q boundary layer growth coordinate defined in § 2 - Q =(W–1+S) K - r radial distance from shock tube axis - S boundary layer integral defined by equation A6 - t time - t =/ , dimensionless ratio of test times - T =l/l , Roshko's dimensionless ratio of hot flow lengths - u axial flow velocity in laboratory coordinate system, see figure 1a - u* =u _e–u₂, perturbation axial flow velocity - U shock velocity - U ₀ initial shock velocity at the diaphragm - U* =U–U ₀, perturbation shock velocity - v axial flow velocity in shock-fixed coordinate system, see figure 1b - w radial flow velocity - W =U ₀/(U ₀–u ₂), density ratio across the shock - x axial distance from shock tube diaphragm - x _s, x _s axial distance between shock wave and diaphragm - t = _I/ , dimensionless ratio of test times - X =l _I/l , Roshko's dimensionless ratio of hot flow lengths - y =(D/2)–r, radial distance from the shock tube wall - ratio of specific heats - boundary layer thickness; undefined - boundary layer displacement thickness - boundary layer thickness defined by equation A2 - characteristic direction defined by dx/dt = (u ₂–a ₂) - =(M ₀ ² +1)/(M ₀ ² –1) - viscosity - characteristic direction defined by dx/dt=(u ₂+a ₂) - density - * = _te–₂, perturbation density - Prandtl number - shock tube test time - =M ₀ ² /(M ₀ ² –1) Suffices 1 conditions in the undisturbed flow ahead of the shock - 2 conditions immediately behind an unattenuated shock - 3 conditions in the expanded driver gas - 4 conditions in the undisturbed driver gas - e conditions between the shock and the interface, averaged across the inviscid core flow - i conditions at the interface - I denotes the predictions of ideal shock tube theory - asymptotic conditions given when x _s and t - s conditions at or immediately behind the shock - w conditions at the shock tube wall - a, b, b ¹, c, d, d ¹, f, f ¹, g, g ¹, j, k, k ¹ conditions at the points indicated in figure 2     

Solution of regular beam equations in arbitrary emission conditions on a curvilinear surface

The regular beam equations are solved analytically for the case of emission from an arbitrary surface in conditions of total space charge (-mode) and in a given external magnetic field H (§2) for temperature-limited emission (T-mode), in an external magnetic field H (§3); and for emission with nonzero initial velocity (§4). The emitter is taken as the coordinate surface x¹=0 in an orthogonal system x¹ (i = =1,2,3), while the current density J and field on it are given functions j(x², x³), (x², x³. The solution is written as series in (x¹) with coefficients dependent on x², x³, determined from recurrence relations. For emission in the -mode and H 0, =1/3; for temperature-limited emission, =1/2; with nonzero initial velocity, =1. The results are extended to the case of a beam in the presence of a moving background of uniform density (5).     

The stokes problems for a conducting fluid with a suspension of particles

The Stokes problems of an incompressible, viscous, conducting fluid with embedded small spherical particles over an infinite plate, set into motion in its plane by impulse and by oscillation, in the presence of a transverse magnetic field, are studied. The velocities of the fluid and of the particles and the wall shear stress are obtained. The stress is found to increase due to the particles and the magnetic field, with the effect of the particles diminishing as the field strength is increased.Nomenclature H ₀ strength of the imposed magnetic field - k density ratio of particles to fluid (per unit volume of flow field) - m _e ² H ₀ ² / - t time - y co-ordinate normal to the plate - u fluid velocity - v particle velocity - _e magnetic permeability of the fluid - kinematic viscosity of the fluid - electric conductivity of the fluid - fluid density - particle relaxation time - frequency of oscillation of the plate     

Solution of the equations for a regular beam emitted by a curvilinear surface in the nonstationary case

An analytical solution is given of the equations for a regular beam (§1) emitted by an arbitrary surface in the nonstationary case and the - and T-limited states for nonzero initial velocity (§ §2-4). It is assumed that the emitter is the coordinate surface x¹=0 in the orthogonal system x¹ (i=1, 2, 3), and the current density J, the electric field , and the magnetic field H are given functions J(t, x²,x³), (t, x², x³), and H (x¹, x², x³). The solution is given in the form of series in terms of (X¹) with coefficients that are functions of t, x², and x³. These coefficients are determined from recurrence relations ( =1/3, 1/2, 1, depending on the emission conditions). Plane, cylindrical, and spherical diodes are considered in § 5 in the case in which the high-frequency component of the current density J is not small in comparison with its constant components.     

