Non-unique solutions of the boundary layer equations for the flow near the equator of a rotating sphere in a rotating fluid

Summary In this paper it is shown, using a numerical technique, that axially-symmetric solutions of the boundary layer equations which describe the rotating flow near the equator of a rotating sphere are not unique. In certain regimes it is found that at least three possible solutions are possible. When the sphere and fluid rotate with almost the same angular velocity it is shown that the approach to solid body rotation is a non linear process.With 5 Figures     

Unsteady nonsimilar compressible boundary layer flow over a rotating sphere

Summary The problem of determining the development of the nonsimilar viscous compressible boundary layer flow with time over a rotating sphere is considered when both free stream velocity and rotation vary arbitrarily with time. Variations of skin friction, heat transfer and displacement thickness with mass injection, rotation, wall enthalpy and time are discussed.     

Effect of large blowing rates on the compressible boundary layer flow at the stagnation point of a rotating sphere

Summary The effect of a large surface blowing (injection) rate on the steady laminar compressible boundary-layer flow at the forward stagnation point of a rotating sphere has been studied. The resulting coupled nonlinear ordinary differential equations have been solved using two methods, namely, the method of matched asymptotic expansion and the implicit finite difference scheme in combination with the quasilinearization technique. It is found that the boundary layer thickness increases considerably with the blowing rate. The location of the dividing streamline moves away from the surface with increasing blowing rates, but moves towards the surface when the total enthalpy at the wall or the rotation parameter increases. For large blowing rates and small rotation parameter the surface heat transfer and the surface shear stress in the tangential direction tend to zero, but the longitudinal shear stress remains finite but small. Also, for this case, the longitudinal shear stress at the wall is approximately found to be directly proportional to the sum of the total enthaply at the wall and to the square of the rotation parameter and inversely proportional to the blowing rates. The rotation parameter induces overshoot in the longitudinal velocity, and the magnitude of the velocity overshoot increases significantly with rotation and blowing parameters. However, there is no overshoot in the longitudinal velocity in the absence of rotation whatever may be the values of the blowing parameter.     

Stability of flow of a structurally viscous fluid in the boundary layer on a flat plate

The method of small perturbations has been used to study the hydrodynamical stability of flow of a structurally viscous fluid in the boundary layer on a flat plate.     

Slow viscoelastic fluid flow past a rotating sphere

The problem of axially symmetric flow of a particular type of non-Newtonian fluid past a rotating sphere due to a uniform stream at infinity is investigated. The presence of a region of reversed flow is found under certain conditions depending on the angular velocity of the sphere, the speed of the uniform stream and radius of the sphere. This region which is attached to the rear portion of the sphere is found to depend strongly on the viscoelasticity of the fluid. The vortex is seen to move towards the sphere as the viscoelastic parameter increases while the other parameters are kept fixed. As this viscoelastic parameter approaches a critical value, the vortex is found to disappear.     

Study on boundary layer transition of a rotating disk

Summary Behaviour of spiral vortices being generated in transition regime of a disk rotating in otherwise undisturbed fluid is experimentally studied in detail. Through visualizations of the transition regime by using close-up camera, new striped flow patterns originating along the axis of spital vortices are found to be ring-like vortices which occur on the surfaces of each spiral vortices. Mechanism of the spiral vortex is clarified by cutting the vortices by strobo slit light. It is also found out experimentally that the phase velocity of the vortices is zero.     

Nonlinear critical layers in the boundary layer on a rotating disk

The work of Gregory, Stuart and Walker (1955, Proc R Soc Ser A 406:93–106) and Hall (1986, Phil Trans R Soc London Ser A 248:155–199) is extended to include nonlinear effects for the stationary cross-flow vortex. It is shown that amplitude-dependent neutral modes are described by a forced Haberman equation. The corrections to the neutral wavenumbers and waveangles are derived and it is suggested that the nonlinear neutral modes can have wavenumbers decreased by an O(1) amount as compared to linear theory.     

