Stokes Flow

6 Key excerpts on "Stokes Flow"

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  Advanced Transport Phenomena

  Analysis, Modeling, and Computations

  • P. A. Ramachandran(Author)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  Fur-thermore, the flow may become turbulent at some distance from the starting point, which adds another level of complexity to the problem. 525 15.1 Low-Reynolds-number (Stokes) flows Low-Reynolds-number (Stokes) flows ............................................................................... 15.1 The simplified version of the N–S equation where the inertial terms are neglected is known as Stokes flow or creeping flow. This simplified representation applies when the flow Reynolds number is small. The situation is encountered in a number of important appli-cations such as the study of (i) micro-circulation, (ii) suspensions and emulsions and their rheological behavior, (iii) colloidal dispersions, (iv) flow in micro-fluidic devices, (v) flow in porous media, etc. First, we present some basic properties of Stokes flow. Then general solutions are presented, and we show how the solution to complex flows can be constructed by the superposition of these general solutions. The application to the basic problem of Stokes flow past a sphere is presented. The derivations and other mathematical details pertinent to the general solutions are not presented here, since this is an introductory treatment of Stokes flow. Some of these are left as exercises. 15.1.1 Properties of Stokes Flow The differential equation describing Stokes flow can be represented as μ ∇ 2 v − ∇ p = 0 (15.1) and ∇ · v = 0 (15.2) The term p can be treated as the modified pressure P to make the problem general; the notation p is used here for simplicity. In the above simplification we note that the convective terms are dropped because the inertia effects are small due to the small value of the Reynolds number, while the time derivative is dropped since the viscous diffusion time in this case is small compared with the observation time (the Strouhal number is small). This amounts to making a pseudo-steady-state assumption for the time variable.

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  Incompressible Flow

  • Ronald L. Panton(Author)
  • 2024(Publication Date)
  • Wiley

    (Publisher)

  Eq. 21.7 for prescribed geometry and boundary conditions on the velocity is mathematically unique.

  Perhaps one of the most useful mathematical properties of Stokes Flows is a direct result of the linearity. Consider what happens if we reverse the velocity (

  ) of a certain flow problem. All equations and boundary conditions are still satisfied. The stresses change direction (

  ), and the pressure changes sign

  . Thus, Stokes Flows are reversible in the sense that the reverse flow is also a Stokes Flow. These facts allow one to argue that a velocity pattern about a symmetrical object or in a symmetrical flow channel must also be symmetrical. The corresponding pressure distribution is antisymmetric. An application of this principle to symmetrical objects in an infinite fluid shows that these objects have no wakes. The downstream flow has the same streamline pattern and velocity magnitudes as the upstream pattern. Viscous diffusion of the vorticity proceeds upstream and downstream with equal effectiveness. Figure 21.1 shows the symmetric flow over a block. A wake is a phenomenon for

  .

  Figure 21.1

   Viscous flow over a block shows symmetry at

  . Reproduced with permisison from Taneda., 1979 / The Physical Society of Japan.

  From a mathematical standpoint the separation-of-variables techniques are useful in linear problems described in orthogonal coordinate systems. Series expansions in eigenfunctions are developed for Stokes Flow in Happel and Brenner (1963 and 1983) and by Leal (2007).

  Confined flows in general lead to well-structured solutions to low-Reynolds-number equations. For flows where the domain is infinite, we shall find that the situation is quite different. Stokes Flows on an unbounded domain are not uniformly valid from a mathematical standpoint. The difficulty is analogous to the one we discovered for high-Reynolds-number flows. Stokes Flows on an infinite domain turn out to be singular at infinity. A perturbation theory that includes Stokes Flows and gives the correct behavior at infinity is quite different for two- or three-dimensional objects. We discuss these equations in Sections to 21.12.

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  Physical Fluid Dynamics

  • P McCormack(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R S E V E N SLOW VISCOUS FLOW 7.1 Introduction This chapter deals with flows at low Reynolds numbers. In the first part, namely that dealing with Stokes Flows, it follows that the inertia or transport terms in the Navier-Stokes equation can be neglected. In other words, vorticity is transferred essentially by diffusion. A linearization of the equations of motion is effected, which reduces them to the biharmonic equation. This equation has been studied extensively in classical elas-ticity. It is shown that, in problems involving infinite regions, the solutions of the Stokes equations give rise to nonnegligible inertia terms, which means that in these cases the Stokes equations are not consistent. The difficulty is resolved in the case when there is a uniform flow at large distances from an obstacle (see Section 7.6 on Oseen flows). In this case it is shown *hat the inertia terms may be approximated by assuming that vorticity is transported with speed U 0 in the χ direction only. This approximation gives good results at large dis-tances; at small distances from the obstacle the solu-tions of the Oseen flows and Stokes Flows differ by a negligible amount, of the order of the Reynolds number. 276 7.2 Stokes FlowS 277 7.2 Stokes Flows At Reynolds numbers very much smaller than unity, the transport terms in the Navier-Stokes equations are very much smaller than the diffusion terms. This may be seen by considering the orders of magnitude of representative terms of these types. Thus, a typical transport term is u du/dx which has order u 2 /L, where L is the length over which significant changes in the flow take place (it might for example be the size of an ob-stacle or the diameter of a pipe). In the same way a typical diffusion term is ν d 2 u/dx 2 , which is of order vUjL 2 . The ratio transport terms _ u 2 /L ^ UL ^ (7 1) diffusion terms vu/L 2 ν where Re^ = ULjv is the Reynolds number of the flow and where U is a representative velocity in the flow.

