An Asymptotic Solution for Laminar Flow of an Incompressible Fluid Between Rotating Disks

1968 ◽  
Vol 35 (1) ◽  
pp. 155-159 ◽  
Author(s):  
L. Matsch ◽  
W. Rice
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Radial Velocity ◽  
Velocity Component ◽  
Tangential Velocity ◽  
Outer Radius ◽  
Creeping Flow ◽  
Arbitrary Distribution ◽  
Rotating Disks ◽  
Inner Radius

Laminar flow is considered between parallel rotating disks having a circular exhaust hole at an inner radius and supplied with fluid at the outer radius with pressure higher than the available sink pressure. The problem statement for asymptotic (fully developed) flow is formulated. A procedure for perturbing a creeping flow solution and an iteration scheme are developed to produce a solution for higher Reynolds numbers. The solution depends on two parameters, a Reynolds number and a mass flow parameter, and is asymptotic in the sense that a third parameter would be necessary for a solution with an arbitrary tangential velocity component specified at the outer radius of the disks and/or an arbitrary distribution of the radial velocity component between the disks. From computations conducted by digital computer, the region having uninflected radial velocity profiles is delineated. Typical results are presented for the velocity components as functions of Reynolds number, the average radial component of velocity at the entrance, and the inner radius of the disks.

Theoretical Investigation on Inflow Between Two Rotating Disks

2017 ◽  
Vol 139 (11) ◽  
Author(s):  
Achhaibar Singh
Keyword(s):  
Reynolds Number ◽  
Radial Velocity ◽  
Tangential Velocity ◽  
Tangential Direction ◽  
Speed Ratio ◽  
Rotating Disks ◽  
Practical Applications ◽  
Parabolic Profile ◽  
Pressure Decrease ◽  
Radial Convection

Mathematical relations are obtained for velocities and pressure distribution for a fluid entering the peripheral clearance of a pair of rotating concentric disks that converges and discharges through an opening at the center. Both, the flows in the gap of corotating disks and in the gap of contrarotating disks can be predicted using the present analytical solutions. The prediction of instability of radial velocity for corotating disks at the speed ratio of unity is very important for practical applications. The radial velocity profile is similar to a parabolic profile exactly at speed ratio of unity. The profile drastically changes with the small difference of ±1% in the disks’ rotation. The radial convection was observed in the tangential velocity at a low radius. Centrifugal force caused by disk rotation highly influences the flow resulting in backflow on the disks. The pressure consists of friction losses and convective inertia. Therefore, the pressure decrease is high for increased speed ratio, throughflow Reynolds number, and rotational Reynolds number. The pressure decrease for the flow between contrarotating disks is lesser than that for the flow between corotating disks due to decreased viscous losses in the tangential direction. This study provides valuable guidance for the design of devices where disks are rotated independently by highlighting the instabilities in the radial velocity at the speed ratio of unity.

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.

scholarly journals Transient Thermoelastic Analysis of Pressurized Rotating Disks Subjected to Arbitrary Boundary and Initial Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Mohammad Zamani Nejad ◽  
Azam Afshin
Keyword(s):  
Thermal Stresses ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Separation Of Variables ◽  
Outer Radius ◽  
Rotating Disks ◽  
Arbitrary Boundary ◽  
Inner Radius ◽  
Transient Thermal

This paper focuses on exact analytical solution of transient thermoelastic behaviors of rotating pressurized disks subjected to arbitrary boundary and initial conditions. The pressure, inner radius, and outer radius are considered constant. The basic thermoelasticity theory under generalized assumptions is used to solve the thermoelastic problem. Using the method of the separation of variables, the relations of temperature and transient thermal stresses in the radial direction are obtained. In the case study, the disk is considered under heat flux. Some useful discussions and numerical examples are presented. The analytical results were compared with those of the finite element method and good agreement was found. The relations obtained in this paper can be applied to any arbitrary boundary and initial conditions.

