Effects of MHD free convection and mass transfer on the flow past an oscillating infinite coaxial vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past oscillating infinite coaxial vertical circular cylinder. The analytical expressions for velocity, temperature and concentration of the fluid are obtained by using perturbation technique.

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Einwirkungen von freier MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden unendlichen koaxialen vertikalen Zylinder

Zusammenfassung Die Einwirkungen der freien MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden, unendlichen, koaxialen, vertikalen Zylinder wurden untersucht. Die analytischen Ausdrücke der Geschwindigkeit, Temperatur und Fluidkonzentration sind durch die Perturbationstechnik erhalten worden.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r _a ,r _b radius of inner and outer cylinder - a, b dimensionless inner and outer radius - r,r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z,z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

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     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

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     

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].     

Power spectra of fluid velocities measured by laser Doppler velocimetry

The power spectrum and the correlation of the laser Doppler velocimeter velocity signal obtained by sampling and holding the velocity at each new Doppler burst are studied. Theory valid for low fluctuation intensity flows shows that the measured spectrum is filtered at the mean sample rate and that it contains a filtered white noise spectrum caused by the steps in the sample and hold signal. In the limit of high data density, the step noise vanishes and the sample and hold signal is statistically unbiased for any turbulence intensity.List of symbols A cross-section of the LDV measurement volume, m² - A empirical constant - B bandwidth of velocity spectrum, Hz - C concentration of particles that produce valid signals, number/m³ - d _m diameter of LDV measurement volume, m - f(₁, ₂ | u) probability density of t _i; and t _j given (t) for all t, Hz² - probability density for t _j-t_i, Hz - n (t, t) number of valid bursts in (t, t) = N + n - N (t, t) mean number of valid bursts in (t, t) - N _e mean number of particles in LDV measurement volume - valid signal arrival rate, Hz - mean valid signal arrival rate, Hz - R _uu time delayed autocorrelation of velocity, m²/s² - S _u power spectrum of velocity, m²/s²/Hz - t ₁, t ₂ times at which velocity is correlated, s - t _i, t _j arrival times of the bursts that immediately precede t ₁ and t ₂, respectively, s - t _ij t _j–t _i s - T averaging time for spectral estimator, s - T _u integral time scale of u (t), s - T Taylor's microscale for u (t), s - u velocity vector = U + u, m/s - u fluctuating component of velocity, m/s - U mean velocity, m/s - u _m sampled and held signal, m/s Greek symbols (t) noise signal, m/s - _m (t) sampled and held noise signal, m/s - bandwidth of spectral estimator window, radians/s - time between arrivals in pdf, s - Taylor's microscale of length = UT m - kinematic viscosity - ₁, ₂ arrival times in pdf, s - root mean square of noise signal, m/s - _u root mean square of u, m/s - delay time = t ₂ - t ₁ s - B duration of a Doppler burst, s - circular frequency, radians/s - _c low pass frequency of signal spectrum radians/s Other symbols ensemble average - conditional average - ^ estimate     

Theory of single well tests in a leaky aquifer. Analytical solution for non steady flow

The evaluation of a pump test or a slug test in a single well that completely penetrates a leaky aquifer does not yield a unique relation between the hydraulic properties of the aquifer, independent of the testing conditions. If the flow is transient, the drawdown is characterized by a single similarity parameter that does not distinguish between the storativity and the leakage factor. If the flow is quasi stationary, the drawdown is characterized by a single similarity parameter that does not distinguish between the transmissivity and the leakage factor. The general non steady solution, which is derived in closed form, is characterized bythree similarity parameters.Nomenclature a e 0.8905 = auxiliary parameter - b thickness of the aquifer - b _c thickness of the semipervious stratum - B() auxiliary function - f(s),g(s) auxiliary functions in the complex plane - F(t),G(t) auxiliary functions of time - h undisturbed level of the phreatic surface - K conductivity of the aquifer - K _c conductivity of the semipervious stratum - m ₀ leakage factor - m dimensionless leakage factor - N(s) auxiliary function in the complex plane - Q _w (t) discharge flux - Q steady discharge flux - Q ₀ constant discharge flux during limited time - Q(t) dimensionless discharge flux - r ₀ radius of the well - r radial coordinate - r dimensionless radial coordinate - s complex variable - s ₀ pole - S storativity of the aquifer - S _n n'th part of an integration contour - t time - t dimensionless time - T transmissivity of the aquifer - ,,,,, dimensionless parameters - Euler's number - dummy variable - ₁(), ₂() auxiliary functions - (r, t) drawdown - ₀(t) drawdown in the well - (r, t) dimensionless drawdown - ₀(t) dimensionless drawdown in the well     

