Models for viscous dissipation of energy obtained from the Lagrangian form of the equations of motion

1974 ◽  
Vol 17 (4) ◽  
pp. 679 ◽  
Author(s):  
Mary Attermeyer
Keyword(s):  
Viscous Dissipation ◽  
Equations Of Motion ◽  
Dissipation Of Energy ◽  
Lagrangian Form

scholarly journals Superposition of Motions in The Space of Lagrange Variables

2021 ◽  
Author(s):  
YURIY ALYUSHIN
Keyword(s):  
Conservation Law ◽  
Equations Of Motion ◽  
External Exposure ◽  
Helical Rolling ◽  
Combined Motion ◽  
Vector Addition ◽  
Lagrangian Form ◽  
Deformable Bodies ◽  
Lagrange Variables ◽  
Energy Conservation Law

The technique of superposition of motions in the space of Lagrange variables is described, which allows us to obtain the equations of combined motion by replacing the Lagrange variables of superimposed (external) motion with Euler variables of nested (internal) motion. The components of velocity and acceleration in the combined motion obtained as a result of differentiating the equations of motion in time coincide with the results of vector addition of the velocities and accelerations of the particles involved in the superimposed motions at each moment of time. Examples of motion and superposition descriptions for absolutely solid and deformable bodies with equations for the main kinematic characteristics of motion, including for robot manipulators with three independent drives, pressing with torsion, bending with tension, and cross– helical rolling, are given. For example, given the fragment of calculation of forces in the kinematic pairs shown the advantages of the description of motion in Lagrangian form for the dynamic analysis of lever mechanisms, allows to determine the required external exposure when performing the energy conservation law at any time in any part of the mechanism.

An example of minimum energy dissipation in viscous flow

1959 ◽  
Vol 251 (1267) ◽  
pp. 550-564 ◽  
Keyword(s):  
Maximum Rate ◽  
Equations Of Motion ◽  
Minimum Energy ◽  
Vertical Tube ◽  
Approximate Theory ◽  
Eccentricity Ratio ◽  
Dissipation Of Energy ◽  
Rapid Fall ◽  
The Times ◽  
Minimum Energy Dissipation

An approximate theory is developed to describe the behaviour of a heavy ball passing slowly down a vertical tube having a diameter only slightly exceeding the diameter of the ball, and filled with a viscous fluid. It is shown that according to this theory the equations of motion can be satisfied when the ball takes up any degree of eccentricity in the tube and that any given eccentricity requires a particular velocity of rotation about a horizontal axis. It is found that the eccentricity ratio corresponding to minimum dissipation of energy for given velocity of descent (i.e. to maximum rate of fall for a given weight of ball) is about 0.98, and that the velocity is then rather more than twice the velocity corresponding to zero eccentricity. Experiments are described in which it was shown that provided conditions were such that the ball descended very slowly, the minimum dissipation prediction was verified within the expected accuracy, but that for larger clearances and more rapid fall the predicted angular velocity and eccentricity were not achieved within the times for which observation was possible.

The trapping and vertical focusing of internal waves in a pycnocline due to the horizontal inhomogeneities of density and currents

1985 ◽  
Vol 158 ◽  
pp. 199-218 ◽  
Author(s):  
S. I. Badulin ◽  
V. I. Shrira ◽  
L. Sh. Tsimring
Keyword(s):  
Wave Packet ◽  
Viscous Dissipation ◽  
Internal Waves ◽  
Equations Of Motion ◽  
Local Maximum ◽  
Velocity Shear ◽  
Vertical Shear ◽  
Small Scale ◽  
The Mean ◽  
Mean Current

This paper studies the propagation of a wave packet in regions where the central packet frequency ω is close to the local maximum of the effective Väisälä frequency Nf(z) = N(z)/[1 − k·U(z)/ω], where k is the central wavevector of the packet and U is the mean current with a vertical velocity shear. The wave approaches the layer ω = Nfm asymptotically, i.e. trapping of the wave takes place. The trapping of guided internal waves is investigated within the framework of the linearized equations of motion of an incompressible stratified fluid in the WKB approximation, with viscosity, spectral bandwidth of the packet, vertical shear of the mean current and non-stationarity of the environment taken into account. As the packet approaches the layer of trapping, the growth of the wavenumber k is restricted only by possible wave-breaking and viscous dissipation. The growth of k is accompanied by the transformation of the vertical structure of internal-wave modes. The wave motion focuses at a certain depth determined by the maximum effective Väisälä frequency Nfm. The trapping of the wave packet results in power growth of the wave amplitude and steepness. At larger times the viscous dissipation becomes a dominating factor of evolution as a result of strong slowing down of the packet motion.The role of trapping in the energy exchange of internal waves, currents and small-scale turbulence is discussed.

scholarly journals Effect of Tapering on Natural Frequencies of Rotating Beams

2007 ◽  
Vol 14 (3) ◽  
pp. 169-179 ◽  
Author(s):  
A. Bazoune
Keyword(s):  
Eigenvalue Problem ◽  
Free Vibration ◽  
Excellent Agreement ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Generalized Eigenvalue ◽  
Stiffness Matrices ◽  
Wide Range ◽  
Lagrangian Form ◽  
Stiffening Effect

The problem of free vibration of a rotating tapered beam is investigated by developing explicit expressions for the mass, elastic and centrifugal stiffness matrices in terms of the taper ratios. This investigation takes into account the effect of tapering in two planes, the effect of hub radius as well as the stiffening effect of rotation. The equations of motion are derived; the associated generalized eigenvalue problem is defined in conjunction with a suitable Lagrangian form and solved for a wide range of parameter changes. The effect of tapering on the natural frequencies of the beam is examined with all parameter changes present. Results are compared with those available in literature and are found to be in excellent agreement.

