Technology & Engineering
Navier Stokes Cylindrical
The Navier-Stokes equations in cylindrical coordinates are a set of partial differential equations that describe the motion of fluid substances. They are used to analyze fluid flow in cylindrical geometries, such as pipes and tubes. These equations take into account the conservation of mass, momentum, and energy, and are fundamental in the study of fluid dynamics.
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8 Key excerpts on "Navier Stokes Cylindrical"
- Kansari Haldar(Author)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
5 Navier–Stokes Equations in Cylindrical Polar Coordinates 5.1 Introduction Fundamental problems regarding the flow of fluid in a circular tube frequently exist in physics, chemistry, biology, medicine, and engineering. These prob-lems are nonlinear in character and can be analyzed by means of momentum balance. Their most important feature is that they can be described by the Navier–Stokes equations in cylindrical polar coordinates. When the fluid enters a circular tube from an external device, the velocity profile at the entry of the tube is flat. Immediately after the entry, the ve-locity profile adjacent to the wall is affected by the friction exerted by the surface of the tube, on the flowing fluid. As the fluid moves further along the tube, this flat portion gradually diminishes, and it ultimately disappears at a point where the flow is fully developed with an asymptotic parabolic veloc-ity profile. The distance between the entry of the tube and this point is called entry length . To summarize, one can say that the flow develops and assumes a parabolic velocity profile beyond this entry length. Theoretically, the length of the circular pipe should be infinite when a steady laminar flow with constant fluid density is considered. Their measure is to avoid the end effects. 5.2 Equations of Motion Consider an axially symmetrical flow of viscous incompressible fluid. To study this type of flow, we used the equations of motion in cylindrical co-ordinates ( r, θ , z ), where the z -axis is taken as the axis of symmetry. Let u, v , and w be the components of velocity, where u is the radial velocity component perpendicular to the symmetrical axis, v is the rotational component, and w is the velocity component parallel to the z -axis. For the axisymmetric case, the value of v is taken to be zero i.e., v = 0, and the motion is considered to be two-dimensional, so that u, w , and the pressure p are independent of θ. 85
eBook - PDF- Wm.H. Corcoran(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Many solutions of the equations under this restriction are discussed in texts on hydrodynamics (3, 6) and aerodynamics. Such problems are, however, usually of little importance in chemical engineering. V-IO. Navier-Stoke s Equation s in Cylindrica l Coordinate s In distinction to the situation which exists for ordinary differential equations, the task of bringing the solution of a partial differential equation into agreement with the boundary conditions is usually of the same order of 1 See Appendix II for vector form of Navier-Stokes equations and generalized •coordinate transformations. 1 0 . CYLINDRICAL COORDINATES difficulty as that of solving the equation originally. Consequently any simplification in the boundary conditions is of great convenience. For example, in case the boundary to the flow is a cylinder, cylindrical instead of Cartesian coordinates can be used, and the boundary conditions are satisfied at r = r 0 (V.90) instead of at V* 2 + y 2 = r Q (V.91) where r 0 is the radius of the cylinder. This apparently slight simplification is often of great practical importance; and so the Navier-Stokes equations together with the accelerations and the angular velocities will be given in cylindrical coordinates: du, _ dP 1 d η Idr u, ΒΗ φ dru x Η 4[£(5 (' ) + ' (' ) < / Λ I BP 1 d [η ldru r Βΐ4 φ Bru x ι Γ a / a« 0 ι a / a«* , a / awA] / Χ Γ Λ ο ν _ _ 9P 1 d [ η (dr u, 9ϋφ dr u x Y Ρ Ί θ = Ρ + Jl^TdT + Βφ ~dx) (V.94) , ι Γ d I du x ι a / 9w, , a / where a^, a^ r w 0 a^ r a« r % 2 / Λ Γ Λ Κ . -jz-= + w r — Η τ-Γ + , (V.95) <20 a0 dr r Βψ dx r Ίθ - Βθ + W f ar + r Βψ + Ux dx + r ' du x du x du x ΗφΒΗχ du x 1θ = Ί θ + Μ ^ + Τ 3 φ + Μ ^ ( ν · 9 7) 143 144 V. G E N E R A L EQUATIONS O F F L U I D MOTION and , 0)φ CO, 1 / 1 du x duA ' = ~2 7 ~Βφ ~ ~d%) y 1 ldu r du x 2dx~~ ~dr) ' 1 1 ιΒτΗφ du r 2 7 ~d7~ ~ YfiJ (V.98) (V.99) (V.100) V-ll .
