The developing tangential velocity profile for axial flow in an annulus with a rotating inner cylinder

1968 ◽  
Vol 307 (1488) ◽  
pp. 55-69 ◽  
Keyword(s):  
Velocity Profile ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Axial Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Taylor Vortices ◽  
Flow Through ◽  
Momentum Integral ◽  
Tangential Velocity Profile

A method is described of predicting the growth of a tangential velocity profile in fully developed laminar axial flow through a concentric annulus when the inner surface is rotated at speeds which are insufficient to generate Taylor vortices. The treatment, which is based on simplification and subsequent solution of the Navier-Stokes equations, as Fourier-Bessel series, appears preferable to momentum-integral techniques through greater simplicity of expression and in requiring fewer assumptions about the developing tangential profile. The validity of the predictions is best at high axial Reynolds number.

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.

Tangential flow development for laminar axial flow in an annulus with a rotating inner cylinder

1972 ◽  
Vol 328 (1572) ◽  
pp. 123-141 ◽  
Keyword(s):  
Velocity Profile ◽  
Tangential Velocity ◽  
Axial Flow ◽  
Axial Distance ◽  
Taylor Vortices ◽  
Developing Flow ◽  
Axial Velocity Profile ◽  
Displacement Thickness ◽  
Core Rotation ◽  
Tangential Velocity Profile

The numerical finite-difference procedure of Gosman et al. (1969) is used to predict the growth of the tangential velocity profile and boundary-layer displacement thickness across an isothermal laminar axial flow through a concentric annulus when the inner cylinder is rotated at speeds which are insufficient to generate Taylor vortices. Solutions are obtained for fully developed and for developing axial flow over the ranges 0.05 < R 1 /R 2 < 0.98, 0.0002 < l < 1.0 and 100 < Re < 1700. The axial velocity profile is predicted to be insensitive to core rotation and, if varied, to influence only marginally the development of the tangential velocity profile; this is such that its dimensionless displacement thickness is related to dimensionless axial distance by a power law except near full development and at very low Reynolds number. Predictions at high Re accord extremely well with measurements. Astill’s (1964) stability criterion for the onset of vortices in tangential developing flow is accordingly presented afresh in terms of system parameters readily available to the designer.

scholarly journals Application of Navier – Stokes Equation to Solve Fluid Flow Problems

2020 ◽  
pp. 101-114
Author(s):  
Ebiendele Ebosole Peter ◽  
Adamu Bala
Keyword(s):  
Fluid Flow ◽  
Laminar Flow ◽  
Velocity Profile ◽  
Stokes Equations ◽  
The Other ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
Flow Through

The Navier – Stokes equations were used to obtain the velocity profile for two different fluid flow problems, firstly to a laminar flow through a pipe and secondly to flow of incompressible fluid between two boundaries, one boundary is the air and the other boundary moving with a velocity, inclined at an angle . The velocity profiles were obtained and presented in a diagram of showing how the fluid flow through the channels.

scholarly journals Transient Pressure-Driven Electroosmotic Flow through Elliptic Cross-Sectional Microchannels with Various Eccentricities

Computation ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 27
Author(s):  
Nattakarn Numpanviwat ◽  
Pearanat Chuchard
Keyword(s):  
Laplace Transform ◽  
Electroosmotic Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Mathieu Functions ◽  
Navier Stokes Equations ◽  
Flow Rates ◽  
Elliptic Cross ◽  
Flow Through ◽  
Relative Errors

The semi-analytical solution for transient electroosmotic flow through elliptic cylindrical microchannels is derived from the Navier-Stokes equations using the Laplace transform. The electroosmotic force expressed by the linearized Poisson-Boltzmann equation is considered the external force in the Navier-Stokes equations. The velocity field solution is obtained in the form of the Mathieu and modified Mathieu functions and it is capable of describing the flow behavior in the system when the boundary condition is either constant or varied. The fluid velocity is calculated numerically using the inverse Laplace transform in order to describe the transient behavior. Moreover, the flow rates and the relative errors on the flow rates are presented to investigate the effect of eccentricity of the elliptic cross-section. The investigation shows that, when the area of the channel cross-sections is fixed, the relative errors are less than 1% if the eccentricity is not greater than 0.5. As a result, an elliptic channel with the eccentricity not greater than 0.5 can be assumed to be circular when the solution is written in the form of trigonometric functions in order to avoid the difficulty in computing the Mathieu and modified Mathieu functions.