Heat and Mass Transfer on the MHD Fluid Flow Due to a Porous Rotating Disk With Hall Current and Variable Properties

2010 ◽  
Vol 133 (2) ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Viscous Dissipation ◽  
Joule Heating ◽  
Hall Current ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Integration Scheme ◽  
Shear Stresses ◽  
Similarity Transformations ◽  
Fluid Properties

The steady magnetohydrodynamics (MHD) laminar compressible flow of an electrically conducting fluid on a porous rotating disk is considered in the present paper. The governing equations of motion are reduced to a set of nonlinear differential equations by means of similarity transformations. The fluid properties are taken to be strong functions of temperature and Hall current that also readily accounts for the viscous dissipation and Joule heating terms. Employing a highly accurate spectral numerical integration scheme, the effects of viscosity, thermal conductivity, Hall current, magnetic field, suction/injection, viscous dissipation, and Joule heating on the considered flow are examined. The quantities of particular physical interest, such as the torque, the wall shear stresses, the vertical suction velocity, and the rate of heat transfer are calculated and discussed.

REDUCTIONS OF CHERN–SIMONS THEORY RELATED TO INTEGRABLE SYSTEMS WHICH HAVE GEOMETRIC APPLICATIONS

2004 ◽  
Vol 18 (09) ◽  
pp. 1261-1275 ◽  
Author(s):  
PAUL BRACKEN
Keyword(s):  
Integrable Systems ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Simons Theory ◽  
Gauge Potential ◽  
Pure Gauge ◽  
Zero Curvature ◽  
Lagrangian Form ◽  
Chern Simons ◽  
Chern Simons Theory

The Chern–Simons functional is introduced in terms of chiral fields and then studied here. The current can be regarded as a non-Abelian pure gauge potential so that the zero-curvature equations are of Lagrangian form for pure non-Abelian Chern–Simons theory. The equations of motion are developed and a formalism which connects the zero curvature equations with a related moving trihedral is introduced. The moving frame equations are written down for the system and a correspondence between these equations and several related elementary integrable systems is described in the same formalism as well.

scholarly journals Some Observations of Wind Velocity Autocorrelations in the Lowest Layers of the Atmosphere

1955 ◽  
Vol 8 (4) ◽  
pp. 535 ◽  
Author(s):  
RJ Taylor
Keyword(s):  
Viscous Dissipation ◽  
Wind Velocity ◽  
Fair Agreement ◽  
Inertial Subrange ◽  
Time Interval ◽  
Mean Square ◽  
Shearing Stress ◽  
Local Isotropy ◽  
Dissipation Of Energy

From wind velocity observations in the lowest layers of the atmosphere, mean square velocity differences over a time interval at a point fixed in space are derived and their variation with the time interval is considered. The magnitude of these mean square differences is related to the rate of viscous dissipation of energy and to the shearing stress. On the average, fair agreement with predictions from the dimensional arguments of the theory of local isotropy is shown even though the results pertain to eddy sizes outside. the inertial subrange as usually defined.

Analysis of Secondary Flows and Heat Transfer in Viscoelastic Flow With Viscous Dissipation in Non-Circular Tubes

2012 ◽  
Author(s):  
Mario F. Letelier ◽  
Fernando N. Zapata ◽  
Dennis A. Siginer ◽  
Juan Stockle
Keyword(s):  
Viscous Dissipation ◽  
Equations Of Motion ◽  
Viscoelastic Behavior ◽  
Weissenberg Number ◽  
Secondary Flows ◽  
Perturbation Approach ◽  
Axially Symmetric ◽  
Cross Sectional ◽  
Non Linear ◽  
Linear Viscoelastic

The steady laminar non-isothermal flow of a non-linear viscoelastic fluid in tubes or arbitrary contour is analyzed under constant wall heat flux. Viscous dissipation is taken into account. Non-linear viscoelastic behavior is modeled by means of the Phan-Thien-Tanner model. The equations of motion and energy are solved analytically through a perturbation approach coupled with a one-to-one mapping of the boundary to map the circular shape into various axially symmetric cross-sectional shapes with the Weissenberg number as the perturbation parameter Longitudinal and transversal velocity components are determined for several cross-sectional shapes, for which the temperature field and the Nusselt number variation with the Prandtl, Brinkman and Reynolds numbers are computed.

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