Sphere uniform flow

Amongst the assumptions we have made in developing the model are the following that Pick s law is applicable to the diffusion processes, the gel particles are isotropic and behave as hard spheres, the flow rate is uniform throughout the bed, the dispersion in the column Ccui be approximated by the use of an axial dispersion coefficient cuid that polymer molecules have an independent existence (i.e. very dilute solution conditions exist within the column). Our approach borrows extensively many of the concepts which have been developed to interpret the behaviour of packed bed tubular reactors (5). 

The boundary conditions are the same as for steady motion considered in Chapters 1, 3, and 4, i.e., uniform flow remote from the particle, no slip and no normal flow at the particle boundary, and, for fluid particles, continuity of tangential stress at the interface. For a sphere the normal stress condition at the interface is again formally redundant, but indicates whether a fluid particle will remain spherical. 

A sphere of radius R is tethered by a narrow string in a steady uniform flow of incompressible viscous fluid (Fig. 3.14). Under certain circumstances (i.e., very low Reynolds number, Re = pUD/p) the creeping flow may be analyzed assuming that the viscous terms in the Navier-Stokes equations dominate over the acceleration terms. 

The first term on the right-hand side of Eq. (3.80) is the total drag force in the opposite direction of Up, including both Stokes drag and Oseen drag. The second term represents a lift force in the direction perpendicular to Up. Thus, the lift force or Magnus force for a spinning sphere in a uniform flow at low Reynolds numbers is obtained as... 

Heat Transfer of a Single Sphere in a Uniform Flow... 

For forced convective heat transfer over a sphere in a uniform flow, a frequently used empirical relation was proposed by Ranz and Marshall (1952) as... 

Formost gases, Pris nearly constant ( 0.7). Therefore, for convective heat transfer of an individual solid sphere in a gaseous medium, Nup is only a function of Rep. An alternative estimation of Nup for thermal convection of a sphere in a uniform flow was recommended by McAdams (1954) as... 

The term F represents the contribution from the uniform flow, whereas the term G accounts for the deviation from the parallel flow due to the presence of the sphere. 

Sphere, assisting flow, uniform surface temperature 3.5... 

Maxey, M.R. and Riley, J.J. (1983), Equation of motion for a small rigid sphere in a non uniform flow, Phys. Fluids, 26(4), 883. 

The fluid-particle interaction closures applied in the modern single particle momentum balances originate from the classical work on the Newton s second law as applied to a small rigid sphere in an unsteady, non-uniform flow limited to Stokesian flow conditions Rep [Pg.554]

The standard drag curve refers to a plot ot Cp as a function of Re for a smooth rigid sphere in a steady uniform flow field. The best fit of the cumulative data that have been obtained for this drag coefficient is shown in Fig 5.2. Numerous parameterizations have been proposed to approximate this curve (e.g., many of them are listed by [22]). 

Rigid spheres sometimes experience a lift force perpendicular to the direction of the flow or motion. For many years it was believed that only two mechanisms could cause such a lift. The first one described is the so-called Magnus force which is caused by forced rotation of a sphere in a uniform flow field. This force may also be caused by forced rotation of a sphere in a quiescent fluid. The second mechanism is the Saffman lift. This causes a particle in a shear flow to move across the flow field. This force is not caused by forced rotation of the particle, as particles that are not forced to rotate also experience this lift (i.e., these particles may also rotate, but then by an angular velocity induced by the flow field itself). 

A rotating sphere in uniform flow will experience a lift which causes the particle to drift across the flow direction. This is called the Magnus effect (or force). The physics of this phenomenon are complex. 

The rotation of a rigid sphere will cause the surrounding fluid to be entrained. When the sphere is placed in a uniform flow, this results in higher fluid velocity on one side of the particle, and lower velocity on the other side. 

For steady flow of an aerosol around a collector such as a cylinder or sphere, a force balance can be written on a particle by considering it to be fixed in a uniform flow of elocity u — uy ... 

The limit Pe 0 yields the pure conduction heat transfer case. However, for a fluid in motion, we find that the pure conduction limit is not a uniformly valid first approximation to the heat transfer process for Pe 1, but breaks down far from a heated or cooled body in a flow. We discuss this in the context of the Whitehead paradox for heat transfer from a sphere in a uniform flow and then show how the problem of forced convection heat transfer from a body in a flow can be understood in the context of a singular-perturbation analysis. This leads to an estimate for the first correction to the Nusselt number for small but finite Pe - this is the first small effect of convection on the correlation between Nu and Pe for a heated (or cooled) sphere in a uniform flow. 