Approximate Calculation of Pressure Drop in Laminar Flow of Generalized Newtonian Fluid Through Channels with Insert

1995 ◽  
Vol 60 (9) ◽  
pp. 1476-1491
Author(s):  
Václav Dolejš ◽  
Petr Doleček ◽  
Ivan Machač ◽  
Bedřich Šiška
Keyword(s):  
Reynolds Number ◽  
Pressure Drop ◽  
Laminar Flow ◽  
Numerical Solution ◽  
Newtonian Fluid ◽  
Calculation Method ◽  
Approximate Calculation ◽  
Creeping Flow ◽  
Experimental Results ◽  
Generalized Newtonian Fluid

An equation of Rabinowitsch-Mooney type has been suggested for approximate calculation of pressure drop in flow of generalized Newtonian fluid through channels with insert both in the region of creeping flow and at higher values of the Reynolds number, and this calculation method has been verified for four types of insert using own numerical solution and experimental results as well as literature data.

Laminar Inward Flow of an Incompressible Fluid Between Rotating Disks, With Full Peripheral Admission

1968 ◽  
Vol 35 (2) ◽  
pp. 229-237 ◽  
Author(s):  
K. E. Boyd ◽  
W. Rice
Keyword(s):  
Reynolds Number ◽  
Flow Rate ◽  
Stokes Equations ◽  
Complete Solution ◽  
Applied Pressure ◽  
Tangential Velocity ◽  
Tangential Component ◽  
Rotating Disks ◽  
Outer Periphery ◽  
Inlet Conditions

The laminar flow of an incompressible Newtonian fluid, radially inward between parallel co-rotating disks is considered. The through-flow is supported by an externally applied pressure difference between the outer periphery and a circular fluid exhaust hole at an inner radius. The fluid supplied at the outer periphery is considered with arbitrary velocity components, such that the tangential component may be greater or less than the disk peripheral velocity. A sufficiently complete problem statement is formulated from the Navier-Stokes’ equations. The problem has three parameters: a Reynolds number, a flow-rate parameter, and a peripheral tangential velocity component parameter. A numerical method of solution is detailed and typical numerical results are given illustrating the phenomena that occur in the inlet region for various inlet conditions. It is shown that the solution becomes the asymptotic solution given by previous investigators at interior radii following the inlet. Correspondence between the complete solution given herein and the earlier asymptotic solutions is established as dependent on corresponding values of Reynolds number and flow rate only. The results are discussed from the point of view of application of the solution in the development of multiple-disk turbines.

Local Heat Transfer in Enclosed Co-rotating Disks With Axial Throughflow

1994 ◽  
Vol 116 (1) ◽  
pp. 66-72 ◽  
Author(s):  
S. Y. Kim ◽  
J. C. Han ◽  
G. L. Morrison ◽  
E. Elovic
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Axial Flow ◽  
Local Heat Transfer ◽  
Disk Surface ◽  
Local Heat ◽  
Outer Radius ◽  
Rotating Disks ◽  
Transfer Coefficients ◽  
Heat Transfer Coefficients

Local heat transfer in enclosed co-rotating disks with axial through flow is investigated. The rotating cavity has two plane disks and a cylindrical rim (shroud). The ratio of the rim span to the disk outer radius is 0.4 and the ratio of the disk inner radius to outer radius is 0.25. The objectives of this study are to investigate the effects of axial coolant flow rate, rotation speed, and disk surface temperature on the local heat transfer coefficients inside the disk cavity. Both uniform disk surface heat flux and uniform disk surface temperatures are tested for axial flow Reynolds numbers between 2500 and 25,000 rotational Reynolds numbers between 0 and 5.11 × 105, and rotational Grashof numbers between 5 × 106 and 1.3 × 1010. The results show that the local heat transfer coefficients for the nonrotating cavity increase with increasing axial flow Reynolds number. In general, the local Nusselt numbers at large radii of the disks and rim increase with increasing rotational Reynolds number. However, the local Nusselt numbers at small radii of the disks initially decrease and then increase with increasing rotational Reynolds number. The uniform heat flux condition provides slightly higher heat transfer coefficients than those for the uniform wall temperature condition.

