Shear Stokes flow

Freely rotating cylinder. Now let us consider a circular cylinder freely floating in an arbitrary linear shear Stokes flow (Re —i 0). The velocity distribution for such a flow remote from the cylinder is still given by relations (2.7.8). 

The solution of the hydrodynamic problem on a freely rotating cylinder in an arbitrary shear Stokes flow with the boundary conditions (2.7.8) and (2.7.10) has the form [93,166]... 

For a spherical particle in an axisymmetric shear Stokes flow (Re — 0), numerical results for the mean Sherwood number can be well approximated in the entire range of Peclet numbers by the expression (4.7.9), where the asymptotic value Shpoo must be taken from the first row in Table 4.4. As a result, we obtain the formula... 

Let us investigate convective mass transfer to the surface of a solid sphere freely suspended in an arbitrary plane shear Stokes flow. In this case, the fluid velocity distribution remote from the particle is given by formulas (4.5.1) with... 

An arbitrary shear Stokes flow past a fixed cylinder is described by the stream function (2.7.9). We restrict our discussion to the case 0 2 fi < 1, in which there are four stagnation points on the surface of the cylinder. Qualitative streamline patterns for a purely straining flow (at CIe 0) and a purely shear flow (at CIe = 1) are shown in Figure 2.10. 

Freely rotating cylinder. Now let us consider convective mass transfer to the surface of a circular cylinder freely suspended in an arbitrary linear shear Stokes flow (Re -> 0). In view of the no-slip condition, the cylinder rotates at a constant angular velocity equal to the angular velocity of the flow at infinity. The fluid velocity distribution is described by formulas (2.7.11). The streamline pattern qualitatively differs from that for the case of a fixed cylinder. For 0 0, there are no stagnation points on the surface of the cylinder and there exist two qualitatively different types of flow. For 0 < Ifigl < 1, there are both closed and open streamlines in the flow, the region filled with closed streamlines is adjacent to the surface of the cylinder, and streamlines far from the cylinder are open (Figure 2.11). For Ifl l > 1, all streamlines are open. 

Drop, bubble Axisymmetric shear Stokes flow Analytical, DBLA 0 [170]... 

Drop, bubble Plane shear Stokes flow 3.8 2.8... 

The dependence of the auxiliary Sherwood number Sho on the Peclet number Pe for a translational Stokes flow past a spherical particle or a drop is determined by the right-hand sides of (4.6.8) and (4.6.17). In the case of a linear shear Stokes flow, the values of Sho are shown in the fourth column in Table 4.4. 

We start with the governing equations of the Stokes flow of incompressible Newtonian fluids. Using an axisymraetric (r, z) coordinate system the components of the equation of motion are hence obtained by substituting the shear-dependent viscosity in Equations (4.11) with a constant viscosity p, as... 

Liquid viscosities have been observed to increase, decrease, and remain constant in microfluidic devices as compared to viscosities in larger systems. ° Deviations from the no-slip boundary condition have been observed to occur at high shear rates. One important deviation from no-slip conditions occurs at moving contact lines, such as when capillary forces pull a liquid into a hydrophilic channel. The point at which the gas, liquid, and solid phases move along the channel wall is in violation of the no-slip boundary condition. Ho and Tai review discrepancies between classical Stokes flow theory and observations of flow in microchannels. No adequate theory is yet available to explain these deviations from classical behavior. ... 

S. Wakiya, Application of bipolar co-ordinates to the two-dimensional creeping-motion of a liquid. I. Flow over a projection or a depression on a wall, J. Phys. Soc. Jpn. 39, 111 3-20 (1975) M. E. O Neill, On the separation of a slow linear shear flow from a cylindrical ridge or trough in a plane, Z. Angew. Math. Phys. 28,438-48 (1977) A. M. J. Davis and M. E. O Neill, Separation in a Stokes flow past a phase with a cylindrical ridge or trough, Q. J. Mech. Appl. Math. 30, 355-68 (1977). 

G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech. 44, 419-40 (1970) R. G. Cox, The motion of long slender bodies in a viscous fluid, Part 1, General theory, J. Fluid Mech. 44, 791-810, (1970) Part 2, Shear Flow, J. Fluid Mech. 45, 625-657 (1971) J. B. Keller and S. T. Rubinow, Slender-body theory for slow viscous flow, J. Fluid Mech. 75, 705-14 (1976) R. E. Johnson, An improved slender-body theory for Stokes flow, J. Fluid Mech. 99, 411-31 (1980) A. Sellier, Stokes flow past a slender particle, Proc. R. Soc. London Ser. A 455, 2975-3002 (1999). 