Figure 7-2. Illustration of the decomposition of the problem of a freely rotating sphere in a simple shear flow as the sum of three simpler problems (a) a sphere rotating in a fluid that is stationary at infinity, (b) a sphere held stationary in a uniform flow, and (c) a nonrotating sphere in a simple shear flow that is zero at the center of the sphere. The angular velocity Cl in (a) is the same as the angular velocity of the sphere in the original problem. The translation velocity in (b) is equal to the undisturbed fluid velocity evaluated at the position of the center of the sphere. The shear rate in (c) is equal to the shear rate in the original problem.

The lift on a nonrotating sphere in a uniform flow is also zero because of the symmetry of the problem. Again, because no direction perpendicular to U can be distinguished from any other, the only possibility is that there can be no force on the sphere in any of these directions. The only force acting on the sphere in this case will be a drag force that is collinear with U. 

Figure 7-11. A schematic representation of the domain for a uniform flow past an arbitrary, axisymmetric body. For the case of a solid sphere, this is Stokes problem.

As an example of the application of (7-131), we consider creeping flow past an arbitrary axisymmetric body with a uniform streaming motion at infinity. For the case of a solid sphere, this is known as Stokes problem. In the present case, we begin by allowing the geometry of the body to be arbitrary (and unspecified) except for the requirement that the symmetry axis be parallel to the direction of the uniform flow at infinity so that the velocity field will be axisymmetric. A sketch of the flow configuration is shown in Fig. 7 11. We measure the polar angle 9 from the axis of symmetry on the downstream side of the body. Thus ij = I on this axis, and ij = — 1 on the axis of symmetry upstream of the body. 

Here, we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition,... 

A second straightforward example, solved previously by other means, is Stokes original problem of uniform flow past a stationary sphere. To apply the methods of the preceding subsection to this problem, it is convenient to transform to the disturbance flow problem,... 

To this point, the solution is essentially identical to that obtained for uniform flow past a sphere in Section A. Beyond this, however, the two differ. To satisfy boundary conditions at the surface of a sphere, a nonzero value of a is necessary. However, for a point force, a = 0. The simplest way to see this is to note that c is dimensionless, whereas a must have dimensions of (length)2. But a point force has no characteristic length scale, so a = 0. Thus,... 

An especially powerful result is obtained if we apply the formula (8-215) to a stationary solid sphere of radius a in the undisturbed flow u00(x). In this case, the solution for uniform flow past a stationary solid sphere yields the general result... 

Figure 9-3. A schematic representation of the solution domain for heat transfer from a sphere in a uniform flow in the limit of small Peclet numhers, that is, Pe —> 0.

Figure 9-4. Contours of constant temperature (isotherms) for heat transfer from a sphere in a uniform flow at low Peclet numbers according to Eq. (9-51). Note that the sphere appears in this representation, in which Ic=k/Uoo, as a point source (sink) of heat (the radius of the sphere, a, is vanishingly small compared with k/Uoo in the limit as Pe - 0). In the inner region, near to the sphere, the isotherms at leading order of approximation are still spherical as illustrated in Fig. 9-2. The three contours plotted are o = 0.25, 2.75, and 5.25.

D. UNIFORM FLOW PAST A SOLID SPHERE AT SMALL, BUT NONZERO, REYNOLDS NUMBER... 

D. Uniform Flow Past a Solid Sphere at Small, but Nonzero, Reynolds Number... 

We now seek a solution of (9 7) and (9-8) for small values of the Peclet number, Pe , by using the matched asymptotic expansion procedure that was detailed for uniform flow past a sphere in Section C. Although the reader may not immediately see that the derivation of an asymptotic solution for this new problem necessitates use of the matched asymptotic expansion technique, an attempt to develop a regular expansion for 9 for Pe 1 leads to a Whitehead-type paradox similar to that encountered for the uniform-flow problem. 

As in the uniform-flow problem, the solution domain divides into two parts. In the so-called inner region, the sphere diameter provides an appropriate characteristic length scale, and the dimensionless form of (9-7) is applicable, along with the boundary condition (9-8a) at the sphere surface. We assume, in this region, that an asymptotic expansion for 9 exists in the form... 

H. Heat Transfer From a Solid Sphere in Uniform Flow for Re -C I and Pe I... 

H. HEAT TRANSFER FROM A SOLID SPHERE IN UNIFORM FLOW FOR Re < I AND Pe I... 

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