The Developing Laminar Flow and Pressure Drop in the Entrance Region of Annular Ducts

1964 ◽  
Vol 86 (4) ◽  
pp. 827-833 ◽  
Author(s):  
E. M. Sparrow ◽  
S. H. Lin
Keyword(s):  
Pressure Drop ◽  
Laminar Flow ◽  
Circular Tube ◽  
Outer Radius ◽  
Radius Ratio ◽  
Entrance Region ◽  
Flow Development ◽  
Inner Radius ◽  
Wide Range ◽  
Plate Channel

A new analytical method has been applied for determining the developing laminar flow in the hydrodynamic entrance region of annular ducts. Detailed results are presented for the development of the velocity distribution and the pressure drop over a wide range of annulus radius ratios r1/r2 (r1 = inner radius of annulus, r2 = outer radius of annulus). It is found that the pressure drop and flow development in annular ducts with radius ratios substantially less than unity is quite similar to that in a parallel-plate channel (r1/r2 → 1). On the other hand, the results far an annular duct with radius ratio as small as 0.001 depart significantly from those for a circular tube (r1/r2 = 0). The hydrodynamic entrance length, measured as a multiple of the hydraulic radius, increases as the duct radius ratio decreases at a fixed Reynolds number.

Steady Viscous Flow Between Two Porous Disks With Stretching Motion

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Tiegang Fang ◽  
Xin He
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Reynolds Number ◽  
Radial Velocity ◽  
Large Mass ◽  
Creeping Flow ◽  
Similarity Equation ◽  
Core Flow ◽  
Mass Injection ◽  
Radial Profiles

In this work, an exact solution to the steady-state Navier–Stokes (NS) equations is presented for viscous flows between two stretchable disks with mass transpiration effects. The governing momentum equations were converted into an ordinary differential equation by a similarity transformation technique. The similarity equation was solved numerically and the effects of Reynolds number and the mass transpiration parameter were investigated. At very low Reynolds numbers (i.e., R → 0), a creeping flow was observed with a parabolic radial velocity profile and a cubic function profile for the vertical velocity. With the increase of the Reynolds number, the flow shows a boundary layer behavior near the wall with a constant velocity core flow in the centerline region between the two disks for mass suction or lower mass injection. The effects of the mass transpiration on the flow are quite different and interesting. With strong suction, the radial profiles also show boundary layer type characteristics with a core flow. But for large mass injection, the radial velocity approaches to a linear profile under higher Reynolds number. These results are a rare case of an exact solution to the NS equations and are useful as a benchmark problem for the validation of three-dimensional (3D) numerical computation code.

Entropy generation and heat transfer analysis for ferrofluid flow between two rotating disks with variable conductivity

2021 ◽  
pp. 095440622199118
Author(s):  
Anupam Bhandari
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Differential Equations ◽  
Entropy Generation ◽  
Heat Transfer Analysis ◽  
Entropy Generation Rate ◽  
Geometrical Parameters ◽  
Rotating Disks ◽  
Coupled Differential Equations ◽  
Lower Disk

Present model analyze the flow and heat transfer of water-based carbon nanotubes (CNTs) [Formula: see text] ferrofluid flow between two radially stretchable rotating disks in the presence of a uniform magnetic field. A study for entropy generation analysis is carried out to measure the irreversibility of the system. Using similarity transformation, the governing equations in the model are transformed into a set of nonlinear coupled differential equations in non-dimensional form. The nonlinear coupled differential equations are solved numerically through the finite element method. Variable viscosity, variable thermal conductivity, thermal radiation, and volume concentration have a crucial role in heat transfer enhancement. The results for the entropy generation rate, velocity distributions, and temperature distribution are graphically presented in the presence of physical and geometrical parameters of the flow. Increasing the values of ferromagnetic interaction number, Reynolds number, and temperature-dependent viscosity enhances the skin friction coefficients on the surface and wall of the lower disk. The local heat transfer rate near the lower disk is reduced in the presence of Harman number, Reynolds number, and Prandtl number. The ferrohydrodynamic flow between two rotating disks might be useful to optimize the use of hybrid nanofluid for liquid seals in rotating machinery.

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