A direct formulation of correlations for wall-proximity corrections of hot-wire measurements

Correlations for corrections to hot-wire data for the effects of wall proximity within the viscous sublayer are usually presented in the form u/u = F (y u /). The application of such correlations requires a prior knowledge of the wall shear stress; alternatively, the correlation must be used in an iterative fashion. It is shown in the present note that any such correlation may be recast with no loss of generality in the explicit form u/u _m = f (y u _m/), which is more convenient for use.List of symbols u difference between measured and actual velocities, u _m – u - u _m measured velocity - u shear velocity, - u ⁺ on-dimensional velocity, u/u - y distance from wall - y ⁺ non-dimensional distance from wall, y u / - fluid density - fluid kinematic viscosity - _s wall shear stress     

Robustness of Exponential Dichotomies in Infinite-Dimensional Dynamical Systems

In this paper we examine the issue of the robustness, or stability, of an exponential dichotomy, or an exponential trichotomy, in a dynamical system on an Banach space W. These two hyperbolic structures describe long-time dynamical properties of the associated time-varying linearized equation _t +A=B(t) , where the linear operator B(t) is the evaluation of a suitable Fréchet derivative along a given solution in the set K in W. Our main objective is to show, under reasonable conditions, that if B(t)=B(, t) depends continuously on a parameter and there is an exponential dichotomy, or exponential trichotomy, at a value ₀, then there is an exponential dichotomy, or exponential trichotomy, for all near ₀.We present several illustrations indicating the significance of this robustness property.     

An experimental study of the lateral migration of a droplet in a creeping flow

The distribution of droplets in a plane Hagen-Poiseuille flow of dilute suspensions has been measured by a special LDA technique. This method assumes a well defined relation between the velocity of the droplets and their lateral position in the channel. The measurements have shown that the droplet distribution is non-uniform and depends on the viscosity ratio between the droplets and the carrier liquid. The results have been compared with a theory by Chan and Leal describing the lateral migration of suspended droplets.List of symbols a particle radius, m - d half width of the channel, m - Re flow Reynolds number, = 2 _m · d · /µ - flow velocity, m/s - _m flow velocity at the channel axis, m/s - We Weber number, = 2 _m ^Emphasis>/2 · d · / - x distance from center line (x = 0) of the channel, m - non-dimensional distance from the channel center line, _x ^d - y distance along the channel (y = 0 at channel inlet), m - non-dimensional distance along the channel, = y/2d - non-dimensional, normalized distance along the channel, = · _m · µ/ - interfacial tension, N/m - viscosity ratio of dispersed (droplet) phase to viscosity of continuous phase - µ viscosity of continuous phase, Pa · s - density of continuous phase, kg/m³ - phase density difference, kg/m³ Experiments were performed at Max-Planck-Institut, Göttingen     

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

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Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

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Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

Momentum balance in highly distorted turbulent pipe flows

An in depth study into the development and decay of distorted turbulent pipe flows in incompressible flow has yielded a vast quantity of experimental data covering a wide range of initial conditions. Sufficient detail on the development of both mean flow and turbulence structure in these flows has been obtained to allow an implied radial static pressure distribution to be calculated. The static pressure distributions determined compare well both qualitatively and quantitatively with earlier experimental work. A strong correlation between static pressure coefficient Cp and axial turbulence intensity is demonstrated.List of symbols C _p static pressure coefficient = (pw-p)/1/2 - D pipe diameter - K turbulent kinetic energy - (r, , z) cylindrical polar co-ordinates. / 0 - R, y pipe radius, distance measured from the pipe wall - U, V axial and radial time mean velocity components - mean value of u - u, u/ , / - u, , w fluctuating velocity components - axial, radial turbulence intensity - turbulent shear stress - u friction velocity, (u ² = ₀/p) - ₀ wall shear stress - ^* boundary layer thickness A version of this paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Stability of a laminar boundary layer in an incompressible fluid with variable physical properties