Rarefied MHD laminar radial channel flow

The steady axisymmetrical laminar flow of slightly rarefied electrically conducting gas between two circular parallel disks in the presence of a transverse magnetic field is analytically investigated. A solution is obtained by expanding the velocity and the pressure distribution in terms of a power series of 1/r. The effect of rare-faction is taken to be manifested by slip of the velocity at the boundary. Velocity, induced magnetic field, pressure and shear stress distributions are determined and compared with the case of no rarefaction.Nomenclature b outer radius of channel - C _f skin friction coefficient, _w /(Q ²/t ⁴) - H ₀ impressed magnetic field - H _r ^* induced magnetic field in the radial direction - H _r induced dimensionless magnetic field in the radial direction, H _r ^* /H ₀ - M Hartmann number, H ₀ t(/)^1/2 - P dimensionless static pressure, P*t ⁴/Q ² - P* static pressure - P _b dimensionless pressure at outer radius of channel - P ₀ reference dimensionless pressure - Q source discharge - R gas constant - Rm magnetic Reynolds number, Q/t - Re Reynolds number, Q/t - 2t channel width - T absolute gas temperature - u dimensionless radial component of the velocity, u*t ²/Q - u* radial component of the velocity - w dimensionless axial component of the velocity, w*t ²/Q - w* axial component of the velocity - z, r dimensionless axial and radial directions, z*/t and r*/t, respectively - z*, r* axial and radial direction, respectively - molecular mean free path - magnetic permeability - coefficient of kinematic viscosity - density - electrical conductivity     

Regular reflection of plane detonation waves from an elastic half-space

An effective method for the approximate solution of the Eq. [1] for the intensity of a reflected shock wave in the case of oblique incidence of a detonation wave on an elastic half-space is described; the elastic half-space is described by a certain specific form of the equation of state. Formulas relating the front and particle velocities behind the transmitted wave front to physical parameters are derived. Values of the wave intensity and other quantities determined with the aid of a Ural-2 computer are cited.The author of [1, 2] investigated the regular reflection of shock waves from the boundary between two bodies. In the present paper we solve the analogous problem in the case of oblique incidence of a detonation wave on an elastic half-space. The detonation wave deforms the elastic half-space, which assumes the position OK₁ (Fig. 1) forming the angle to the initial direction KO of the halfspace boundary. We assume that the acoustic stiffness of the halfspace is larger than the acoustic stiffness of the explosive. In this case, both reflected wave 2 and transmitted wave 3 are shock waves [3]. Let us denote the velocities of propagation of the detonation, reflected, and transmitted waves by U_i(i=1, 2, 3), respectively; let the pressure be p_i and let the density bep _i(i=0, 1, 2, 3, 4). The quantities U₁, ₁, ₀, and ₄ are given. We determine the intensities of waves 2 and 3, their velocities of propagation, and the angles ₂, ₃ and . The parameters are constant within each of the domains a, b, c, d, and e. In domains a and e the medium is stationary, i.e., u₀=u₄ =0. The basic equations of the problem express the conditions at the wave fronts and the dynamic and kinematic relationships.     