Effects of MHD free convection and mass transfer on the flow past a vibrating infinite vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past a vibrating infinite isothermal and constant heat flux vertical circular cylinder. The expressions for velocity, temperature, concentration and skin-friction of the fluid are obtained in closed form by using Laplace transform technique. The effects ofPr (Prandtl number),Sc (Schmidt number),Gr (Grashof number,Gr>0 implies cooling andGr<0 heating of the cylinder),Gm (modified Grashof number),M (magnetic field parameter) and variation of time on velocity distribution have been studied graphically. The results presented in this paper agree with the results of Lien and Chen when magnetic parameter approaches zero.

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Effekte der freien MHD Konvektion und der Stoffübertragung auf die Strömung längs eines vibrierenden unendlich langen vertikalen Kreiszylinders

Zusammenfassung Es werden die Effekte der freien MHD Konvektion und der Stoffübertragung auf die Strömung längs eines vibrierenden unendlich langen vertikalen isothermen Kreiszylinders mit konstanter Wärmestromdichte untersucht. Es werden geschlossene Ausdrücke für die Geschwindigkeit, Temperatur, Konzentration und Wandreibung des Fluides mittels der Laplace-Transformation erhalten. Die Effekte der Prandtl-ZahlPr, Schmidt-ZahlSc, Grashof-ZahlGr (Gr>0 bedeutet kühlen,Gr<0 heizen), der modifizierten Grashof-ZahlGm, des ParametersM für das magnetische Feld und das zeitliche Verhalten der Geschwindigkeitsverteilung wurden graphisch untersucht. Die Ergebnisse dieser Untersuchung stimmen mit denen von Lien und Chen überein, wenn der Parameter für das magnetische Feld nahe bei 0 liegt.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r ₀ radius of the circular cylinder - r ₀ dimensionless radius - r, r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z, z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Momentum and energy balances for dispersed two-phase flow

Summary The equations of motion and the mechanical energy balances for two-phase flow systems are derived by integration over a volume containing a large number of elements of the dispersed phase.List of symbols A, A boundary of volumes V, V - dA, dA surface element of A, A - A _s boundary of particles in V - dA _s surface element of A _s - F force per unit volume of the system - g –gz=gravity vector - g acceleration by gravity - I unit tensor - p pressure - Q dissipation in the continuous phase - Q _s dissipation in the dispersed phase - R compression work in the continuous phase - R _s compression work in the dispersed phase - t time - u velocity of continuous phase - u _s velocity of dispersed phase - u magnitude of u - u _s magnitude of u _s - V volume in the two-phase system - V part of V occupied by the continuous phase - W work done by F - z vertical coordinate - local volume fraction of the dispersed phase - pI–=stress tensor of the continuous phase - _s turbulent particle stress tensor - density of the continuous phase - _s density of the dispersed phase - shearing-stress tensor of the continuous phase - _s turbulent particle shearing-stress tensor - nabla operator - u, u _s velocity gradient tensor - substantial derivative (Shell Internationale Research Maatschappij N.V.)(Bataafse Internationale Petroleum Maatschappij N.V.)     

