Asymmetric flow above a rotating disk

1985 ◽  
Vol 157 ◽  
pp. 471-492 ◽  
Author(s):  
C.-Y. Lai ◽  
K. R. Rajagopal ◽  
A. Z. Szeri
Keyword(s):  
Angular Velocity ◽  
Stokes Equations ◽  
Rotating Disk ◽  
Navier Stokes ◽  
Arbitrary Parameter ◽  
Asymmetric Flow ◽  
Navier Stokes Equations ◽  
Von Karman ◽  
Axisymmetric Solutions ◽  
Von Kármán

In this paper we generalize the von Kármán solution for flow above a single rotating disk, to include non-axisymmetric solutions. These solutions contain an arbitrary parameter; for zero value of the parameter the asymmetric flow degenerates into the classical von Kármán solution. Thus the classical solution is never isolated when considered within the scope of the full Navier–Stokes equations; there are asymmetric solutions in every neighbourhood of the von Kármán solution. Calculations are reported here for s = 0, 0.02 and 0.06, where s represents the ratio of angular velocity of the fluid at infinity to the angular velocity of the disk. A subset of the solutions obtained here corresponds to flow induced by the rotation of a disk when the latter is placed in a fluid that is moving with a constant uniform velocity.

Numerical Integration of the Navier-Stokes Equations for a Disk Rotating in a Housing

1985 ◽  
Vol 40 (8) ◽  
pp. 789-799 ◽  
Author(s):  
A. F. Borghesani
Keyword(s):  
Boundary Layer ◽  
Cylindrical Cavity ◽  
Boundary Layer Thickness ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Rotating Disk ◽  
Infinite Medium ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Viscous Torque

The Navier-Stokes equations for the fluid motion induced by a disk rotating inside a cylindrical cavity have been integrated for several values of the boundary layer thickness d. The equivalence of such a device to a rotating disk immersed in an infinite medium has been shown in the limit as d → 0. From that solution and taking into account edge effect corrections an equation for the viscous torque acting on the disk has been derived, which depends only on d. Moreover, these results justify the use of a rotating disk to perform accurate viscosity measurements.

scholarly journals Determining the Tangential Velocity profile of a Rotating Fluid in a Static Container using Navier–Stokes equations

2020 ◽  
Author(s):  
RAJDEEP TAH ◽  
SARBAJIT MAZUMDAR ◽  
Krishna Kant Parida
Keyword(s):  
Angular Velocity ◽  
Velocity Profile ◽  
Velocity Gradient ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Navier Stokes ◽  
Rotating Fluid ◽  
Navier Stokes Equations ◽  
Similar Principle ◽  
Tangential Velocity Profile

The shape of the liquid surface for a fluid present in a uniformly rotating cylinder is generally determined by making a Tangential velocity gradient along the radius of the rotating cylindrical container. A very similar principle can be applied if the direction of the produced velocity gradient is reversed, for which the source of rotation will be present at the central axis of the cylindrical vessel in which the liquid is present. Now if the described system is completely closed, the angular velocity will decrease as a function of time. But when the surface of the rotating fluid is kept free, then the Tangential velocity profile would be similar to that of the Taylor-Couette Flow, with a modification that; due to formation of a curvature at the surface, the Navier-Stokes law is to be modified. Now the final equation may not seem to have a proper general solution, but can be approximated to certain solvable expressions for specific cases of angular velocity.

scholarly journals Vortex motion of a continuous medium depending on the pressure change

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
Alexander Braginsky
Keyword(s):  
Angular Velocity ◽  
Continuous Medium ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Vortex Motion ◽  
Pressure Change ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Least Action ◽  
Force Components

Abstract In this paper, we study the vortex motion of a continuous medium, which is described by forces obtained from the principle of least action. It is shown that in a continuous medium the vortex force components are proportional to the velocity and pressure gradient components. This article gives a description of the 2D vortex motion of air in zones of high and low pressure. If the pressure decreases, the angular velocity of rotation of the continuous medium increases, whereas if the pressure increases, the angular velocity fades. The lifting force is obtained due to the vortex movement of air in the form of a funnel. It is shown that the vortex force contains a vortex term of the Euler hydrodynamic equations with a relative factor equal to the velocity of the continuous medium squared divided by the sound velocity squared. To describe the motion of a continuous medium correctly it is necessary to replace the forces obtained by Euler with the forces obtained from the minimum of action in the equations of motion. It is concluded that vortex motions and turbulence are described by the obtained equations of motion, and not by the Navier–Stokes equations. Most likely, this is related to the Problem of the Millennium description of turbulence announced at the International Congress of Mathematics in 2000.

