The flow due to a rotating disc

1934 ◽  
Vol 30 (3) ◽  
pp. 365-375 ◽  
Author(s):  
W. G. Cochran
Keyword(s):  
Angular Velocity ◽  
Viscous Fluid ◽  
Equations Of Motion ◽  
Steady Motion ◽  
Constant Angular Velocity ◽  
Incompressible Viscous Fluid ◽  
Polar Coordinates ◽  
Rotating Disc ◽  
Cylindrical Polar Coordinates ◽  
Image Position

1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Kármán. If r, θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r, θ and z increasing, respectively, and p is the pressure, then Kármán shows that the equations of motion and continuity are satisfied by taking

scholarly journals IV. On the second approximation to the "oseen” solution the motion of a viscous fluid

1928 ◽  
Vol 227 (647-658) ◽  
pp. 93-135 ◽  
Keyword(s):  
Viscous Fluid ◽  
Radial Flow ◽  
Equations Of Motion ◽  
Incompressible Viscous Fluid ◽  
Large Radius ◽  
Sufficient Distance ◽  
Uniform Stream

In a recent paper the author obtained expressions for the forces on a stationary cylinder in a steady stream of incompressible viscous fluid and showed that the force transverse to the stream follows the well-known Kutta-Joukowski law, whereas the force in the direction of the stream itself is given by a similar law, involving, instead of the circulation, an outward radial flow, compensated by an intake along a “tail” behind the cylinder. These results were obtained by considering the motion at a distance from the cylinder, and assuming that the velocities of disturbance from the uniform stream were so small that, at a sufficient distance, their squares and products could be neglected both in the equations of motion and in the integrals round a circle of large radius, in terms of which the forces on the cylinder were expressed.

The stability of viscous fluid flow between rotating cylinders

1937 ◽  
Vol 33 (1) ◽  
pp. 41-61 ◽  
Author(s):  
S. Goldstein
Keyword(s):  
Angular Velocity ◽  
Wave Length ◽  
Steady Motion ◽  
Circular Cylinders ◽  
Critical Flow ◽  
Average Value ◽  
Time Period ◽  
Component Velocity ◽  
The Stability ◽  
Image Position

The stability of the motion of viscous incompressible fluid, of density ρ and kinematic viscosity ν, between two infinitely long coaxial circular cylinders, of radiiaanda+d, whered/ais small, is investigated mathematically by the method of small oscillations. The inner cylinder is rotating with angular velocity ω and the outer one with angular velocity αω, and there is a constant pressure gradient parallel to the axis. The fluid therefore has a component velocityWparallel to the axis, in addition to the velocity round the axis. A disturbance is assumed which is symmetrical about the axis and periodic along it. The critical disturbance, which neither increases nor decreases with the time, is periodic with respect to the time (except whenW= 0, when the critical disturbance is a steady motion). As Reynolds number of the flow we take ||d/ν, whereis the average value ofWacross the annulus, and we denote bylthe wave-length of the disturbance along the axis, by σ/2π the time period of the critical flow, bycthe wavelength of the critical flow, byωcthe critical value of ω, and we putapproximately, if α is not nearly equal to 1.

Some physical interpretations of potentials representing supersonic motion of compressible fluids

1949 ◽  
Vol 45 (2) ◽  
pp. 246-250
Author(s):  
R. K. Tempest
Keyword(s):  
Incompressible Flow ◽  
Velocity Potential ◽  
Compressible Fluids ◽  
Polar Coordinates ◽  
Supersonic Speed ◽  
Small Perturbations ◽  
Stream Flows ◽  
Cylindrical Polar Coordinates ◽  
Image Position ◽  
Supersonic Motion

1. Compressible and incompressible flow. Small perturbations in an otherwise uniform stream of compressible fluid moving at supersonic speed are described by the approximate linearized equation for the velocity potential. When the stream flows in the z-direction, the equation assumes the formwhere M is the Mach number of the flow and α2 is positive. In cylindrical polar coordinates (r, z), the equation may be written aswhich is Laplace's equation in coordinates (iαr, z). We may therefore relate potentials of incompressible flow which are solutions of (1·2) to potentials of compressible flow which are solutions of (1·1).