Annular Axial Flow-Induced Vibration of Cylindrical Shells by Flu¨gge’s Shell Theory

2006 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Makoto Katou
Keyword(s):  
Shell Theory ◽  
Stokes Equations ◽  
Cylindrical Shells ◽  
Axial Flow ◽  
Navier Stokes ◽  
Complex Eigenvalue ◽  
Flow Induced Vibration ◽  
Navier Stokes Equations ◽  
Flowing Fluid ◽  
Narrow Passage

The unstable phenomena of thin cylindrical shells subjected to annular axial flow are investigated. In this paper, the analytical model is composed of an elastic axisymmetric shell and a rigid one which are arranged co-axially. Considering the fluid structure interaction between shells and fluid flowing through an annular narrow passage, the coupled equation of motion is derived using Flu¨gge’s shell theory and Navier-Stokes equations. The unstable phenomena of thin cylindrical shells are clarified by using the root locus based on the complex eigenvalue analysis. The numerical parameter studies on the shells with a freely supported end and a rigid one, and with both simply supported ends, are performed taking the dimensins of shells, the characteristics of flowing fluid so on as parameters. The influence of these parameters on the threshold of instability of the coupled vibration between thin cylindrical shells and annular axial flowing fluid are investigated and discussed.

Similarity solutions for viscous vortex cores

1992 ◽  
Vol 238 ◽  
pp. 487-507 ◽  
Author(s):  
Ernst W. Mayer ◽  
Kenneth G. Powell
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Flow ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Axial Pressure Gradient ◽  
Self Similar ◽  
Similar Solutions

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

Effect of a Time-Dependent Stenosis on Flow Through a Tube

1968 ◽  
Vol 90 (2) ◽  
pp. 248-254 ◽  
Author(s):  
D. F. Young
Keyword(s):  
Stokes Equations ◽  
Flow Characteristics ◽  
Time Dependent ◽  
Arterial System ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
Dependent Growth ◽  
Abnormal Tissue

A common occurrence in the arterial system is the narrowing of arteries due to the development of atherosclerotic plaques or other types of abnormal tissue development. As these growths project into the lumen of the artery, the flow is disturbed and there develops a potential coupling between the growth and the blood flow through the artery. A discussion of the various possible consequences of this interaction is given. It is noted that very small growths leading to mild stenotic obstructions, although not altering the gross flow characteristics significantly, may be important in triggering biological mechanisms such as intimal cell proliferation or changes in vessel caliber. An analysis of the effect of an axially symmetric, time-dependent growth into the lumen of a tube of constant cross section through which a Newtonian fluid is steadily flowing is presented. This analysis is based on a simplified model in which the convective acceleration terms in the Navier-Stokes equations are neglected. Effect of growth on pressure distribution and wall shearing stress is given and possible biological implications are discussed.

Improved κ-ɛ Modelling of Impeller Flow Performance of a Mixed-Flow Pump under off-Design Operating States

1993 ◽  
Vol 207 (4) ◽  
pp. 219-229 ◽  
Author(s):  
S M Fraser ◽  
Y Zhang
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Careful Analysis ◽  
Navier Stokes ◽  
Mean Flow ◽  
Mixed Flow ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
The Mean ◽  
Flow Pump

Three-dimensional turbulent flow through the impeller passage of a model mixed-flow pump has been simulated by solving the Navier-Stokes equations with an improved κ-ɛ model. The standard κ-ɛ model was found to be unsatisfactory for solving the off-design impeller flow and a converged solution could not be obtained at 49 per cent design flowrate. After careful analysis, it was decided to modify the standard κ-ɛ model by including the extra rates of strain due to the acceleration of impeller rotation and geometrical curvature and removing the mathematical ill-posedness between the mean flow turbulence modelling and the logarithmic wall function.

Measurement and Observation of Jet Thrust for Water Flow Through Micro-Orifice

2012 ◽  
Vol 134 (8) ◽  
Author(s):  
Akiomi Ushida ◽  
Tomiichi Hasegawa ◽  
Takehiro Hoshina ◽  
Shouta Kudou ◽  
Hiroshige Uchiyama ◽  
...  
Keyword(s):  
Elastic Stress ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Biological Applications ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Numerical Predictions ◽  
Orifice Flow ◽  
Flow Through ◽  
The Many

Owing to the many potential industrial and biological applications of microfluid mechanics, it has recently become an attractive research topic. However, researchers have mainly concentrated on microchannel flows and studies investigating micro-orifice flows are rare cases. In the present study, the results from experiments conducted on flows through micro-orifices with diameters of 100 μm, 50 μm, and 25 μm are presented. In these experiments, the thrust and diameter of observed outflow jets are measured. The resultant thrust and diameter of the jets for the 100 μm orifice flow agree with the numerical predictions obtained via the Navier–Stokes equations. Conversely, for an orifice with a diameter of 50 μm or less, it is found that the thrust is lower than that predicted and the existence of jet swell becomes apparent. With the estimated elastic stress proportional to squared mean velocity, a change in the elasticity of the water as it flows through a micro-orifice is strongly suggested.

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.

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