A class of nonlinear,completely integrable abstract wave equations

IfL is a positive self-adjoint operator on a Hubert spaceH, with compact inverse, the second-order evolution equation int,u+Lu+u _H ² u=0 has an infinite number of first integrals, pairwise in involution. It follows from this that no nontrivial solution tends weakly to 0 inH ast. Under an additional separation assumption on the eigenvalues ofL, all trajectories (u,u) are relatively compact inD(L ^1/2)×H. Finally, if all the eigenvalues are simple, the set of initial values of quasi-periodic solutions is dense in the ball B=(u ₀,u ₀ )D(L ^1/2)×H; L^1/2 u _{0 H} ² +u ² < for sufficiently small.     

Free convection effects on the oscillatory flow of an electrically conducting rarefied gas past an infinite, vertical plate with constant suction

An analysis of a two-dimensional, unsteady flow of an electrically conducting, viscous, incompressible rarefied gas past an infinite vertical porous plate is carried out under the following assumptions: (i) the suction velocity normal to the plate is constant (ii) the free stream velocity oscillates in time about a constant mean (iii) the plate temperature is constant (iv) the difference between the temperature of the plate and the free stream is moderately large causing the free convection currents (v) first order velocity-slip and the temperature jump boundary conditions (vi) transverse magnetic field (vii) induced magnetic field is negligible.Approximate solutions to the coupled, non-linear equations governing the flow are derived for the mean velocity, mean temperature, mean-skin-friction, mean rate of heat transfer, transient velocity and temperature, fluctuating parts of the velocity profiles, the amplitude and the phase of the skin-friction and the rate of heat-transfer. They are shown graphically followed by a discussion. The effects of ±G (Grashof number), ±E (Eckert number), M (Magnetic field parameter), h ₁ (rarefaction parameter), h ₂ (temperature jump coefficient), (frequency) are discussed for heating (G<0) or cooling (G>0) of the plate by the free convection currents.Nomenclature |B| amplitude of skin-friction - B ₀ applied magnetic field - c _p specified heat at constant pressure - E Eckert number - f ₁ Maxwell's reflection coefficient - f ₂ thermal accommodation coefficient - g _x acceleration due to gravity - G Grashof number - h ₁ rarefaction parameter (L ₁ v ₀/) - h ₂ non-dimensional temperature jump coefficient (L ₂ v ₀/) - k thermal conductivity - K _n Knudsen number - L mean free path - L ₁ (2–f ₁)L/f ₁ - L ₂ - l ₁ characteristic length - M magnetic field parameter - M _r, M _i fluctuating parts of velocity - m - P Prandtl number - p pressure - q rate of heat transfer - q _m mean rate of heat transfer - |Q| amplitude of rate of heat transfer - R suction Reynolds number - T temperature of fluid - T _w temperature of the plate - T temperature of the fluid in free stream - t time - t dimensionless time - U free stream velocity - U dimensionless free stream velocity - U mean of U(t) - u, v velocity components in x, y directions - u dimensionless velocity in x direction - u ₀ mean velocity - u ₁ fluctuating part of velocity - v ₀ suction velocity - x, y coordinate system - x, y dimensionless coordinates - frequency of the free stream oscillations - dimensionless frequency - dimensionless temperature - ₁ fluctuating part of temperature - phase angle of skin-friction - phase angle of rate of heat transfer - density of the fluid in the boundary layer - density of the fluid in the free stream - viscosity - kinematic viscosity - electrical conductivity of the fluid - small positive constant - skin-friction - _m mean skin-friction - specific heat ratio - ₁ coefficient of volume expansion     

The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws

Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u _t+f(u) _x =u _xx on a half-line R⁺, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u)u–u₁ induced by L ¹(R⁺). We prove here that v is asymptotically stable with respect to d: if u ₀–vL ¹, then u(t)–v₁0 as t+. When v is a constant, we show that this property holds if and only if f(v)0. These results complement our study of the Cauchy problem [2].     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

Finite-difference analysis of MHD Stokes problem for a vertical infinite plate in dissipative fluid