The stability of plane-parallel flows of an incompressible fluid with variable kinematic viscosity in the presence of solid walls has been discussed in [1–5]. The stability of Couette flow is considered in [1]. The method of solution, which is the same as that used in [2], differs from the Tollmien-Schlichting method, since the expansion of the solutions in powers of R which is used assumes the smallness of this quantity. A general formulation of the problem of stability of a nonuniform fluid is presented in [3]. Di Prima and Dunn [4] used the Galerkin method to study the stability of the boundary layer relative to vortex-like disturbances in the case of variable kinematic viscosity. Since the development of this sort of disturbance depends only weakly on the form of the velocity profile in the boundary layer, a marked change of the viscosity had little effect on the critical Reynolds number. This same problem is considered in [5], The present author was not able to find in the literature any references to study of the stability of the laminar boundary layer in an incompressible fluid with variable kinematic viscosity relative to disturbances of the Tollmien-Schlichting type, with the exception of mention in [4] of an unpublished work of MacIntosh which showed the essential dependence of the critical Reynolds number on the viscosity gradient.In Section 1 an analysis is made of the effect of variability of the kinematic viscosity on the stability of the boundary layer relative to Tollmien-Schlichting waves under the condition of constant fluid density.Two approaches are possible to the study of the development of disturbances in a heterogeneous fluid. On the one hand, we can assume that the displacement of the fluid particles does not cause changes in the distribution of p(y) and _*(y), i.e., the velocity pulsations are not accompanied by pulsations of and _*. This will be the case if a particle which is characterized by the quantities ₁, ₁ entering a layer with the different values _{2 2}, instantaneously alters its properties so that its density becomes equal to ₂, and its kinematic viscosity becomes equal to ₂. On the other hand, we can consider that a fluid particle moving from layer 1 into layer 2 fully retains the properties which it had in layer 1. In this case the velocity pulsations naturally lead to pulsations of the quantities and _*.In actuality, the phenomenon develops along some intermediate scheme, since the particle alters its properties as it moves in a heterogeneous fluid. The degree of approximation of the process to the first or second scheme depends on the rate of these changes. The analyses below are based on the first scheme.     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

Analyse und Berechnung der Strömungsvorgänge im zylindrischen Scherspalt von Plastifizieraggregaten

Zusammenfassung Dieser Aufsatz zeigt eine Möglichkeit auf, zylindrische Scherteile einer Plastifiziereinheit, auf der strukturviskose Materialien verarbeitet werden, approximativ zu berechnen. Es ist möglich, den Volumenstrom und Druckabfall, die mittlere Schergeschwindigkeit, Scherdeformation und Schubspannung im Scherspalt zu approximieren. Durch diese Gleichungen wird eine Abschätzung der Verteil- und Zerteilvorgänge im Scherelement möglich.

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A method is described for approximating the flow in cylindrical shearing gaps of plasticating extruder, which is applicable to shear thinning materials. It is possible to calculate the through-put and pressure drop as well as the shear rate, strain and shear stress in the gap. With these equations the distribution and separation process in shearing gaps can be evaluated.

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D Zylinderdurchmesser - d ₁ Schnecken-Kerndurchmesser der Meteringzone - d _s Durchmesser des zylindrischen Scherteils - K Konstante im Potenzfließgesetz - K _0T Koeffizient des Potenzfließgesetzes - L ₁ Länge der Anlaufschräge - L _s Länge des zylindrischen Scherteils - n Fließindex - n ₀ Drehzahl - p Druckabfall über der Scherteillänge - s Scherspalthöhe - T _M Massetemperatur - ₀ Umfangsgeschwindigkeit - _0x Geschwindigkeitskomponente inx-Richtung - _x, _z Geschwindigkeit inx- bzw.z-Richtung als Funktion der Koordinatey - Volumenstrom - x, z Ortskoordinaten - Exponent des Potenzfließgesetzes - Schergeschwindigkeit - mittlere Schergeschwindigkeit - Viskosität - dimensionslose Höhe - Dichte der Schmelze - Schubspannung - _yx, _yz Schubspannungskomponenten - _xx, _zz Normalspannungskomponenten - _ps dimensionsloser Druckgradient - dimensionsloser Volumenstrom - _x, _z dimensionslose Geschwindigkeit inx- bzw.z-Richtung     

Magnetohydrodynamic natural convection heat transfer from radiate vertical porous surfaces