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  Nanotechnology

  Understanding Small Systems, Third Edition

  • Ben Rogers, Jesse Adams, Sumita Pennathur(Authors)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Because Equations 9.22a and 9.22b are linear, pres-sure-driven flow, gravity-driven flow, and electroosmotic flow can all be superposed to generate a solution with combined flows. For example, if we have a pressure-driven flow in one direction and electroosmotic plug flow in the other, the solution will be a simple addition of both velocity profiles. Nanoscale Fluid Mechanics ◾ 315 9.3.5 Ions and Macromolecules Moving through a Channel Now that we have finally solved for the velocity profile within the channel, we are going to discuss what happens when you put ions and other macromolecules within the channel. This is where a lot of interesting phenomena start to happen—such as interactions between the particles and the wall and different sorts of particle motion due to different fields. We start this section by describing known solutions for low-Reynolds-number flow around a particle with just pressure-driven flow, and then discuss phenomena involving surface charges and electrokinetics. 9.3.5.1 Stokes Flow around a Particle Low Reynolds number, pressure-driven flow is typically known as creeping flow, or Stokes Flow. The equation set used to describe Stokes Flow is the same as Equation 9.23, shown again for reference: 0 2 2 2 2 = -+ +       ∂ ∂ ∝ ∂ ∂ ∂ ∂ P x v x v y x x (9.23a) 0 2 2 2 2 = -+ +       ∂ ∂ ∝ ∂ ∂ ∂ ∂ P y v x v y y y (9.23b) Remember that P is the pressure acting on the fluid (N/m 2 ), and μ is the coefficient of dynamic viscosity (Pa s). Stokes developed these equations, which have subsequently been solved for various types of flow. These flow types include fully developed duct flow (which we derived ourselves in the previous section for two-dimensional parallel flow), flow around immersed bodies, flow in narrow but variable passages, and flow through porous media. The velocity profiles for flow around a spherical particle can be obtained by solving Equation 9.22 in spherical coordinates.

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  Fluid-Structure Interactions in Low-Reynolds-Number Flows

  • Camille Duprat, Howard Stone(Authors)
  • 2015(Publication Date)
  • Royal Society of Chemistry

    (Publisher)

  Unfortunately, this approach is not always so straightfor-ward, as we now describe. Consequently, there are cases where the Stokes equations do not represent a uniformly valid approximation to the Navier– Stokes equation: the inability to solve the two-dimensional problem of uni-form flow past a cylinder—Stokes’s paradox—is the classical example (see e.g. Leal 48 ). Here we just summarize a few ideas, but the subject is much richer and more complex than the brief remarks below indicate. To understand some ways in which inertia can influence the dynamics, we consider steady flow past a particle. The Navier–Stokes equation (2.5), with a characteristic pressure representative of a Stokes Flow and neglecting the body force, can be written in dimensionless form as R e v · V v Z K V p C V 2 v . (2.110) This equation is to be solved with v → K ˆ V as r Z | r |→∞ and with v Z 0 on the particle surface S p ; given that the discussion is in dimensionless terms, here ˆ V is a unit vector in the direction of translation of the particle. Provided the Reynolds number is sufficiently small, we expect that the in-ertia term can be neglected, in which case we have all of the results described in previous sections. If this is true, then we can seek a regular perturbation solution for the velocity distribution in the form v ( r , R e ) Z v 0 ( r ) C R e v 1 ( r ) C O ( R e 2 ) . (2.111) However, in the far field r 1, where the velocity approaches a uniform flow, we know that the disturbance velocity decays as r K 1 . Thus, we compare Low-Reynolds-Number Flows 73 the inertia and viscous terms in eqn (2.110) and observe that O ( R e v · V v ) O ( V 2 v ) Z O ( R e r K 2 ) O ( r K 3 ) Z R e r for r 1. (2.112) This ratio is no longer small when r O R e K 1 1.

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  Physics of Continuous Matter

  Exotic and Everyday Phenomena in the Macroscopic World

  • B. Lautrup(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  That is why bacteria do not sprout wings but rely on oars or screws to move around. In this chapter we shall first study creeping flow around moving bodies far from containing boundaries. The behavior of drag at higher Reynolds numbers is included for completeness. Afterward we turn to lubrication theory based on creeping flow in narrow gaps. 17.1 Stokes Flow At low Reynolds number, Re 1 , the advective acceleration can be left out of the Navier– Stokes equations for incompressible flow (15.21), so that they for steady flow become r p D r 2 v ; r v D 0; (17.1) an approximation usually called Stokes Flow . Gravity, g D r ˆ , has for simplicity been left out but is easy to include by replacing p by p C 0 ˆ . From the divergence of the first equation it follows that the pressure must satisfy the Laplace equation, r 2 p D 0 . 288 PHYSICS OF CONTINUOUS MATTER Creeping flow is mathematically (and numerically) much easier to handle than general flow because of the absence of non-linear terms that tend spontaneously to break the natural symmetry of the solutions in time as well as space with turbulence as the extreme result. The linearity of the creeping flow equations may sometimes be used to express solutions to complicated flow problems as linear superpositions of simpler solutions. Drag and lift on a moving body Consider a solid body cruising with constant velocity U through a static fluid. The body creates a temporary disturbance in the fluid that disappears again some time after the body has passed a fixed observation point. Seen from the body, the fluid appears to move in a steady pattern that at sufficiently large distances becomes uniform, described by the vector U (see the margin figure). Newtonian relativity guarantees, as we have discussed before, that these situations are physically equivalent, so we may use the Stokes Flow equations (17.1). . . . . . . . . . . . . . . . . . . . . . . . . . . .

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