Finite-difference solution of MHD flow past an impulsively started vertical infinite plate in an electrically conducting fluid has been presented on taking into account the viscous dissipative heat. Results for velocity and temperature are shown graphically whereas the numerical values of the skin-friction and the rate of heat transfer are entered in the table. The results are discussed in terms of the parameters M (the Hartmann number), G (the Grashof number, G>0, cooling of the plate by free convection, G<0, heating of the plate by free convection currents), E (the Eckert number) and P (the Prandtl number).Nomenclature B ₀ applied magnetic field - c _p specific heat at constant pressure - g acceleration due to gravity - k thermal conductivity - t time - T temperature of the fluid near the plate - T temperature of the fluid far away from the plate - U ₀ velocity of the plate - u velocity of the fluid - coefficient of volume expansion - kinematic viscosity - scalar electrical conductivity - coefficient of viscosity - density of the fluid     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

Finite difference solution of unsteady natural convection boundary layer flow over an inclined plate with variable surface temperature

The unsteady natural convection boundary layer flow over a semi-infinite inclined plate is considered with the wall temperatureT _w ,(x) (=T +ax ⁿ )varying as the power of the axial coordinate. The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson type. Numerical results are obtained for different values of Prandtl number, Grashof number and n for different angles of inclination. The steadystate velocity and temperature profiles, local and average skin frictions and Nusselt numbers are shown graphically. The effects of the angle of inclination and exponent n on velocity and temperature profiles, skin friction and Nusselt number have been discussed. The velocity, temperature and Nusselt number of the present study are compared with the available results and a good agreement is found to exist between the two.

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Differenzlösung für nichtstationäre natürliche Grenzschicht-Konvektionsströmung an einer geneigten Platte mit veränderlicher Oberflächentemperatur

Zusammenfassung Die nichtstationäre natürliche Grenzschicht-Konvektionsströmung an einer halbunendlichen geneigten Platte wird unter Zugrundelegung der GesetzmäßigkeitT _w ,(x) (=T +ax ⁿ für die Wandtemperatur als Funktion der Achsialkoordinate untersucht, und zwar mit Hilfe eines impliziten Differenzverfahrens vom Crank-Nicolson Typ und bei Variation der Prandtl- und Grashof-Zahlen, des Exponenten n und des Neigungswinkels. Graphisch dargestellt sind die Geschwindigkeits-und Temperaturprofile im stationären Zustand, die örtlichen und gemittelten Reibungsbeiwerte und der Nusselt-Zahlen. Der Einfluß des Neigungswinkels und des Exponenten n auf diese Größe wird diskutiert. Im Vergleich mit den Ergebnissen aus anderen Arbeiten konnte gute Übereinstimmung festgestellt werden.

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Nomenclature g acceleration due to gravity - Gr _L Grashof number at x=L - L length of the plate - n exponent in the power law variation of the wall temperature - Nu _x local Nusselt number - Nu _X dimensionless local Nusselt number - average Nusselt number - dimensionless average Nusselt number - p pressure - Pr Prandtl number - t time - t dimensionless time - T temperature - Tw temperature on the plate - T dimensionless temperature - u x-velocity component - U dimensionlessX-velocity component - v y-velocity component - V dimensionlessY-velocity component - x spatial coordinate along the plate - X dimensionless spatial coordinate along the plate - y spatial coordinate normal to the plate - Y dimensionless spatial coordinate normal to the plate Greek symbols thermal diffusivity - ß volumetric coefficient of thermal expansion - t dimensionless time-step - X dimensionless finite difference grid spacing in theX-direction - Y dimensionless finite difference grid spacing in theY-direction - angle of inclination of plate with horizontal - kinematic viscosity - density - _x local skin friction - _X dimensionless local skin friction - average skin friction - dimensionless average skin friction     

The response of a turbulent boundary layer to different shaped transverse grooves