Discontinuities in helium II

The jump conditions at surface of discontinuity are derived for the two-fluid model of helium II from postulated balance laws for the total energy, the linear momentum of the superfluid and from an entropy production inequality. These conditions are used to discuss a contact surface and the propagation of a weak shock.Nomenclature A Helmholtz free energy function (A=U–ST) - F _i ^s external body force acting on the superfluid - f _i ^s superfluid acceleration (= _t v _i ^s +v _k ^s u _i,k ^s ) - G _i ^s density of supply of linear momentum to the superfluid - g _i ^s diffusive force of the superfluid (=G _i ^s –m ^s v _i ^s ) - m ^s short notation for _t ^s +( ^s v _k ^s ), _k - qk heat flux vector per unit area per unit time - r heat supply function per unit mass unit per time - S entropy per unit mass of helium II - T absolute temperature (> 0) - t _i ^s superfluid stress vector ( _i ^s =n _{k ki} ^s ) - U internal energy per unit mass of helium II - v _i ^s superfluid velocity - total mass density - ^s superfluid mass density - _ij ^s superfluid stress tensor     

Measurement of the turbulent diffusivity in the near field of variable density jets using conditional velocimetry

Pulsed laser Mie scattering and laser Doppler velocimetry (LDV), both conditioned on the origin of the seed particles, have been successively performed in turbulent jets with variable density. In the early stages of the jet developments, significant differences are measured between the ensemble average LDV data obtained by jet seeding and those obtained by seeding the ambient air. Careful analysis of the marker statistics shows that this difference is a quantitative measure of the turbulent mixing. The good agreement with gradient–diffusion modelling suggests the validity of a general diffusion equation where the velocities involved are expressed in terms of ensemble conditional Favre averages. This operator accounts for all events (including intermittent ones) and for variations in the density of the marked fluid whose velocity is still specified by the binary origin of the marker.List of symbols D_L laminar diffusivity, m²/s - D_T turbulent diffusivity, m²/s - d diameter of the jet nozzle, m - F_r Froude number - J diffusion vector, m/s - k global sensitivity of the detection system for one particle (signal level) - N_P number of seed particles in the probe volume - N_P,i number of seed particles in sample i - N_P⁽ⁱ⁾ value of N_P in channel i - N_B number of Doppler bursts - count rate of bursts, s^–1 - N_v number of validated Doppler bursts - count rate of validated bursts, s^–1 - N_id number of ideal particles - N_id* number of marked ideal particles - P* probability that an ideal particle be marked by a seed particle - P(z) probability density function for z, m³/kg - probability to have k seed particles in the probe volume - probability of having k seed particle conditioned on a given value of z - r radial coordinate, m - R =⁽¹⁾/⁽²⁾, density ratio - S₁ local signal level with jet seeding - S₁⁽¹⁾ reference signal level in pure stream 1 with jet seeding - s₁ = S_1/S₁⁽¹⁾, normalized signal - v_c volumic capacity of the probe volume, m³ - V velocity vector, m/s - V_x axial velocity component, m/s - V_r radial velocity component, m/s - V_P particulate velocity vector, m/s - V_Pj velocity vector of particle j, m/s - V_Pij velocity vector of the jth particle in sample i, m/s - V_i velocity vector of the marked flow for realization i, m/s - V_1,i velocity vector of the flow such it is marked in realization i by particles issuing only from stream 1, m/s - x axial coordinate, m - Y_i local mass fraction of species i - Z mixture fraction:local mass fraction of jet fluid - Z_i mixture fraction for realization iGreek local density, kg/m³ - _i local density for realization i, kg/m³ - ⁽¹⁾ density in stream 1 (density of the jet fluid), kg/m³ - ₁ time of flight of jet seed particles to reach the probe volume, s - _B duration of a Doppler burst, sAverages <A> ensemble average of A - Ā time average of A - Favre average, , ( ) the present notation is only due to printing problems - A Favre fluctuation,      

Flow of a multicomponent gas between parallel permeable surfaces

The present paper gives an exact solution of the equations describing the flow of a multicomponent gas between two parallel permeable planes, one of which moves relative to the other with constant velocity (i. e., we study a flow of the Couette type).Notation y coordinate - u, v velocity components - density - c_i mass concentration of i-th component - I_i diffusional flux of i-th component - H enthalpy - T temperature - m molecular weight - viscosity coefficient - heat conduction coefficient - c_p mixture specific heat - D_ij the binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - N total number of components - n number of components in injected gas - l distance between planes Indices i, j component numbers - w applies to quantities for y=0 - ^* applies to quantities for y=l     

