DOCSLIB.ORG

Electrorheological Properties of Polypyrrole and Its Composite ER Fluids

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Electrorheological Properties of Polypyrrole and Its Composite ER Fluids J. Ind. Eng. Chem., Vol. 13, No. 6, (2007) 879-894 REVIEW Electrorheological Properties of Polypyrrole and its Composite ER Fluids † Do-Heyoung Kim and Young Dae Kim Faculty of Applied Chemical Engineering, Chonnam National University, Kwangju 500-757, Korea Received October 31, 2007 Abstract: Electrorheological (ER) fluids are suspensions of polarizable nonconducting or semiconducting par- ticles in a nonconducting continuous phase of low relative polarizability. In the absence of an electric field, they have the properties of suspensions of neutral solid particles. Upon the application of an electric field, an organized structure of particles is formed and the ER fluids exhibit a remarkable change in rheological prop- erties, including a drastic increase in apparent viscosity as well as yield stress. Various mechanisms have been proposed to explain the ER behavior to understand the ER behaviors and design effective ER fluids. Polypyrrole (PPy) is one of the most promising semiconducting polymers because it has higher conductivity and environmental stability than many other semiconducting polymers. PPy and its composites have been ex- tensively used as ER materials and their ER fluids showed promising ER responses. ER properties of PPy based ER fluids (PPy, PPy copolymer, PPy coated particles, and PPy nanocomposites, etc.) and the ER be- haviors of PPy based ER fluids such as shear, yield, and transient stress behavior and additive effect are reviewed. Keywords: electrorheological fluid, electrorheology, semiconducting polymer, polypyrrole, yield stress, semi- conducting polymer composite Introduction The main limitation of ER technology development is a 1) lack of effective ER fluids [11]. Electrorheological (ER) response is defined as the dra- During the past decade there has been an increasing matic change in rheological properties of a suspension of amount of interest in designing effective ER fluids: theo- small particles due to the application of a large electric retically and synthetically. The general requirements of field transverse to the direction of flow. ER fluids are an effective ER fluid are the followings [11,13]: 1) there typically composed of nonconducting or semiconducting should be a marked rheological properties change on the particles dispersed in a nonconducting continuous phase. application of an electric field, 2) the off-field viscosity A large ER effect was first reported by Winslow in 1949 of the ER fluid should be low, 3) the current flow should [1], and has been reviewed in several publications [2-12]. be zero or low to minimize power loss as well as heating The simplicity of engineering designs based on ER mate- effects, 4) there should be a broad operating temperature rials has facilitated the development of specifications for range (hence anhydrous ER fluids have an advantage), 5) a broad range of devices, such as dampers, clutches, and there should be tunability of the particle properties to adaptive structures [12]. Although many ER devices control the ER properties as well as the suspension sta- have been brought successfully to the prototype stage, bility properties, and 6) there should be a strong ER ef- and despite much industrial activity, the anticipated com- fect in both dc and ac fields. mercialization of these devices has yet to be realized. The continuous phase of an ER fluid is usually a non- conducting liquid phase such as insulating oils. In some cases the continuous phase properties strongly affect the † To whom all correspondence should be addressed. ER response [14-16]. Useful continuous phases generally (e-mail: [email protected]) 880 Do-Heyoung Kim and Young Dae Kim have as many of the following properties as possible: 1) pyrrole [38,40] and semiconducting polymer composites high boiling point and low freezing point (in other words, [41-44] have been studied as high-performance anhydrous it should have a wide working temperature range), 2) low ER materials, and they showed superior physical properties, viscosity to keep the viscosity of the ER fluid at a low such as high polarizability and environmental stability. level at zero electric field, 3) high electrical resistance ICP constitute a class of polymers with particular inter- and high dielectric breakdown strength, 4) chemical and est owing to their physical and chemical properties. PPy thermal stability to prevent degradation on storage and is one of the most promising ICP because it has higher service, 5) a high density (particle sedimentation might conductivity and environmental stability in the con- not occur until the densities of both the liquid and the ductive state than many other semiconducting polymers solid match each other), 6) hydrophobicity and low mois- and hence PPy and its composites are extensively used as ture absorbability from the environment, and 7) low tox- ER materials. To design effective ER fluids by employ- icity and low cost [17,18]. ing PPy or PPy derivatives, many research groups fo- Various mechanisms have been proposed to explain the cused on the preparation of semiconducting PPy-based ER response. The inter-electrode circulation proposes composite materials. Heterogeneous semiconducting pol- that the inter-electrode circulation of particles between ymer composites, especially for semiconducting polymer the electrodes, due to the particle charge change by elec- coated organic or inorganic composites and semicon- trochemical reactions at the electrode surface, lead to the ducting polymer-organic or semiconducting polymer-in- ER response [7,19]. The electro-osmosis suggests that organic nanocomposites, have drawn the attention over the ER response arise from the formation of a water last few years, giving rise to a host of various composites bridge between the particles [20]. The surfactant bridge and nanocomposites with interesting physical properties model proposes that surfactants enhance the ER response and important application potential [10,45,46]. at low surfactant concentration by the increased surface In this paper, ER properties of PPy based ER fluids polarization and then lead to the nonlinear ER behavior (PPy, PPy copolymer, PPy coated particles, and PPy due to the increased conduction through the surfactant nanocomposites, etc.) are reviewed. Also, the ER behav- bridge formed between the particles [21,22]. The electric iors of PPy based ER