Impacts of Uniform Magnetic Field and Internal Heated Vertical Plate on Ferrofluid Free Convection and Entropy Generation in a Square Chamber
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Lecture 18 Ocean General Circulation Modeling
Lecture 18 Ocean General Circulation Modeling 9.1 The equations of motion: Navier-Stokes The governing equations for a real fluid are the Navier-Stokes equations (con servation of linear momentum and mass mass) along with conservation of salt, conservation of heat (the first law of thermodynamics) and an equation of state. However, these equations support fast acoustic modes and involve nonlinearities in many terms that makes solving them both difficult and ex pensive and particularly ill suited for long time scale calculations. Instead we make a series of approximations to simplify the Navier-Stokes equations to yield the “primitive equations” which are the basis of most general circu lations models. In a rotating frame of reference and in the absence of sources and sinks of mass or salt the Navier-Stokes equations are @ �~v + �~v~v + 2�~ �~v + g�kˆ + p = ~ρ (9.1) t r · ^ r r · @ � + �~v = 0 (9.2) t r · @ �S + �S~v = 0 (9.3) t r · 1 @t �ζ + �ζ~v = ω (9.4) r · cpS r · F � = �(ζ; S; p) (9.5) Where � is the fluid density, ~v is the velocity, p is the pressure, S is the salinity and ζ is the potential temperature which add up to seven dependent variables. 115 12.950 Atmospheric and Oceanic Modeling, Spring '04 116 The constants are �~ the rotation vector of the sphere, g the gravitational acceleration and cp the specific heat capacity at constant pressure. ~ρ is the stress tensor and ω are non-advective heat fluxes (such as heat exchange across the sea-surface).F 9.2 Acoustic modes Notice that there is no prognostic equation for pressure, p, but there are two equations for density, �; one prognostic and one diagnostic. -
1.2 Porous Media Flow
Research Collection Doctoral Thesis Experimental and numerical investigation of porous media flow with regard to the emulsion process Author(s): Benedikt Hövekamp, Tobias Publication Date: 2002 Permanent Link: https://doi.org/10.3929/ethz-a-004511272 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 14836 Experimental and Numerical Investigation of Porous Media Flow with regard to the Emulsion Process A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH¨ for the degree of Doctor of Technical Sciences presented by Tobias Benedikt Hov¨ ekamp Dipl.-Ing. born June 18, 1970 citizen of Germany accepted on the recommendation of Prof. Dr.-Ing. E. Windhab, examiner Prof. Dr. K. Feigl, co-examiner. 2002 c 2002 Tobias Hov¨ ekamp Laboratory of Food Process Engineering (ETH Zurich)¨ All rights reserved. Experimental and Numerical Investigation of Porous Media Flow with regard to the Emulsion Process ISBN: 3-905609-17-7 LMVT Volume: 16 Published and distributed by: Laboratory of Food Process Engineering Swiss Federal Institute of Technology (ETH) Zurich¨ ETH Zentrum, LFO CH-8092 Zurich Switzerland http://www.vt.ilw.agrl.ethz.ch Printed in Switzerland by: bokos druck GmbH Badenerstrasse 123a CH-8004 Zurich¨ Rien n’est plus fort qu’une idee´ dont l’heure est venue Victor Hugo To Barbara Danksagung Die vorliegende Dissertation wurde erst durch die Mithilfe und Unterstutzung¨ vieler Men- schen moglich.¨ Gerne mochte¨ ich mich an dieser Stelle bei allen bedanken, die zum Gelingen dieser Arbeit beigetragen haben. -
Entropy Generation in Mhd Flow of Viscoelastic Nanofluids with Homogeneous-Heterogeneous Reaction, Partial Slip and Nonlinear Thermal Radiation
Journal of Thermal Engineering, Vol. 6, No. 3, pp. 327-345, April, 2020 Yildiz Technical University Press, Istanbul, Turkey ENTROPY GENERATION IN MHD FLOW OF VISCOELASTIC NANOFLUIDS WITH HOMOGENEOUS-HETEROGENEOUS REACTION, PARTIAL SLIP AND NONLINEAR THERMAL RADIATION M. Almakki1, H. Mondal2*, P. Sibanda3 ABSTRACT We investigate the combined effects of homogeneous and heterogeneous reactions in the boundary layer flow of a viscoelastic nanofluid over a stretching sheet with nonlinear thermal radiation. The incompressible fluid is electrically conducting with an applied a transverse magnetic field. The conservation equations are solved using the spectral quasi-linearization