Among references that discuss closed-streamline patterns and eddies in low-Reynolds-number flows, the reader may wish to refer to Ref. 13, Chap. 7, and D. J. Jeffrey and J. D. Sherwood, Streamline patterns and eddies in low-Reynolds-number flow, J. Fluid Mech. 96, 315-34 (1980) A. M. J. Davis and M. B. O Neill, The development of viscous wakes in a Stokes flow when a particle is near a large obstacle, Chem. Eng. Sci. 32, 899-906 (1977) A. M. J. Davis and M. B. O Neill, Separation in a slow linear shear flow past a cylinder and a plane, J. Fluid Mech., 81, 551-64 (1977). 

Note that since the problem of Stokes flow is linear, one can find the velocity and pressure fields in translational-shear flows as the superposition of solutions describing the translational flow considered in Section 2.2 and shear flows considered in the present section. 

By virtue of the no-slip condition on the surface, a sphere freely suspended in a plane shear flow will rotate at a constant angular velocity fl equal to the flow rotation velocity at infinity. The solution of the corresponding three-dimensional hydrodynamic problem on a particle in a Stokes flow is given in [343]. 

Fixed cylinder. Let us consider diffusion to the surface of a fixed circular cylinder in a steady-state linear shear Stokes (Re - 0) flow in the plane normal to the cylinder axis. The velocity distribution of such a flow remote from the... 

It follows from (4.11.3) that in the region -1 < fl < +1, the mean Sherwood number varies only slightly (the relative increment in the mean Sherwood number as Iflfil varies from 0 to 1 is at most 1.3%). In the special cases of purely straining (CIe = 0) and purely shear (Ifi l = 1) linear Stokes flow past a circular cylinder, formula (4.11.3) turns into those given in [342, 343]. 

It was shown in [ 166,351 ] that Eq. (5.1.5) provides several valid initial terms of the asymptotic expansion of the Sherwood number as Pe —> 0 for any kinetics of the surface chemical reaction. (Specifically, one obtains three valid terms for the translational Stokes flow and four valid terms for an arbitrary shear flow.)... 

Under the Stokes flow and particle separation assumptions, the viscous force between two approaching particles should scale as fiaU, with n the Newtonian viscosity of the medium and U the approach velocity. With U ay, where y is the applied shear rate, the energy dissipation within the gap between the particles scales as fJi-a y. We have here assumed that the interaction frequency between the particles is of the order of y. This will be true so long as the particle concentration is not so high that we are close to the maximum packing fraction for which flow can occur, a point which is discussed in greater detail in the following section. 

The various expressions found in the literature for the Stokes flow of a sphere of radius a under the force of gravity in a power-law liquid have been simplified by the present author to give the average shear rate and shear stress as... 

Finally, an important aspect for polymer processing is the fact that the uniqueness of Stokes flow is lost when dealing with viscoelastic liquids. As a consequence, the breakup of droplets in systems with a viscoelastic matrix has been found to be largely dependent upon the shear flow history, with more... 

Saffman PG (1965) The lift on a small sphere in a slow shear flow. J Ruid Mech 22 385-400 Schlauch E, Ernst M, Seto R, Briesen H, Sommerfeld M, Behr M (2013) Comparison of three simulation methods for colloidal aggregates in Stokes flow finite elements, lattice Boltzmann and Stokesian dynamics. Comput Huids 86 199-209... 

Dense Compaction at high shear rates the aggregate tend rotate very quickly around the axis of shear. Little restructuring is seen from already very compromised structures, but internal rearrangement occurs. Care has to be taken whether assumptions about the Stokes flow still hold valid. 

The increase in viscosity with the volume fraction of solids in the suspension does not in itself imply any non-Newtonian behavior, as the stress can remain strictly linear in the shear rate (Guazzelli and Morris 2012). In fact, when we consider the hard-sphere suspension in a Newtonian liquid under Stokes flow conditions, the expectation is that the rheology should be Newtonian. This is a basic result of dimensional analysis since there is no intrinsic rate associated with either the fluid or the particles, the only rate is that set by the flow shear rate. 

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