Magnetohydrodynamic natural convection heat transfer from radiate vertical surfaces with fluid suction or injection is considered. The nonsimilarity parameter is found to be the conductive fluid injection or suction along the streamwise coordinate = V{4x/² g(T _w – T )}^1/4. Three dimensionless parameters had been found to describe the problem: the magnetic influence number N = B ² _y /V ², the radiation-conduction parameter R _d = k _R /4aT ³ , and the Gebhart number Ge _x = gx/cp to represent the effect of the viscous dissipation. It is found that increasing the magnetic field strength causes the velocity and the heat transfer rates inside the boundary layer to decrease. Its apparent that increasing the radiation-conduction parameter decreases the velocity and enhances the heat transfer rates. The Gebhart number, i.e, the viscous dissipation had no effect on the present problem.Nomenclature a Stefan-Boltzmann constant - B _y Magnetic field flux density Wb/m² - Cf _x Local skin friction factor - c _p Specific heat capacity - f Dimensionless stream function - Ge _x Gebhart number, gx/cp - g Gravitational acceleration - k Thermal Conductivity - L Length of the plate - N Magnetic influence number, B ² _y /V ² - p Pressure - Pr Prandtl number - q _r Radiative heat flux - q _w (x) Local surface heat flux - Q _w (x) Dimensionless Local surface heat flux - R _d Planck number (Radiation-Conduction parameter), k _R /4aT ³ - T Temperature - T Free stream temperature - T _w Wall temperature - u, v Velocity components in x- and y-directions - V Porous wall suction or injection velocity - V _w Porous wall suction or injection velocity - x, y Axial and normal coordinates - Thermal diffusivity Greek symbols _R Roseland mean absorption coefficient, 4/3R _d - Coefficient of thermal expansion - Nonsimilarity parameter, V{4x/² g(T _w – T )}^1/4 - Peseudo-similarity variable - Dimensionless temperature - _w Ratio of surface temperature to the ambient temperature, T _w /T - Dynamice viscosity - Kinemtic viscosity - Fluid density - Electrical conductivity - _w Local wall shear stress - Dimensional stream function     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

The chaotic response of a fluttering panel: The influence of maneuvering

The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.Nomenclature a plate length - a _r u _r /h - D plate bending stiffness - E modulus of elasticity - g acceleration due to gravity - h plate thickness - j₁,j₂,j₃ base vectors of the body frame of reference - K spring constant - M Mach number - n 1 + ₀/g - N ₁ applied in-plane force - p–p aerodynamic pressure - P pa ⁴/Dh - q ₀/2 - Q _r generalized Lagrangian forces - R rotation matrix - R ₄ N, a ²/D - t time - kinetic energy - u plate deflection - u displacement of the structure - u _r modal amplitude - v₀ velocity - x coordinates in the inertial frame of reference - z coordinates in the body frame of reference - Ka/(Ka+Eh) - - elastic energy - 2qa ³/D - a/_mh - Poisson's ratio - material coordinates - air density - _m plate density - - _r prescribed functions - _r sin(r z/a) - angular velocity - a/v₀ - skew-symmetric matrix form of the angular velocity     

Flow of viscous and viscoelastic fluids through and around porous bodies

The slow flow of a viscous fluid through and around porous spheres is considered. The numerical simulation uses a special mixture of computational techniques: quadratic approximation and expansion in power series. The resulting calculations predict the evolution of the main features of the flow if the boundary conditions are varying, particularly if the tangential velocity is neglected or if a viscous filtration velocity is assumed at the sphere surface. The cases of full and hollow spheres with uniform and non uniform permeabilities are considered, the external impermeable walls of the flow being concentric spheres or cylinders. Some influence of viscoelastic properties of the fluid is also given.Nomenclature AA _n , A_n, B_n, b_n, C_n, c_n, D_n constants of integration - C _n (t) Gegenbauer functions with degree n and order –1/2 - e shell thickness - K, K* permeability - P _n (t) Legendre functions - Q _v volumetric rate of flow - p, p ₀, p _e pressure, far away pressure, average pressure - R* sphere radius - r, spherical coordinates - Re Reynolds' number (see equation 37) - s, t sinus and cosinus - V ₀ ^* uniform velocity - v velocity component - We Weissenberg's number (see equation (37)) - permeability coefficient - thickness coefficient - structural coefficient - diameter ratio sphere-cylinder - * dynamic viscosity of the fluid - stream functions - normal stress ( _rr ) - tangential stress ( ) - ₀ ^* relaxation time of the fluid     

Thermal radiation effects on MHD-Rayleigh flow with constant surface heat flux

Thermal radiation heat transfer effects on the Rayleigh flow of gray viscous fluids under the effect of a transverse magnetic field are investigated. The free convection heat transfer problem from constant surface heat flux moving plate is selected for study. It is found that the increasing of the magnetic field number M= H₀² / U₀²decreased velocities inside boundary layer, the increasing of the conduction–radiation parameter Rd=k__R/4aT³ decreased both temperatures and heat transfer rates. It is also found that the increasing of the dimensionless surface heat flux parameter q₀^*=q₀ /(kU₀T₎ increased the temperatures inside the boundary layer and increased the heat transfer rates. Comparison with previous works shows excellent agreement. Different transient velocity profiles, temperature profiles and local Nusselt numbers against different dimensionless groups are drawn.     