Calculation of a laminar boundary layer on a rotating porous disk

Averaging of noninertial terms over the boundary-layer section in the equations of motion is used to study the effect of suction and injection on the hydrodynamic flow near a rotating disk.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 49, No. 6, pp. 1022–1026, December, 1985.     

Spiral vortices in boundary layer transition regime on a rotating disk

Summary Behavior of spiral vortices on a disk rotating in still fluid is studied theoretically and experimentally in detail. A linear stability analysis, in which effects of streamline curvature and Coriolis force are considered, gives a critical Reynolds number at the onset of instability close to the one measured here by using a hot wire probe. Gradient of the vortex axis is determined under a condition of the maximum amplification. Flow patterns in the transition regime are experimentally visualized. The results show that the number of the spiral vortices is 31 or 32 as mean value and the gradient of the vortex axis decreases from 14° to 7° as the local Reynolds number is increased.

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Spiral-Wirbel im Umschlagbereich der Grenzschicht an einer rotierenden Scheibe

Zusammenfassung Das Verhalten der spiralen Wirbel an einer in ruhender Flüssigkeit rotierenden Scheibe wird theoretisch und experimentell eingehend untersucht. Eine lineare Stabilitätstheorie, in welcher der Einfluß von Krümmung der Stromlinien und Coriolis-Kraft in Betracht gezogen wird, ergibt eine kritische Reynolds-Zahl des Indifferenzpunktes, die gut mit dem mit einem Hitzdrahtgerät gemessenen Wert übereinstimmt. Der Gradient der Wirbelachse wird unter der Bedingung der maximalen Anfachung bestimmt. Der Strömungsvorgang im Umschlagbereich wird experimentell sichtbar gemacht. Dadurch ergibt sich, daß die Anzahl der an der Scheibe auftretenden Wirbel 31 oder 32 im Mittel beträgt, und der Gradient der Wirbelachse mit zunehmender lokaler Reynolds-Zahl von 14° bis zu 7° abnimmt.

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With 12 Figures

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Dedicated to Professor Dr. Henry Görtler on the occasion of his seventieth birthday     

Interaction of drops with boundary layer on rotating surface

The interaction of a drop with a boundary layer on the surface of a rotating disk is investigated experimentally.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 31, No. 6, pp. 1068–1073, December, 1976.     

Stability of the free-convection boundary layer on a vertical permeable plate

The hydrodynamic stability of flow in a free-convection boundary layer on a permeable plate is numerically investigated. The effect of the plate nonisothermicity and the injection blowing intensity on the critical parameters is determined.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 48, No. 3, pp. 382–387, March, 1985.     

Analysis of forced convection micropolar boundary layer about a permeable sphere

The heat transfer characteristics and local skin friction for forced convection flow of micropolar fluid over an isothermal or a constant-heat-flux surface of sphere with surface mass transfer have been studied. The problem was formulated by applying a suitable variables transformation and the solutions were obtained by an implicit finite difference method. Numerical results were carried out for a wide range of mass transfer parameters as the Prandtl number at 0.7 or 7 with several values of material parameters of the micropolar fluid. The variations of the local friction factor and local Nusselt number are plotted and discussed.     

Perturbed motion of a viscous fluid layer on the surface of a rotating cylinder

We investigate the plane flow of a viscous fluid layer plane flow on the surface of a cylinder rotating with a constant angular velocity with account for inertia forces and the acceleration of gravity.Lomonosov Moscow State University, Moscow; Bauman Moscow State Technical University; Belarusian State University, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 6, pp. 689–694, June, 1994.     

On the stability of a hot rotating layer of micropolar fluid

The convective stability of a horizontal layer of incompressible micropolar fluid heated from below and rotating about a vertical axis has been investigated on the basis of linear theory, using normal mode analysis. The boundaries are assumed to be free. After introducing the corrections to the basic equations considered by Sastry and Rao [1], it has been found that the rotation has a destabilizing effect which contradicts the earlier assertion presented in [1].Moreover, microinertia, which does not affect the stability of a hot horizontal layer of incompressible micropolar fluid in the absence of rotation [2], is found to have destabilizing effect.     