Fast Diffusion to Self-Similarity: Complete Spectrum,Long-Time Asymptotics,and Numerology

The complete spectrum is determined for the operator on the Sobolev space W^1,2(Rⁿ) formed by closing the smooth functions of compact support with respect to the norm Here the Barenblatt profile is the stationary attractor of the rescaled diffusion equation in the fast, supercritical regime m the same diffusion dynamics represent the steepest descent down an entropy E(u) on probability measures with respect to the Wasserstein distance d₂. Formally, the operator H=HessE is the Hessian of this entropy at its minimum , so the spectral gap H:=2–n(1–m) found below suggests the sharp rate of asymptotic convergence: from any centered initial data 0u(0,x)L¹(Rⁿ) with second moments. This bound improves various results in the literature, and suggests the conjecture that the self-similar solution u(t,x)=R(t)^–n(x/R(t)) is always slowest to converge. The higher eigenfunctions – which are polynomials with hypergeometric radial parts – and the presence of continuous spectrum yield additional insight into the relations between symmetries of Rⁿ and the flow. Thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass-preserving dilation (or still better, affine image). The strange numerology of the spectrum is explained in terms of the number of moments of .Dedicated to Elliott H. Lieb on the occasion of his 70th birthday.     

Magnetohydrodynamic laminar source flow between two parallel disks

The steady axisymmetrical laminar source flow of an incompressible conducting fluid between two circular parallel disks in the presence of a transverse magnetic field is analytically investigated. A solution is obtained by expanding the velocity and the pressure distribution in terms of a power series of 1/r. Velocity, induced magnetic field, pressure and shear stress distributions are determined and compared with the case of the hydrodynamic solution. Pressure is found to be a function of both r and z in the general case and the flow is not parallel. At high magnetic fields, the velocity distribution degenerates to a uniform core surrounded by a boundary layer near the disks.Nomenclature C _f skin friction coefficient - H ₀ impressed magnetic field - H _r induced magnetic field in the radial direction, H _r /H ₀ - M Hartmann number, H ₀ t(/)^1/2 - P dimensionless static pressure, P*t ⁴/Q - P* static pressure - P ₀ reference dimensionless pressure - Q source discharge - R outer radius of disks - Rm magnetic Reynolds number, Q/t - Re Reynolds number, Q/t - 2t channel width - u dimensionless radial component of the velocity, u*t ²/Q - u* radial component of the velocity - w dimensionless axial component of the velocity, w*t ²/Q - w* axial component of the velocity - z, r dimensionless axial and radial directions, z*/t and r*/t, respectively - z*, r* axial and radial direction, respectively - magnetic permeability - coefficient of kinematic viscosity - density - electrical conductivity - ² LaPlacian operator in axisymmetrical cylindrical coordinates     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

On Perturbative Expansions to the Stochastic Flow Problem

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, _f ². Historically, perturbation schemes have involved the assumption that _f ²<1. Here it is shown that _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, _f ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, , defined. Since the processes f and f can often be considered independent, further assumptions on f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of f in terms of _f and various length scales. The ratio of the integral scale in the main direction of flow ( _x ) to the total domain length (L^*), _x ²=_x/L^*, plays an important role in the convergence of the perturbation scheme. For _x smaller than a critical value _c, _x < _c, the scheme's perturbation parameter is =_f/_x for one- dimensional flow, and =_f/_x ² for two-dimensional flow with mean flow in the x direction. For _x > _c, the parameter =_f/_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.     

On the effects of uniform high suction on the rotationally symmetric flow of a conducting liquid near a stationary disc in the presence of a transverse magnetic field