Axisymmetric Stagnation—Point Flow and Heat Transfer of a Viscous Fluid on a Rotating Cylinder With Time-Dependent Angular Velocity and Uniform Transpiration

2006 ◽  
Vol 129 (1) ◽  
pp. 106-115 ◽  
Author(s):  
A. B. Rahimi ◽  
R. Saleh
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Reynolds Number ◽  
Angular Velocity ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Flow And Heat Transfer ◽  
Self Similar

The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite rotating circular cylinder with transpiration U0 are investigated when the angular velocity and wall temperature or wall heat flux all vary arbitrarily with time. The free stream is steady and with a strain rate of Γ. An exact solution of the Navier-Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by the use of appropriate transformations for the most general case when the transpiration rate is also time-dependent but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the angular velocity of the cylinder and its wall temperature or its wall heat flux vary as specified time-dependent functions. In particular, the cylinder may rotate with constant speed, with exponentially increasing/decreasing angular velocity, with harmonically varying rotation speed, or with accelerating/decelerating oscillatory angular speed. For self-similar flow, the surface temperature or its surface heat flux must have the same types of behavior as the cylinder motion. For completeness, sample semi-similar solutions of the unsteady Navier-Stokes equations have been obtained numerically using a finite-difference scheme. Some of these solutions are presented for special cases when the time-dependent rotation velocity of the cylinder is, for example, a step-function. All the solutions above are presented for Reynolds numbers, Re=Γa2∕2υ, ranging from 0.1 to 1000 for different values of Prandtl number and for selected values of dimensionless transpiration rate, S=U0∕Γa, where a is cylinder radius and υ is kinematic viscosity of the fluid. Dimensionless shear stresses corresponding to all the cases increase with the increase of Reynolds number and suction rate. The maximum value of the shear stress increases with increasing oscillation frequency and amplitude. An interesting result is obtained in which a cylinder rotating with certain exponential angular velocity function and at particular value of Reynolds number is azimuthally stress-free. Heat transfer is independent of cylinder rotation and its coefficient increases with the increasing suction rate, Reynolds number, and Prandtl number. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration.

An Asymptotic Solution for the Laminar Flow of a Thin Film on a Rotating Disk

1973 ◽  
Vol 40 (1) ◽  
pp. 43-47 ◽  
Author(s):  
J. W. Rauscher ◽  
R. E. Kelly ◽  
J. D. Cole
Keyword(s):  
Thin Film ◽  
Asymptotic Solution ◽  
Radial Direction ◽  
Stokes Equations ◽  
Fluid Flows ◽  
Rotating Disk ◽  
Navier Stokes ◽  
Rossby Number ◽  
Navier Stokes Equations ◽  
Higher Order Terms

An analysis on the basis of the Navier-Stokes equations is made of the flow of a liquid in a thin film on a rotating disk. Fluid flows outward in the radial direction due to the centrifugal force. An asymptotic solution valid for small values of the Rossby number is presented. Higher-order terms correcting for the effects of convection, Coriolis acceleration, radial diffusion, surface curvature, and surface tension are found and discussed on a physical basis.

Inward Flow Between Stationary and Rotating Disks

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
Achhaibar Singh
Keyword(s):  
Laminar Flow ◽  
Flow Field ◽  
Velocity Distribution ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Rotating Disk ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Rotating Disks ◽  
Navier Stokes Equations

The present study predicts the flow field and the pressure distribution for a laminar flow in the gap between a stationary and a rotating disk. The fluid enters through the peripheral gap between two concentric disks and converges to the center where it discharges axially through a hole in one of the disks. Closed form expressions have been derived by simplifying the Navier– Stokes equations. The expressions predict the backflow near the rotating disk due to the effect of centrifugal force. A convection effect has been observed in the tangential velocity distribution at high throughflow Reynolds numbers.

Melting From a Horizontal Rotating Disk

1989 ◽  
Vol 56 (1) ◽  
pp. 47-50 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Heat Transfer ◽  
Numerical Integration ◽  
Stokes Equations ◽  
Rotating Disk ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Relative Importance ◽  
Governing Equations ◽  
Navier Stokes Equations ◽  
Transfer Rates

Melting of a disk is facilitated by rotation. The problem is governed by a nondimensional parameter α which represents the relative importance of injection (melt) rate and rotation times viscosity. The nonlinear governing equations are solved by perturbations for small α and numerical integration for arbitrary α. Torque and heat transfer rates are found. The solution is one of the rare exact similarity solutions of the Navier-Stokes equations.

A note on viscous flow induced by half-line sources bounded by conical surfaces

2019 ◽  
Author(s):  
Prabakaran Rajamanickam ◽  
Adam D Weiss
Keyword(s):  
Radial Velocity ◽  
Line Source ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
The Other ◽  
Half Line ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Axisymmetric Solutions ◽  
Conical Surfaces

Summary In this article, axisymmetric solutions of the Navier–Stokes equations governing the flow induced by a half-line source when the fluid domain is bounded by a conical wall are discussed. Two types of boundary conditions are identified; one in which the radial velocity along the axis is prescribed, and the other in which the radial velocity along the axis is obtained as an eigenvalue of the problem. The existence of these solutions is limited to a range of Reynolds numbers, and the transition from one case to the other is discussed in detail.

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