Turbulent flow in a boundary layer on the inlet and outlet sides of a rotating channel

Expressions are obtained for approximately determining the shear stresses on the outlet and inlet sides of a rotating channel on the basis of a turbulence energy balance equation, A. N. Kolmogorov's hypothesis, and the Monin-Obukhov similitude theory.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 6, pp. 977–986, December, 1981.     

Axisymmetric thermal boundary layer of a micropolar fluid on a cylinder

The laminar micropolar thermal boundary layer in axial flow along a long, thin circular cylinder is investigated using the theory of micropolar fluids formulated by Eringen. The governing energy equation has been solved numerically by the power series approach. Missing values of the thermal functions are tabulated for a wide range of the material parameters, the transverse curvature parameter, and Prandtl number of the fluid.     

Longitudinal surface curvature effects on boundary layer of a micropolar fluid

Summary The effect of longitudinal surface curvature on steady two-dimensional incompressible laminar boundary layer of a micropolar fluid has been considered. Van Dyke's first oder perturbation analysis is applied to the full equations of motion derived in curvilinear coordinate system which facilitates to carry out boundary layer approximation for flow past a curved surface. This results into two systems of partial differential equations which are called the zeroth order and the first order boundary layer equations. The zeroth order equations are the usual boundary layer equations for a micropolar fluid. The first order equations take into account the longitudinal surface curvature effect explicitly. Similar solution of the governing equations exists if (i) the inviscid flow velocity on the surface varies linearly along the surface and (ii) the longitudinal surface curvature is constant. Numerical results are presented illustrating the dependence of the important flow quantities of both zeroth order and first order boundary layers on the micropolar fluid parameters. The results have been compared with the corresponding results for a Newtonian fluid. It has been found that the skin friction decreases and the wall couple stress increases for convex side of the surface and vice versa for the concave side.

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Einfluß der longitudinalen Oberflächenkrümmung auf die Grenzschicht einer mikropolaren Flüssigkeit

Zusammenfassung Der Einfluß einer longitudinalen Oberflächenkrümmung auf die stationäre, zweidimensionale, inkompressible, laminare Grenzschicht einer mikropolaren Flüssigkeit wird untersucht. Van Dyke's Strömungsanalyse erster Ordnung wird angewendet auf die Bewegungsgleichungen in krummlinigen Koordinaten, die einfach die Grenzschichtnäherung für die Strömung hinter einer gekrümmten Oberfläche aufzeigen. Dies führt zu zwei Systemen partieller Differentialgleichungen, die die Grenzschichtgleichungen nullter und erster Ordnung genannt werden. Die Gleichungen nullter Ordnung sind die üblichen Grenzschichtgleichungen für eine mikropolare Flüssigkeit. Die Gleichungen erster Ordnung berücksichtigen den Einfluß der longitudinalen Oberflächenkrümmung explizit. Ähnliche Lösungen der Grundgleichungen existieren wenn: (i) sich die reibungsfreie Strömungsgeschwindigkeit an der Oberfläche linear entlang der Oberfläche ändert und (ii) die longitudinale Oberflächenkrümmung konstant ist. Numerische Ergebnisse werden angegeben, um die Abhängigkeit der wichtigsten Strömungsgrößen der Grenzschicht nullter und erster Ordnung von den Parametern der mikropolaren Flüssigkeit aufzuzeigen. Die Ergebnisse werden verglichen mit den entsprechenden Ergebnissen für eine Newtonsche Flüssigkeit. Es wurde festgestellt, daß für eine konvexe Fläche die Wandreibung abnimmt und die gekoppelten Spannungen an der Wand zunehmen; das Umgekehrte gilt für eine konkave Fläche.

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With 2 Figures     

Influence of thermal radiation on the laminar boundary layer of a nonabsorbing fluid

The influence of thermal radiation on the laminar boundary layer of a nonabsorbing fluid with variable thermophysical properties flowing around a heat emitting surface is investigated under natural and forced convection conditions.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 20, No. 6, pp. 1008–1014, June, 1971.