In the present paper an attempt has been made to find out effects of uniform high suction in the presence of a transverse magnetic field, on the motion near a stationary plate when the fluid at a large distance above it rotates with a constant angular velocity. Series solutions for velocity components, displacement thickness and momentum thickness are obtained in the descending powers of the suction parameter a. The solutions obtained are valid for small values of the non-dimensional magnetic parameter m (= 4 _e ² H ₀ ² /) and large values of a (a2).Nomenclature a suction parameter - E electric field - E _r , E , E _z radial, azimuthal and axial components of electric field - F, G, H reduced radial, azimuthal and axial velocity components - H magnetic field - H _r , H , H _z radial, azimuthal and axial components of magnetic field - H ₀ uniform magnetic field - H* displacement thickness and momentum thickness ratio, */ - h induced magnetic field - h _r , h , h _z radial, azimuthal and axial components of induced magnetic field - J current density - m nondimensional magnetic parameter - p pressure - P reduced pressure - R Reynolds number - U ₀ representative velocity - V velocity - V _r , V , V _z radial, azimuthal and axial velocity components - w ₀ uniform suction through the disc. - density - electrical conductivity - kinematic viscosity - _e magnetic permeability - a parameter, (/)^1/2 z - a parameter, a - * displacement thickness - momentum thickness - angular velocity     

Stability characteristics of condensate films

The linear stability theory is used to study stability characteristics of laminar condensate film flow down an arbitrarily inclined wall. A critical Reynolds number exists above which disturbances will be amplified. The magnitude of the critical Reynolds number is in all practical situations so small that a laminar gravity-induced condensate film can be expected to be unstable. Several stabilizing effects are acting on the film flow; at an inclined wall these effects are due to surface tension, gravity and condensation mass transfer.

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Zusammenfassung Mit Hilfe der linearen Stabilitätstheorie werden die Stabilitätseigenschaften laminarer Kondensatfilme an einer geneigten Wand untersucht. Es zeigt sich, daß Kondensatfilme in jedem praktischen Fall ein unstabiles Verhalten aufweisen. Der stabilisierende Einfluß von Oberflächenspannung, Schwerkraft und Stoffübertragung durch Kondensation bewkkt jedoch, daß Störungen in bestimmten Wellenlängenbereichen gedämpft werden.

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Nomenclature c=c^*/u₀ complex wave velocity, celerity, dimensionless - c^*=c _r ^* + i c _i ^* complex wave velocity, celerity, dimensional - c_p specific heat at constant pressure - g gravitational acceleration - h_fg latent heat - k thermal conductivity of liquid - p^* pressure - p=p^*/u₀ ² dimensionless pressure - Pe=Pr Re^* Peclet number - Pr Prandtl number - Re^*=u₀ / Reynolds number (defined with surface velocity) - S temperature perturbation amplitude - t^* time - t=t^* u₀/ dimensionless time - T temperature - T_s saturation temperature - T_w wall temperature - T=T_s-T_w temperature drop across liquid film - u^*, v^* velocity components - u=u^*/u₀ dimensionless velocity components - v=v^*/u₀ dimensionless velocity components - u₀ surface velocity of undisturbed film flow - v _g ^* vapor velocity - x^*, y^* coordinates - x=x^*/ dimensionless coordinates - y=y^*/ dimensionless coordinates Greek Symbols =^* wave number, dimensionless - ^*=2 /^* wave number dimensional - ^* wave length, dimensional - =^*/ wave length, dimensionless - local thickness of undisturbed condensate film - kinematic viscosity, liquid - density, liquid - _g density vapor - surface tension - = (1 +) film thickness of disturbed film, Fig. 1 - stream function perturbation amplitude - angle of inclination Base flow quantities are denoted by^–, disturbance quantities are denoted by.     

Method of calculating supersonic three-dimensional flow past smooth bodies

A method is suggested in [1] for calculating supersonic flow past smooth bodies that uses an analytic approximation of the gasdynamic functions on layers and the method of characteristics for calculating the flow parameters at the nodes of a fixed grid. In the present paper this method is discussed for three-dimensional flows of a perfect gas in general form for cylindrical and spherical coordinate systems; relations are presented for calculating the flow parameters at the layer nodes, results are given for the calculation of the flow for specific bodies, and results are shown for a numerical analysis of the suggested method. Three-dimensional steady flows with plane symmetry are considered. In the relations presented in the article all geometric quantities are referred to the characteristic dimension L, the velocity components u, v, w and the sonic velocitya are referred to the characteristic velocity W, the density is referred to the density of the free stream, and the pressure p is referred to w².