method. This analysis is carried out in order to enhance the system performance, with the source of entropy generation and the impact of Bejan number on viscoelastic nanofluid due to a partial slip in homogeneous and heterogeneous reactions flow using the spectral quasi-linearization method. Various fluid parameters of interest such as entropy generation, Bejan number, fluid velocity, shear stress heat and mass transfer rates are studied quantitatively, and their behaviors are depicted graphically. A comparison of the entropy generation due to the heat transfer and the fluid friction is made with the help of the Bejan number. Among the findings reported in this study is that the entropy generation has a significant impact in controlling the rate of heat transfer in the boundary layer region. Keywords: Nanofluids, Viscoelastic Fluid, Homogeneous—Heterogeneous Reactions, Nonlinear Thermal Radiation. INTRODUCTION In recent years, nanofluids have attracted a considerable amount of interest due to their novel properties that make them potentially useful in a number of industrial applications including transportation, power generation, micro manufacturing, thermal therapy for cancer treatment, chemical and metallurgical sectors, heating, cooling, ventilation, and air-conditioning. -
Vortex Stretching in Incompressible and Compressible Fluids
Vortex stretching in incompressible and compressible fluids Esteban G. Tabak, Fluid Dynamics II, Spring 2002 1 Introduction The primitive form of the incompressible Euler equations is given by du P = ut + u · ∇u = −∇ + gz (1) dt ρ ∇ · u = 0 (2) representing conservation of momentum and mass respectively. Here u is the velocity vector, P the pressure, ρ the constant density, g the acceleration of gravity and z the vertical coordinate. In this form, the physical meaning of the equations is very clear and intuitive. An alternative formulation may be written in terms of the vorticity vector ω = ∇× u , (3) namely dω = ω + u · ∇ω = (ω · ∇)u (4) dt t where u is determined from ω nonlocally, through the solution of the elliptic system given by (2) and (3). A similar formulation applies to smooth isentropic compressible flows, if one replaces the vorticity ω in (4) by ω/ρ. This formulation is very convenient for many theoretical purposes, as well as for better understanding a variety of fluid phenomena. At an intuitive level, it reflects the fact that rotation is a fundamental element of fluid flow, as exem- plified by hurricanes, tornados and the swirling of water near a bathtub sink. Its derivation from the primitive form of the equations, however, often relies on complex vector identities, which render obscure the intuitive meaning of (4). My purpose here is to present a more intuitive derivation, which follows the tra- ditional physical wisdom of looking for integral principles first, and only then deriving their corresponding differential form. The integral principles associ- ated to (4) are conservation of mass, circulation–angular momentum (Kelvin’s theorem) and vortex filaments (Helmholtz’ theorem). -
Studies on Blood Viscosity During Extracorporeal Circulation
Nagoya ]. med. Sci. 31: 25-50, 1967. STUDIES ON BLOOD VISCOSITY DURING EXTRACORPOREAL CIRCULATION HrsASHI NAGASHIMA 1st Department of Surgery, Nagoya University School of Medicine (Director: Prof Yoshio Hashimoto) ABSTRACT Blood viscosity was studied during extracorporeal circulation by means of a cone in cone viscometer. This rotational viscometer provides sixteen kinds of shear rate, ranging from 0.05 to 250.2 sec-1 • Merits and demerits of the equipment were described comparing with the capillary viscometer. In experimental study it was well demonstrated that the whole blood viscosity showed the shear rate dependency at all hematocrit levels. The influence of the temperature change on the whole blood viscosity was clearly seen when hemato· crit was high and shear rate low. The plasma viscosity was 1.8 cp. at 37°C and Newtonianlike behavior was observed at the same temperature. ·In the hypothermic condition plasma showed shear rate dependency. Viscosity of 10% LMWD was over 2.2 fold as high as that of plasma. The increase of COz content resulted in a constant increase in whole blood viscosity and the increased whole blood viscosity after C02 insufflation, rapidly decreased returning to a little higher level than that in untreated group within 3 minutes. Clinical data were obtained from 27 patients who underwent hypothermic hemodilution perfusion. In the cyanotic group, w hole blood was much more viscous than in the non cyanotic. During bypass, hemodilution had greater influence upon whole blood viscosity than hypothermia, but it went inversely upon the plasma viscosity. The whole blood viscosity was more dependent on dilution rate than amount of diluent in mljkg. -