eBook - PDF- P.G. Ciarlet, CIARLET(Authors)
- 2003(Publication Date)
- Elsevier Science(Publisher)
C HAPTER I The Navier–Stokes Equations for Incompressible Viscous Fluids Introduction: Synopsis The Navier–Stokes equations have been known for more than a century and they still provide the most commonly used mathematical model to describe and study the motion of viscous fluids, including phenomena as complicated as turbulent flow . One can only marvel at the fact that these equations accurately describe phenomena whose length scales (resp. time scale) range from fractions of a millimeter (resp. of a second) to thousands of kilometers (resp. several years). Indeed, the Navier–Stokes equations have been validated by numerous comparisons between analytical or computational results and experimental measurements; some of these comparisons are reported in Chapter IX (see also C ANUTO , H USSAINI , Q UARTERONI and Z ANG [1988, p. 29], L ESIEUR [1990], G UYON , H ULIN and P ETIT [1991]). The content of this chapter is as follows: • In Section 1 we shall briefly discuss the derivation of the Navier–Stokes equations . • In Section 2 we shall address the important issue of the boundary conditions . • In Section 3 we shall discuss the stream function-vorticity formulation of the Navier–Stokes equations. • In Section 4 we shall introduce functional spaces (of the Sobolev type) and use them to derive variational formulations of the Navier–Stokes equations in Section 5. • Finally, in Section 6, we shall mention some mathematical results concerning the existence and/or the uniqueness of the solutions to the Navier–Stokes equations. 1. Derivation of the Navier–Stokes equations for viscous fluids Let Ω be an open and connected region (i.e. a domain ) of R 3 filled with a fluid . The generic point of R 3 will be denoted by x = { x i } 3 i = 1 while d x will denote the elementary volume d x 1 d x 2 d x 3 .
eBook - PDFAdvanced Transport Phenomena
Analysis, Modeling, and Computations
- P. A. Ramachandran(Author)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
5 Equations of motion and the Navier–Stokes equation Learning objectives In this chapter you will learn • the basic differential equations to describe the fluid motion in general (equations of motion), • the equations for the velocity field for Newtonian fluids (the Navier–Stokes equations), • common boundary conditions needed for flow simulation, • important dimensionless variables for flow analysis and the principle of dynamic similarity, and • how to understand the various fluid behaviors in general and how to model them with some constitutive relations. Mathematical prerequisites No additional mathematical tools are needed for the study of this chapter. You may wish to review the vector calculus, plus your knowledge of tensors and the various operations on tensor quantities. This chapter starts with the development of the differential equation of motion, which is nothing but the statement of the momentum conservation principle. Thus we com-bine the fluid acceleration calculations derived in Chapter 3 and the representation of forces derived in Chapter 4. This leads to a general equation of motion that is based on the conservation principle alone. This is the basic model for the transport of momen-tum, which is also called the equation of (fluid) motion. As you may have guessed already, the divergence of the stress tensor will appear as a term, and the model is not in terms of the velocity alone. The model needs to be closed with appropriate consti-tutive relations between the stress and the strain rate. This requires knowledge of the rheological properties or the flow behavior of the fluid. Hence a discussion of common 185 5.1 Equation of motion: the stress form fluid behaviour is presented, and a classification of the rheological behaviors of fluids is presented next. Fluids obeying a linear relation are referred to as Newtonian fluids.
eBook - PDF- P. A. Lagerstrom(Author)
- 2022(Publication Date)
- Princeton University Press(Publisher)
CHAPTER ONE The Navier-Stokes Equations for a Viscous Heat-Conducting Compressible Fluid B,l. Introduction. Physical foundations. In this chapter a general system of equations for fluid mechanics is developed. It includes the classical Navier-Stokes equations suitably modified to take into account heat conduction and changes in density and temperature, as well as an energy law and one or more equations of state, taken from equilibrium thermodynamics. This system is referred to simply as the Navier-Stokes equations, and the subsequent theoretical treatment of the present section is based on these equations or on approximations to these equations. It is generally believed that the Navier-Stokes equations are capable of describing most phe- nomena observed in fluid mechanics. However, it should be remembered that any useful system of equations for fluid mechanics may always be regarded as an approximation to a more accurate system. There are indications that certain special flow problems arising, for example, in the dynamics of strong shock waves or highly rarefied gases, may require more accurate equations for an adequate treatment. For a further dis- cussion of such problems the reader is referred to I,D, 1,1, and III,H. In principle, the range of validity of the Navier-Stokes equations should be apparent from a derivation of these equations. There are two different schemes for deriving equations for fluid mechanics. One 1 This Section was completed in 1956 and it was not possible to revise it to include newer results and appropriate references to more recent publications. Various people, in particular GALCIT staff members, have rendered invaluable help to the author in preparing this section. H. W. Liepmann worked as co-author during the early stages of writing. Anatol Roshko and Donald Coles are in essence the authors of the articles on experimental results.