fluids such as shear, yield, and tran- double layer proposes that the origin of the ER response sient stress behavior and additive effectives are reviewed. is the overlap of electric double layers [23,24]. The elec- trostatic polarization model explains that the ER re- sponse arises from the electrostatic interactions between ER Mechanisms the particles due to the field induced polarization of the particles [25-33]. A conduction model proposes that the There are many diverse applications of the ER response. ER effect is determined by the conductivity mismatch Although many ER devices have been brought success- between the particle and liquid phase [34,35]. Among fully to the prototype stage, there are currently no com- these mechanisms, the electrostatic polarization model mercially available devices. The main limitation of ER and conduction model seem to be the suitable ex- technology development is a lack of effective fluids planations for the ER behaviors of semiconducting poly- [11,47]. Of primary importance is the development of mer based ER fluids including PPy based ER fluids. suspensions that can perform desired rheological tasks, Activators are often used to activate suspensions. Some for sufficient duration, with minimum power consump- suspensions display little or no ER activity unless small tion, and with acceptable interactions with environment. amount of water or surfactant is added, while other sus- Solutions to these problems require development of new pensions exhibit a significantly enhanced ER response ER fluids and devices, which in turn require under- with activator present [2,36,37]. Enhancing ER activity standing the mechanisms controlling ER activity. Various with activators such as water severely limits the allow- models or mechanisms were proposed previously to ex- able temperature range of operation, promotes corrosion, plain the observed ER phenomena. Although the electro- and increases power consumption. Therefore, it is neces- static polarization mechanism and conduction model ap- sary to design ER fluids which show a high ER response pear to explain most experimental observations of semi- without the limitations imposed by introducing water conducting polymer based ER fluids, other phenomena based activators. would also influence the ER behaviors of semiconduct- To overcome the limitations (thermal stability and corro- ing polymer based ER fluids under some conditions. sion) of water based systems, dry based systems have been investigated with anhydrous particles. Inherent semi- Inter-Electrode Circulation Model conducting polymers (ICP) are most promising ER materi- Inter-electrode circulation model is based on inter-elec- als among various anhydrous materials. Among them, sem- trode circulation of particles [7,19]. Particles in ER fluids iconducting polymers including polyaniline [38,39], poly- often bear a net charge, and therefore move rapidly to- Electrorheological Properties of Polypyrrole and its Composite ER Fluids 881 ward the oppositely charged electrode in a strong electric The water migrates to the particle surface, forming a wa- field. Once at the electrode, ions within the particle pores
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Modeling the General Circulation of the Atmosphere. Topic 1: a Hierarchy of Models
    Modeling the General Circulation of the Atmosphere. Topic 1: A Hierarchy of Models DARGAN M. W. FRIERSON UNIVERSITY OF WASHINGTON, DEPARTMENT OF ATMOSPHERIC SCIENCES TOPIC 1: 1-7-16 Modeling the General Circulation In this class… We’ll study: ¡ The “general circulation” of the atmosphere ¡ Large scale features of the climate Questions like… What determines the precipitation distribution on Earth? GPCP Climatology (1979-2006) Questions like… What determines the precipitation distribution on Earth? GPCP (1979-2006) Questions like… What determines the north-south temperature distribution on Earth? Questions like… What determines the vertical temperature structure on Earth? 350 200 340 ] 330 mb 400 [ 320 310 290 600 290 Pressure 270 280 800 260 280 270 300 300 1000 90˚S 60˚S 30˚S 0˚ 30˚N 60˚N 90˚N Latitude Dry static energy from NCEP reanalysis Questions like… What determines the vertical temperature structure on Earth? 310 320 300 310 400 290 ] 330 280 300 mb 320 [ 600 270 Pressure 290 800 260 280 330 270 1000 90˚S 60˚S 30˚S 0˚ 30˚N 60˚N 90˚N Latitude Moist static energy from NCEP reanalysis Questions like… What determines the location/intensity of the jet streams? -10 -5 0 0 200 ] 25 20 5 mb 400 0 [ 20 15 5 600 10 Pressure 15 5 10 800 1000 90˚S 60˚S 30˚S 0˚ 30˚N 60˚N 90˚N Latitude Zonally averaged zonal winds from NCEP reanalysis Questions like… What determines the intensity of eddies? Questions like… How will these change with global warming? Rainy regions get rainier, dry regions get drier From Lu et al 2006 First: General Circulation
    [Show full text]
  • 6 Fundamental Theorems: Vorticity and Circulation
    EOSC 512 2019 6 Fundamental Theorems: Vorticity and Circulation In GFD, and especially the study of the large-scale motions of the atmosphere and ocean, we are particularly interested in the rotation of the fluid. As a consequence, again assuming the motions are of large enough scale to feel the e↵ects of (in particular, the di↵erential) rotation of the outer shell of rotating, spherical planets: The e↵ects of rotation play a central role in the general dynamics of the fluid flow. This means that vorticity (rotation or spin of fluid elements) and circulation (related to a conserved quantity...) play an important role in governing the behaviour of large-scale atmospheric and oceanic motions. This can give us important insight into fluid behaviour that is deeper than what is derived from solving the equations of motion (which is challenging enough in the first place). In this section, we will develop two such theorems and principles related to the conservation of vorticity and circulation that are particularly useful in gaining insight into ocean and atmospheric flows. 6.1 Review: What is vorticity? Vorticity was previously defined as: @uk !i = ✏ijk (6.1) @xj Or, in vector notation: ~! = ~u r⇥ ˆi ˆj kˆ @ @ @ = (6.2) @x @y @z uvw @w @v @u @w @v @u = ˆi + ˆj + ˆi @y − @z @z − @x @x − @y ✓ ◆ ✓ ◆ ✓ ◆ Physically, the vorticity is two times the local rate of rotation (or “spin”) of a fluid element. Here, it is important to distinguish between circular motion and the rotation of the element. Here, the fluid element moving from A to B on the circular path has no vorticity, while the fluid element moving from C to D has non-zero vorticity.