1 Computational Fluid Dynamics Based Approach for Predicting Heat
Computational Fluid Dynamics Based Approach for Predicting Heat Transfer and Flow Characteristics of Inline Tube Banks with Large Transverse Spacing Niko Pietari Niemelä, Antti Mikkonen, Kaj Lampio, Jukka Konttinen Materials Science and Environmental Engineering, Tampere University, Kalevantie 4, 33100 Tampere, Finland Address correspondence to Niko Pietari Niemelä, Tampere University (TAU), Kalevantie 4, 33100 Tampere, Finland, E-mail: [email protected], Phone Number: +358 40 838 1434 1 ABSTRACT Inline tube configuration is used in various heat exchangers, such as power plant superheaters. The important design parameters are the longitudinal and transverse spacing between the tubes, as they determine the flow and heat transfer characteristics of the tube bank. In superheaters, the transverse tube spacing is often relatively large in order to avoid clogging of the flow passages due to deposition of particulate matter. To design and optimize superheaters is a challenging task, especially because of the complicated vortex-shedding phenomenon. This also complicates the Computational Fluid Dynamics (CFD) modeling, because unsteady simulation approach is required. This paper discusses about various factors that affect the accuracy of prediction of heat transfer and flow characteristics of inline tube banks with large transverse spacing. A suitable CFD model is constructed by comparing different boundary conditions, domain dimensions, domain size, and turbulence models in unsteady simulations. The numerically obtained Nusselt numbers are evaluated against available heat transfer correlations. The correlation of Gnielinski is recommended for tube banks with large transverse spacing, as it agrees within ±13% with the numerically obtained values. The guidelines presented in the paper can serve as reference for future simulations of unsteady flow phenomena. -
Investigation of Air-Cooled Condensers for Waste Heat Driven Absorption Heat Pumps
INVESTIGATION OF AIR-COOLED CONDENSERS FOR WASTE HEAT DRIVEN ABSORPTION HEAT PUMPS A Dissertation Presented to The Academic Faculty by Subhrajit Chakraborty In Partial Fulfillment of the Requirements for the Degree Master of Science in Mechanical Engineering Georgia Institute of Technology May 2017 COPYRIGHT © 2017 BY SUBHRAJIT CHAKRABORTY INVESTIGATION OF AIR-COOLED CONDENSERS FOR WASTE HEAT DRIVEN ABSORPTION HEAT PUMPS Approved by: Dr. Srinivas Garimella, Advisor School of Mechanical Engineering Georgia Institute of Technology Dr. S. Mostafa Ghiaasiaan School of Mechanical Engineering Georgia Institute of Technology Dr. Sheldon M. Jeter School of Mechanical Engineering Georgia Institute of Technology Date Approved: April 25, 2017 To my mother iii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my advisor, Dr. Srinivas Garimella, for his continuous support and guidance. His direction was invaluable in research- and career-related junctures, and I look forward to applying those skills in future endeavors. I would like to thank past and present members of the Sustainable Thermal Systems Lab, particularly, Dr. Alex Rattner, Dr. Jared Delahanty, David Forinash, Dhruv Hoysall, Marcel Staedter, Dr. Darshan Pahinkar, Anurag Goyal, and Allison Mahvi for their technical guidance, willingness to review my work, and answer my questions. I would specifically like to acknowledge Victor Aiello’s help in fabrication of my components and test facility. In addition, I would like to thank Daniel Kromer, Jennifer Lin, Daniel Boman, Khoudor Keniar, Taylor Kunke, Bachir El Fil and Girish Kini for their scientific counsel and comradery. I am very thankful to my family and friends, without their help and support I would not have been able to complete this work. -
