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  • 2

    SINGLE PARTICLE MOTION

    2.1 INTRODUCTION

    This chapter will briefly review the issues and problems involved in con-structing the equations of motion for individual particles, drops or bubblesmoving through a fluid. For convenience we shall use the generic name par-ticle to refer to the finite pieces of the disperse phase or component. Theanalyses are implicitly confined to those circumstances in which the interac-tions between neighboring particles are negligible. In very dilute multiphaseflows in which the particles are very small compared with the global dimen-sions of the flow and are very far apart compared with the particle size, itis often sufficient to solve for the velocity and pressure, ui(xi, t) and p(xi, t),of the continuous suspending fluid while ignoring the particles or dispersephase. Given this solution one could then solve an equation of motion forthe particle to determine its trajectory. This chapter will focus on the con-struction of such a particle or bubble equation of motion.

    The body of fluid mechanical literature on the subject of flows aroundparticles or bodies is very large indeed. Here we present a summary thatfocuses on a spherical particle of radius, R, and employs the following com-mon notation. The components of the translational velocity of the centerof the particle will be denoted by Vi(t). The velocity that the fluid wouldhave had at the location of the particle center in the absence of the particlewill be denoted by Ui(t). Note that such a concept is difficult to extend tothe case of interactive multiphase flows. Finally, the velocity of the particlerelative to the fluid is denoted by Wi(t) = Vi Ui.

    Frequently the approach used to construct equations for Vi(t) (or Wi(t))given Ui(xi, t) is to individually estimate all the fluid forces acting on theparticle and to equate the total fluid force, Fi, to mpdVi/dt (where mp isthe particle mass, assumed constant). These fluid forces may include forces

    52

  • due to buoyancy, added mass, drag, etc. In the absence of fluid acceleration(dUi/dt = 0) such an approach can be made unambiguously; however, inthe presence of fluid acceleration, this kind of heuristic approach can bemisleading. Hence we concentrate in the next few sections on a fundamentalfluid mechanical approach, that minimizes possible ambiguities. The classicalresults for a spherical particle or bubble are reviewed first. The analysis isconfined to a suspending fluid that is incompressible and Newtonian so thatthe basic equations to be solved are the continuity equation

    ujxj

    = 0 (2.1)

    and the Navier-Stokes equations

    C

    {uit

    + ujuixj

    }= p

    xi+ CC

    2uixjxj

    (2.2)

    where C and C are the density and kinematic viscosity of the suspendingfluid. It is assumed that the only external force is that due to gravity, g.Then the actual pressure is p = p Cgz where z is a coordinate measuredvertically upward.

    Furthermore, in order to maintain clarity we confine our attention torectilinear relative motion in a direction conveniently chosen to be the x1direction.

    2.2 FLOWS AROUND A SPHERE

    2.2.1 At high Reynolds number

    For steady flows about a sphere in which dUi/dt = dVi/dt = dWi/dt = 0, itis convenient to use a coordinate system, xi, fixed in the particle as well aspolar coordinates (r, ) and velocities ur, u as defined in figure 2.1.

    Then equations 2.1 and 2.2 become

    1r2

    r(r2ur) +

    1r sin

    (u sin ) = 0 (2.3)

    and

    C

    {urt

    + ururr

    +ur

    ur

    u2

    r

    }= p

    r(2.4)

    +CC

    {1r2

    r

    (r2urr

    )+

    1r2 sin

    (sin

    ur

    ) 2ur

    r2 2r2

    u

    }

    53

  • Figure 2.1. Notation for a spherical particle.

    C

    {ut

    + urur

    +ur

    u

    +urur

    }= 1

    r

    p

    (2.5)

    +CC

    {1r2

    r

    (r2ur

    )+

    1r2 sin

    (sin

    u

    )+

    2r2ur

    ur2 sin2

    }

    The Stokes streamfunction, , is defined to satisfy continuity automatically:

    ur =1

    r2 sin

    ; u = 1

    r sin

    r(2.6)

    and the inviscid potential flow solution is

    = Wr2

    2sin2 D

    rsin2 (2.7)

    ur = W cos 2Dr3

    cos (2.8)

    u = +W sin Dr3

    sin (2.9)

    = Wr cos + Dr2

    cos (2.10)

    where, because of the boundary condition (ur)r=R = 0, it follows that D =WR3/2. In potential flow one may also define a velocity potential, , suchthat ui = /xi. The classic problem with such solutions is the fact thatthe drag is zero, a circumstance termed DAlemberts paradox. The flow issymmetric about the x2x3 plane through the origin and there is no wake.