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
In rarefied systems of particles, drops, or bubbles, the particle–particle interaction can be neglected in the first approximation; then one deals with the behavior of a single particle moving in fluid. In this case, the streamline pattern depends on the particle shape, the flow type (translational or shear), and a number of other geometric factors. E4.6.1. Stokes Equations and Their Solution for the Axisymmetric Case ◮ Stokes equations. One of the main approaches to the analysis and simplification of the Navier–Stokes equations is as follows. One assumes that the nonlinear inertia term ( V ⋅ ∇ ) V is small compared with the linear viscous term ν Δ V and hence can be neglected altogether or taken into account in some special way. This method is well-founded for Re = LU/ν ≪ 1 and is widely used for studying the motion of particles, drops, and bubbles in fluids. Low Reynolds numbers are typical of the following three cases: slow (creeping) flows, highly viscous fluids, and small dimensions of particles. For steady-state flows of viscous incompressible fluid, by neglecting the inertia terms in (E4.2.1.4) and by including all conservative mass forces in the pressure P , we arrive at the Stokes equations ∇ ⋅ V = 0 , μ Δ V = ∇ P . (E4. 6 . 1 . 1 ) The Stokes equations (E4.6.1.1) are linear and hence much simpler than the nonlinear Navier–Stokes equations. For any two solutions { V 1 , P 1 } and { V 2 , P 2 } of (E4.6.1.1), the sum { α V 1 + β V 2 , αP 1 + βP 2 } satisfies the same equations for any constants α and β . E4.6. S PHERICAL P ARTICLES AND C IRCULAR C YLINDERS IN T RANSLATIONAL F LOW 801 In axisymmetric problems, all the variables in the spherical coordinates R , θ , ϕ are independent of ϕ , and the third component of the fluid velocity is zero, V ϕ = 0 . The fluid velocity components V R and V θ can be expressed via the stream function Ψ as follows: V R = 1 R 2 sin θ ∂ Ψ ∂θ , V θ = – 1 R sin θ ∂ Ψ ∂R .
- Heinz-Otto Kreiss, Jens Lorenz(Authors)
- 1989(Publication Date)
- Academic Press(Publisher)
1 The Navier-Stokes Equations In this preliminary chapter we first outline some questions which will be treated in this book. Then we derive the Navier-Stokes equations. Though the deriva- tion will not be used later, it is of interest to understand the underlying logical and physical assumptions, because the mathematical theory of the equations is not complete. There is no existence proof except for small time intervals. Thus it has been questioned whether the N-S equations really describe general flows. If one changes the stress tensor such that diffusion increases when the velocities become large, then existence can be shown. This change of the equa- tions does not seem to be justified physically, however. For example, certain similarity laws - valid for the Navier-Stokes equations - are well-established experimentally, but the modified equations do not allow the corresponding sim- ilarity transformations. Possibly a lack of mathematical ingenuity is the reason for the missing existence proof, and the N-S equations are physically correct. The N-S equations form a quasilinear differential system, and much of our un- derstanding of such systems is gained through the study of linearized equations. These will, in general, have variable coefficients. By fr-eezing the coefficients in such a problem, one obtains systems with constant coefficients. It is much easier to analyse the latter, as will be later shown in Chapter 2. However, the relation between variable-coefficient and constant-coefficient equations is not trivial. The fundamental ideas of linearization and localization are discussed in Section 1.3. 2 Initial-Boundary Value Problems and the Navier-Stokes Equations 1.1.
eBook - PDF- Tomasz W. Dłotko, Yejuan Wang(Authors)
- 2020(Publication Date)
- De Gruyter(Publisher)
6 Navier–Stokes equation in 2D and 3D 6.1 Introduction Following Dan Henry [86], extension of semilinear parabolic equations , namely semi-linear equations with sectorial positive operator in the main part, will be studied in-side the semigroup approach. Such technique was successfully applied to several clas-sical problems, some of them were reported in [32, 86]. The present chapter is devoted to a generalization of that approach to a critical problem . We will deal with the cele-brated 2-D Navier–Stokes equation (N-S equation, for short), an example of a problem critical with respect to the a priori estimate in L 2 . In order, the 3-D N-S equation is su-percritical in a sense specified in Subsection 6.1.3. Its weak solution will be obtained regularizing the Stokes operator by adding to it higher order diffusion term − ϵ (− P Δ ) s , ϵ > 0, s > 5 4 , solving the regularized equation, then letting ϵ → 0. The Dirichlet problem for classical 3-D Navier–Stokes equation considered here has the form: u t = ν Δ u − ∇ p − ( u ⋅ ∇) u + f , div u = 0 , x ∈ Ω , t > 0 , u = 0 , t > 0 , x ∈ 𝜕 Ω , u ( 0 , x ) = u 0 ( x ), (6.1) (see also (6.11)) where ν > 0 is the viscosity coefficient, u = ( u 1 ( t , x ), u 2 ( t , x ), u 3 ( t , x )) denotes velocity, p = p ( t , x ) pressure, f = ( f 1 ( x ), f 2 ( x ), f 3 ( x )) external force, and Ω is a bounded domain with C 2 boundary. It is impossible to recall even the most important publications devoted to that problem, since the corresponding literature is too large; see anyway [28, 31, 63–65, 67, 73, 75, 82, 89, 97, 104, 105, 110, 114, 121, 128, 170, 172] together with the references cited there. In space dimensions 2 and 3, the N-S equation possess local in time regular solu-tions, as stated in Theorems 6.1.4 and 6.1.5.