The Influence of Negative Viscosity on Wind-Driven, Barotropic Ocean
916 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 The In¯uence of Negative Viscosity on Wind-Driven, Barotropic Ocean Circulation in a Subtropical Basin DEHAI LUO AND YAN LU Department of Atmospheric and Oceanic Sciences, Ocean University of Qingdao, Key Laboratory of Marine Science and Numerical Modeling, State Oceanic Administration, Qingdao, China (Manuscript received 3 April 1998, in ®nal form 12 May 1999) ABSTRACT In this paper, a new barotropic wind-driven circulation model is proposed to explain the enhanced transport of the western boundary current and the establishment of the recirculation gyre. In this model the harmonic Laplacian viscosity including the second-order term (the negative viscosity term) in the Prandtl mixing length theory is regarded as a tentative subgrid-scale parameterization of the boundary layer near the wall. First, for the linear Munk model with weak negative viscosity its analytical solution can be obtained with the help of a perturbation expansion method. It can be shown that the negative viscosity can result in the intensi®cation of the western boundary current. Second, the fully nonlinear model is solved numerically in some parameter space. It is found that for the ®nite Reynolds numbers the negative viscosity can strengthen both the western boundary current and the recirculation gyre in the northwest corner of the basin. The drastic increase of the mean and eddy kinetic energies can be observed in this case. In addition, the analysis of the time-mean potential vorticity indicates that the negative relative vorticity advection, the negative planetary vorticity advection, and the negative viscosity term become rather important in the western boundary layer in the presence of the negative viscosity. -
Circulation, Lecture 4
4. Circulation A key integral conservation property of fluids is the circulation. For convenience, we derive Kelvin’s circulation theorem in an inertial coordinate system and later transform back to Earth coordinates. In inertial coordinates, the vector form of the momentum equation may be written dV = −α∇p − gkˆ + F, (4.1) dt where kˆ is the unit vector in the z direction and F represents the vector frictional acceleration. Now define a material curve that, however, is constrained to lie at all times on a surface along which some state variable, which we shall refer to for now as s,is a constant. (A state variable is a variable that can be expressed as a function of temperature and pressure.) The picture we have in mind is shown in Figure 3.1. 10 The arrows represent the projection of the velocity vector onto the s surface, and the curve is a material curve in the specific sense that each point on the curve moves with the vector velocity, projected onto the s surface, at that point. Now the circulation is defined as C ≡ V · dl, (4.2) where dl is an incremental length along the curve and V is the vector velocity. The integral is a closed integral around the curve. Differentiation of (4.2) with respect to this gives dC dV dl = · dl + V · . (4.3) dt dt dt Since the curve is a material curve, dl = dV, dt so the integrand of the last term in (4.3) can be written as a perfect derivative and so the term itself vanishes. -
4. Circulation and Vorticity This Chapter Is Mainly Concerned With
4. Circulation and vorticity This chapter is mainly concerned with vorticity. This particular flow property is hard to overestimate as an aid to understanding fluid dynamics, essentially because it is difficult to mod- ify. To provide some context, the chapter begins by classifying all different kinds of motion in a two-dimensional velocity field. Equations are then developed for the evolution of vorticity in three dimensions. The prize at the end of the chapter is a fluid property that is related to vorticity but is even more conservative and therefore more powerful as a theoretical tool. 4.1 Two-dimensional flows According to a theorem of Helmholtz, any two-dimensional flowV()x, y can be decom- posed as V= zˆ × ∇ψ+ ∇χ , (4.1) where the “streamfunction”ψ()x, y and “velocity potential”χ()x, y are scalar functions. The first part of the decomposition is non-divergent and the second part is irrotational. If we writeV = uxˆ + v yˆ for the flow in the horizontal plane, then 4.1 says that u = –∂ψ⁄ ∂y + ∂χ ∂⁄ x andv = ∂ψ⁄ ∂x + ∂χ⁄ ∂y . We define the vertical vorticity ζ and the divergence D as follows: ∂v ∂u 2 ζ =zˆ ⋅ ()∇ × V =------ – ------ = ∇ ψ , (4.2) ∂x ∂y ∂u ∂v 2 D =∇ ⋅ V =------ + ----- = ∇ χ . (4.3) ∂x ∂y Determination of the velocity field givenζ and D requires boundary conditions onψ andχ . Contours ofψ are called “streamlines”. The vorticity is proportional to the local angular velocity aboutzˆ . It is known entirely fromψ()x, y or the first term on the rhs of 4.1. -
Rheometry CM4650 4/27/2018
Rheometry CM4650 4/27/2018 Chapter 10: Rheometry Capillary Rheometer piston CM4650 F entrance region Polymer Rheology r A barrel z 2Rb polymer melt L 2R reservoir B exit region Q Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 1 © Faith A. Morrison, Michigan Tech U. Rheometry (Chapter 10) measurement All the comparisons we have discussed require that we somehow measure the material functions on actual fluids. 2 © Faith A. Morrison, Michigan Tech U. 1 Rheometry CM4650 4/27/2018 Rheological Measurements (Rheometry) – Chapter 10 Tactic: Divide the Problem in half Modeling Calculations Experiments Dream up models RHEOMETRY Calculate model Build experimental predictions for Standard Flows apparatuses that stresses in standard allow measurements flows in standard flows Calculate material Calculate material functions from Compare functions from model stresses measured stresses Pass judgment on models U. A. Morrison,© Tech Faith Michigan Collect models and their report 3 cards for future use Chapter 10: Rheometry Capillary Rheometer piston CM4650 F entrance region Polymer Rheology r A barrel z 2Rb polymer melt L 2R reservoir B exit region Q Professor Faith A. Morrison Department of Chemical Engineering Michigan Technological University 4 © Faith A. Morrison, Michigan Tech U. 2 Rheometry CM4650 4/27/2018 Rheological Measurements (Rheometry) – Chapter 10 Tactic: Divide the Problem in half Modeling Calculations Experiments Dream up models RHEOMETRY Calculate model Build experimental predictions for Standard Flows apparatuses that stresses in standard allow measurements in standard flows As withflows Advanced Rheology,Calculate material there is way Calculate material functions from Compare functions from too muchmodel stresseshere to cover in measured stresses the time we have. -
Circulation and Vorticity
Lecture 4: Circulation and Vorticity • Circulation •Bjerknes Circulation Theorem • Vorticity • Potential Vorticity • Conservation of Potential Vorticity ESS227 Prof. Jin-Yi Yu Measurement of Rotation • Circulation and vorticity are the two primary measures of rotation in a fluid. • Circul ati on, whi ch i s a scal ar i nt egral quantit y, i s a macroscopic measure of rotation for a finite area of the fluid. • Vorticity, however, is a vector field that gives a microscopic measure of the rotation at any point in the fluid. ESS227 Prof. Jin-Yi Yu Circulation • The circulation, C, about a closed contour in a fluid is defined as thliihe line integral eval uated dl along th e contour of fh the component of the velocity vector that is locally tangent to the contour. C > 0 Î Counterclockwise C < 0 Î Clockwise ESS227 Prof. Jin-Yi Yu Example • That circulation is a measure of rotation is demonstrated readily by considering a circular ring of fluid of radius R in solid-body rotation at angular velocity Ω about the z axis . • In this case, U = Ω × R, where R is the distance from the axis of rotation to the ring of fluid. Thus the circulation about the ring is given by: • In this case the circulation is just 2π times the angular momentum of the fluid ring about the axis of rotation. Alternatively, note that C/(πR2) = 2Ω so that the circulation divided by the area enclosed by the loop is just twice thlhe angular spee dfifhid of rotation of the ring. • Unlike angular momentum or angular velocity, circulation can be computed without reference to an axis of rotation; it can thus be used to characterize fluid rotation in situations where “angular velocity” is not defined easily.