    The real viscous flows around a sphere at large Reynolds numbers,

    54

  • Figure 2.2. Smoke visualization of the nominally steady flows (fromleft to right) past a sphere showing, at the top, laminar separation atRe = 2.8 105 and, on the bottom, turbulent separation atRe = 3.9 105.Photographs by F.N.M.Brown, reproduced with the permission of the Uni-versity of Notre Dame.

    Re = 2WR/C > 1, are well documented. In the range from about 103 to3 105, laminar boundary layer separation occurs at = 84 and a largewake is formed behind the sphere (see figure 2.2). Close to the sphere thenear-wake is laminar; further downstream transition and turbulence occur-ring in the shear layers spreads to generate a turbulent far-wake. As theReynolds number increases the shear layer transition moves forward until,quite abruptly, the turbulent shear layer reattaches to the body, resultingin a major change in the final position of separation ( = 120) and in theform of the turbulent wake (figure 2.2). Associated with this change in flow

    55

  • Figure 2.3. Drag coefficient on a sphere as a function of Reynolds number.Dashed curves indicate the drag crisis regime in which the drag is verysensitive to other factors such as the free stream turbulence.

    pattern is a dramatic decrease in the drag coefficient, CD (defined as thedrag force on the body in the negative x1 direction divided by 12CW

    2R2),from a value of about 0.5 in the laminar separation regime to a value ofabout 0.2 in the turbulent separation regime (figure 2.3). At values of Reless than about 103 the flow becomes quite unsteady with periodic sheddingof vortices from the sphere.

    2.2.2 At low Reynolds number

    At the other end of the Reynolds number spectrum is the classic Stokessolution for flow around a sphere. In this limit the terms on the left-hand sideof equation 2.2 are neglected and the viscous term retained. This solutionhas the form

    = sin2 {Wr

    2

    2+A

    r+ Br

    }(2.11)

    ur = cos {W + 2A

    r3+

    2Br

    }(2.12)

    u = sin {W A

    r3+B

    r

    }(2.13)

    where A and B are constants to be determined from the boundary conditionson the surface of the sphere. The force, F , on the particle in the x1 direction

    56

  • is

    F1 =43R2CC

    {4WR

    +8AR4

    +2BR2

    }(2.14)

    Several subcases of this solution are of interest in the present context. Thefirst is the classic Stokes (1851) solution for a solid sphere in which the no-slipboundary condition, (u)r=R = 0, is applied (in addition to the kinematiccondition (ur)r=R = 0). This set of boundary conditions, referred to as theStokes boundary conditions, leads to

    A = WR3

    4, B = +

    3WR4

    and F1 = 6CCWR (2.15)

    The second case originates with Hadamard (1911) and Rybczynski (1911)who suggested that, in the case of a bubble, a condition of zero shear stresson the sphere surface would be more appropriate than a condition of zerotangential velocity, u. Then it transpires that

    A = 0 , B = +WR

    2and F1 = 4CCWR (2.16)

    Real bubbles may conform to either the Stokes or Hadamard-Rybczynskisolutions depending on the degree of contamination of the bubble surface,as we shall discuss in more detail in section 3.3. Finally, it is of interest toobserve that the potential flow solution given in equations 2.7 to 2.10 is alsoa subcase with

    A = +WR3

    2, B = 0 and F1 = 0 (2.17)

    However, another paradox, known as the Whitehead paradox, arises whenthe validity of these Stokes flow solutions at small (rather than zero)Reynolds numbers is considered. The nature of this paradox can be demon-strated by examining the magnitude of the neglected term, ujui/xj, inthe Navier-Stokes equations relative to the magnitude of the retained termC

    2ui/xjxj. As is evident from equation 2.11, far from the sphere theformer is proportional toW 2R/r2 whereas the latter behaves like CWR/r3.It follows that although the retained term will dominate close to the body(provided the Reynolds number Re = 2WR/C 1), there will always be aradial position, rc, given by R/rc = Re beyond which the neglected term willexceed the retained viscous term. Hence, even if Re 1, the Stokes solutionis not uniformly valid. Recognizing this limitation, Oseen (1910) attemptedto correct the Stokes solution by retaining in the basic equation an approxi-mation to ujui/xj that would be valid in the far field, Wui/x1. Thus

    57

  • the Navier-Stokes equations are approximated by

    W uix1

    = 1C

    p

    xi+ C

    2uixjxj

    (2.18)

    Oseen was able to find a closed form solution to this equation that satisfiesthe Stokes boundary conditions approximately:

    = WR2{r2 sin2

    2R2+R sin2

    4r+

    3C(1 + cos )2WR

    (1 e

    W r2C (1cos )

    )}(2.19)

    which yields a drag force

    F1 = 6CCWR{

    1 +316

    Re

    }(2.20)

    It is readily shown that equation 2.19 reduces to equation 2.11 as Re 0.The corresponding solution for the Hadamard-Rybczynski boundary con-ditio

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