The response of a turbulent boundary layer to three different shaped transverse grooves was investigated at two values of momentum thickness Reynolds numbers ( R =1000 and 3000). A 20-mm wide square, semicircular and triangular groove with depth to width ( d / w) ratio of unity was used. In general, the effects of the grooves are more significant at the higher R , with the most pronounced effects caused by the square groove. An increase in wall shear stress _w was observed just downstream of the groove for all three shapes. The increase in _w is followed by a small decrease in _w below the smooth-wall value before it relaxes back to the corresponding smooth-wall value at x / ₀3. At the higher R , the maximum increase in _w for the square groove is about 50% higher than for the semicircular groove and almost twice that for the triangular groove. The effect of the square groove on U / U ₀, u / U ₀ and v / U ₀ is much more significant than the effect of the semicircular and triangular grooves. There is an increase in the bursting frequency ( f _B⁺) on the grooved-wall compared to the smooth-wall case. The distribution of f _B⁺ downstream of the different shaped grooves is similar to the _w distribution.Symbols C _f skin friction coefficient, C _f2 _w/( ( U ₀)²) - C _f,0 skin friction coefficient on the smooth wall - d groove depth - D _h diameter of the idealized primary eddy inside the groove - D _h,s diameter of the idealized secondary eddies inside the groove - d _i internal layer thickness - E turbulent energy spectrum - f _B bursting frequency - f _B⁺ normalized bursting frequency, f _B⁺ f _B/( u )² - k wave number, k =2f/ U - q _i ⁺ contributing quadrant to the total Reynolds stress – uv , q _i ⁺ – uv _i /( u )², i =1, 2, 3, 4 - R Reynolds number based on , R U ₀ / - R Reynolds number based on , R U ₀ / - U mean velocity in the streamwise direction - U ₀ free stream velocity - U ⁺ normalized U by inner variable, U ⁺ U / u - u root-mean-square of velocity fluctuation in the streamwise direction - u ⁺ normalized u by inner variable, u ⁺ u / u - u friction velocity, u ( _w/ )^0.5 - – uv Reynolds stress - v root-mean-square of velocity fluctuation in the wall-normal direction - w groove width - x streamwise coordinate measured from the groove trailing edge - y wall-normal coordinate - y ⁺ normalized y by inner variables, y ⁺ yu / Greek symbols boundary layer thickness - ₀ boundary layer thickness just upstream of the groove, unless otherwise stated - fluid kinematic viscosity - momentum thickness - fluid density - _w wall shear stress     

The effect of an oscillatory freestream-flow on a NACA-4412 profile at large relative amplitudes and low Reynolds-numbers

Results of an experimental investigation of the flow around a NACA-4412 profile in an oscillating freestream are presented. The experiment took place in an Eiffel-type windtunnel at a chord Reynolds-number of Re = 2 · 10⁵. Measurements of unsteady pressure distributions and boundary-layer profiles as well as flow photographs reveal that even at moderate reduced frequencies significant changes of the flow field may occur, provided that the relative amplitude of the freestream is sufficiently large. So a periodical separation and reattachment of the flow could be observed while in another case the periodical relaminarization of the boundary-layer could be found.List of symbols A relative amplitude of freestream velocity - A _I relative amplitude of first harmonic of the freestream velocity - b span of the airfoil profile - C _A lift-coefficient - C _{A st} lift-coefficient in steady freestream - C _p pressure-coefficient - d profile thickness - f frequency - H ₁₂ shape factor - k reduced frequency - l chord length - p phase-average of pressure - p ₀ total head - p static freestream pressure - p _a ambient pressure - q dynamic head - Re mean Reynolds number - Re ₂ Reynolds number - t current time - T phase time - u velocity in x-direction - u freestream-velocity - u amplitude of freestream-velocity - u _a velocity at boundary-layer edge - u _c cooling-velocity - u fluctuation of velocity in x-direction - u _rms mean square of fluctuation - û nondimensional velocity, Fig. 3 - fluctuation of velocity in y-direction - w fluctuation of velocity in z-direction - x,y,z cartesian coordinates - X _A distance of separation line from leading edge - angle of attack - nondimensional pressure gradient - boundary-layer thickness - ₁ displacement-thickness - ₂ momentum-thickness - kinematic viscosity - angular-velocity - () periodical component - (-) time-average - () stochastic component     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height