The slow motion of a sphere in a rotating, viscous fluid

1964 ◽  
Vol 20 (2) ◽  
pp. 305-314 ◽  
Author(s):  
Stephen Childress
Keyword(s):  
Reynolds Number ◽  
Angular Velocity ◽  
Viscous Fluid ◽  
Classical Problem ◽  
Slow Motion ◽  
Reynolds Numbers ◽  
Constant Angular Velocity ◽  
Axis Of Rotation ◽  
Small Reynolds Numbers ◽  
Analytical Result

The uniform, slow motion of a sphere in a viscous fluid is examined in the case where the undisturbed fluid rotates with constant angular velocity Ω and the axis of rotation is taken to coincide with the line of motion. The various modifications of the classical problem for small Reynolds numbers are discussed. The main analytical result is a correction to Stokes's drag formula, valid for small values of the Reynolds number and Taylor number and tending to the classical Oseen correction as the last parameter tends to zero. The rotation of a free sphere relative to the fluid at infinity is also deduced.

The Design of Current Carrying Mechanisms Operating in a Magnetic Field

1978 ◽  
Vol 100 (3) ◽  
pp. 574-582
Author(s):  
W. S. Reed ◽  
W. D. Smith
Keyword(s):  
Magnetic Field ◽  
Numerical Model ◽  
Angular Velocity ◽  
Experimental Work ◽  
Equations Of Motion ◽  
Constant Angular Velocity ◽  
Constant Voltage ◽  
Constant Torque ◽  
Parametric Studies ◽  
Velocity Input

Extensions to earlier work presented on the electrodynamics of a planar mechanism moving in a magnetic field are considered in this paper. The effects of damping and self-fields are added to the equations of motion. These equations are then integrated numerically for an example design having a constant torque input, a constant angular velocity input, and a constant voltage input. The results of numerous parametric studies are then presented as a basis for design of these devices. Finally the results of experimental work are presented in order to verify the numerical model.

Effects of Partial Slip on the Analytic Heat and Mass Transfer for the Incompressible Viscous Fluid of a Porous Rotating Disk Flow

2011 ◽  
Vol 133 (12) ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Viscous Fluid ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Incompressible Viscous Fluid ◽  
Temperature Fields ◽  
Partial Slip ◽  
Shear Stresses ◽  
Rotating Disk Flow

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a porous disk rotating with a constant angular speed about its axis. The recent study (Turkyilmazoglu, 2009, “Exact Solutions for the Incompressible Viscous Fluid of a Porous Rotating Disk Flow,” Int. J. Non-Linear Mech., 44, pp. 352–357) is extended to account for the effects of partial flow slip and temperature jump imposed on the wall. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions for the flow and temperature fields. Explicit expressions representing the flow properties influenced by the slip as well as a uniform suction and injection are extracted, including the velocity, vorticity and temperature fields, shear stresses, flow and thermal layer thicknesses, and Nusselt number. The effects of variation in the slip parameters are better visualized from the formulae obtained.

The three-dimensional inverse scattering problem for the Helmholtz equation

1973 ◽  
Vol 73 (3) ◽  
pp. 477-488 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Helmholtz Equation ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Radiation Condition ◽  
Polar Coordinates ◽  
Wave Number ◽  
Radial Coordinate ◽  
Cylindrical Polar Coordinates ◽  
Image Position

In this paper we are concerned with solutions of the three-dimensional Helmholtz equation which are of class C2 (i.e. regular) in the exterior of a bounded domain D. In cylindrical polar coordinates (r, z, φ) such solutions satisfy the equationin which we have dimensionalized the radial coordinate r so that the wave number is normalized to unity. If we further assume that u satisfies the Sommerfeld radiation conditionthen u may be regarded as being generated by volume sources, surface sources, or point singularities, all of which are contained in D.

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