    [Show full text]
  • APPH 4200 Physics of Fluids Review (Ch
    APPH 4200 Physics of Fluids Review (Ch. 3) & Fluid Equations of Motion (Ch. 4) 1. Chapter 3 (more notes/examples) 2. Vorticity and Circulation 3. Navier-Stokes Equation Summary: Cauchy-Stokes Decomposition Velocity Gradient Tensor Kinematics Exercise 77 terms of the streamfunction. Differentiating this with respect to x, and following a similar rule, we obtain nctions of rand e, the chain a21/ = cose~ (cose a 1/ _ sine a1/J _ sine ~ (cose a1/ _ sine a1/J. ax2 ar ar r ae r ae ar r ae (3.42) t (:~)e' In a similar manner, -=sma21/ . e a- (. smea1/ -+--cose a1/J +--cose a sme-+--(. a1/ cose a1/J . se + u sine. (3.41) ay2 ar ar r ae r ae -ar r ae (3.43) quation (3.40) implies a1//ar The addition of equations (3.41) and (3.42) leads to Therefore, the polar velocity a21/ a21/ i a (a1/) i a21/ ax2 + ay2 = -; ar ra; + r2 ae2 = 0, which completes the transformation.Problem 3.1 Exercises t the derivative of 1/ gives the 1. A two-dimensional steady flow has velocity components he direction of differentiation. irdinates. The chain rule gives u =y v =x. sin e a1/ --- Show that the streamines are rectangular hyperbolas r ae x2 _ y2 = const. Sketch the flow pattern, and convince yourself that it represents an irrotational flow in a 90° comer. , 2. Consider a steady axisymmetrc flow of a compressibla fluid. The equation ,/ of continuity in cylindrcal coordinatesa a (R, cp, x) is -(pRUR)aR ax+ -(pRux) = O. Show how we can define" a streamfnction so that the equation of contiuity is satisfied automatically.
    [Show full text]
  • Circulation, Vorticity, and Potential Vorticity
    Chapter 4 Circulation, Vorticity, and Potential Vorticity 4.1. What is the circulation about a square of 1000 km on a side for an easterly (that is, westward flowing) wind that decreases 1 in magnitude toward the north at a rate of 10 m s− per 500 km? What is the mean relative vorticity in the square? 1 Solution: Vorticity is: ζ @v @u 0 10 m s− 2 10 5 s 1 D @x − @y D − 5 105 m D − × − − RR 5 1× 12 2 7 2 1 But, C ζdA ζmA 2 10 s 10 m 2 10 m s . D A D D − × − − D − × − 4.2. A cylindrical column of air at 30◦N with radius 100 km expands to twice its original radius. If the air is initially at rest, what is the mean tangential velocity at the perimeter after expansion? Solution: From the circulation theorem C .2 sin φ/ A Constant. Thus, Cfinal 2 sin φ .Ainitial Afinal/ Cinitial. 2 2 C D D 2 − C But Ainitial πri , and Afinal πrf . But rf 2ri. Therefore, Cfinal 2 sin φ 3πri . Now V Cfinal 2πrf , so D D 1 D D − D V 2 sin φ . 3ri=4/ 5:5 m s (anticyclonic). D − D − − 4.3. An air parcel at 30 N moves northward, conserving absolute vorticity. If its initial relative vorticity is 5 10 5 s 1, what is ◦ × − − its relative vorticity upon reaching 90◦N? Solution: Absolute vorticity is conserved for the column: ζ f Constant. Thus, ζfinal ζinitial . finitial ffinal/. Hence, 5 C D 5 1 D C − ζfinal 2 sin .π=2/ 5 10 2 sin .π=6/ 2:3 10 s .
    [Show full text]