Injection moulding of thermoplastics: Freezing during mould filling

The injection moulding of thermoplastic polymers involves, during mould filling, flows of hot melts into mould networks, the walls of which are so cold that frozen layers form on them. Theoretical analyses of such flows are presented here. Br Brinkman number - c _L polymer melt specific heat capacity - c _S frozen polymer specific heat capacity - e exponential function - erf() error function - Gz Graetz number in thermal entrance region - Gz ^* modified Graetz number in thermal entrance region - Gz overall Graetz number - h channel half-height - h ^* half-height of polymer melt region - H mean heat transfer coefficient - k _L polymer melt thermal conductivity - k _S frozen polymer thermal conductivity - ln( ) natural logarithm function - L length of thermal entrance region in pipe or channel - m viscosity shear rate exponent - M(,,) Kummer function - Nu Nusselt number - p pressure - P pressure drop in thermal entrance region - P _f pressure drop in melt front region - Pe Péclet number - Pr Prandtl number - Q volumetric flow rate - r radial coordinate in pipe - R pipe radius - R ^* radius of polymer melt region - Re Reynolds number - Sf Stefan number - t time - T temperature - T _i inlet polymer melt temperature - T _m melting temperature of polymer - T _w pipe or channel wall temperature - U(,,) Kummer function - u _r radial velocity in pipe - u _x axial velocity in channel - u _y cross-channel velocity - u _z axial velocity in pipe - V melt front velocity - w channel width - x axial coordinate in channel - x _f melt front position in channel - y cross-channel coordinate - z axial coordinate in pipe - z _f melt front position in pipe - () gamma function - dimensionless thickness of frozen polymer layer - _i i-th term (i = 1,2,3) in power series expansion of - dimensionless axial coordinate in pipe - _f dimensionless melt front position in pipe - dimensionless cross-channel coordinate - ^* dimensionless half-height of polymer melt region - dimensionless temperature - _i i-th term (i = 0, 1, 2, 3) in power series expansion of - _i first derivative of _i with respect toø - _i second derivative of _i with respect toø - ^* dimensionless wall temperature - thermal diffusivity ratio - - latent heat of fusion - µ viscosity - µ ^* unit shear rate viscosity - dimensionless axial coordinate in channel - _f dimensionless melt front position in channel - dimensionless pressure drop in thermal entrance region - _f dimensionless pressure drop in melt front region - _L polymer melt density - _s frozen polymer density - dimensionless radial coordinate in pipe - ^* dimensionless radius of polymer melt region - ø dimensionless similarity variable in thermal entrance region - dummy variable - dimensionless contracted axial coordinate in thermal entrance region - dimensionless similarity variable in melt front region - ^* constant     

Forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping (1/2 order subharmonic oscillations and entrainment)

Nonlinear forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping are studied. In particular, entrainment phenomena at the critical speeds of 1/2 order subharmonic oscillations of forward and backward whirling modes are investigated. A self-excited oscillation appears in the wide range above the major critical speed. The amplitude of this oscillation reaches a limit value and then a self-sustained oscillation occurs. In the vicinity of a 1/2 order subharmonic oscillation of a forward whirling mode, a self-excited oscillation is entrained by a subharmonic oscillation. In the vicinity of a 1/2 order subharmonic oscillation of a backward whirling mode, either a self-excited oscillation or a subharmonic oscillation occurs.Experiments were made by an elastic rotating shaft with a disc. Nonlinearity in its restoring force was due to an angular clearance of a bearing and internal damping was due to friction between the shaft and an inner ring of the bearing. A self-excited oscillation was observed in the range above the major critical speed and this self-excited oscillation was entrained by a 1/2 order subharmonic oscillation of a forward whirling mode.Nomenclature O–xyz rectangular coordinate system - , _x, _y inclination angle of a shaft and its projections on the xz- and yz-planes - _x, _y inclination angles in rotating coordinates - , polar coordinates - I _p polar moment of inertia of a rotor - I diametral moment of inertia of a rotor - i _p ratio of I _p to I - dynamic unbalance of a rotor - rotating speed (angular velocity) - F magnitude of a dynamic unbalance force, F = (1 – i _p )² - c external damping coefficient - h internal damping coefficient - t time - D _x , D _y internal damping terms in stationary coordinates - D _x , D _y internal damping terms in rotating coordinates - N _x , N _y nonlinear terms in restoring forces     

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.     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress