Spherical Navier-Stokes Equations

7 SPHERICAL NAVIER-STOKES EQUATIONS B.7.1 Mass Continuity  

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Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

However, one difference exists with classical theory in this latter case, the Navier-Stokes equation (443) and the incompressibility condition (444) are assumed to be valid for all distances rict. In this case, it is an easy matter to calculate explicitly the higher-order terms in Eq. (445), and the boundary condition at the B-particle (assumed to be spherical) imposes the condition... 

Thus, analytic solutions for flow around a spherical particle have little value for Re > 1. For Re somewhat greater than unity, the most accurate representation of the flow field is given by numerical solution of the full Navier-Stokes equation, while empirical forms should be used for C. These results are discussed in Chapter 5. 

Within the slip plane the viscosity Is infinitely high and all velocities are zero with respect to the solid. As In 14.6.7), in the Navier-Stokes equation, the Vp and the Vy/ terms are the only ones remaining. Consider a Stern layer, containing only one type of ions, 1, around a spherical particle of radius a, then [4.6.61 reduces to... 

The rotational friction coefficient of a spherical molecule in solution is calculated applying the Navier-Stokes equation for a continuum solvent with a position-dependent viscosity as a model of "microscopic viscosity." The rotational friction coefficient decreases with decreasing surface viscosity. The results are compared with the translational fnction and viscosity B coefficients which are previously obtained from the same model. The B coefficient is most sensitive to a local viscosity change The Gierer-Wirtz model overestimates the effect of the "microscopic viscosity" on the translational friction coefficient comparing with the present results... 

Thus we begin by considering the full Navier-Stokes equation expressed in terms of the streamfunction characteristic velocity and the sphere radius a as a characteristic length scale. Using spherical coordinates, with ij = cos 9, this equation is... 

The continuity and Navier-Stokes equations in cylindrical and spherical coordinate systems are given in Supplement 5. 

In chemical technology, one often meets the problem about a spherically symmetric deformation (contraction or extension) of a gas bubble in an infinite viscous fluid. In the homobaric approximation (the pressure is homogeneous inside the bubble) [306, 312], only the motion of the outer fluid is of interest. The Navier-Stokes equations describing this motion in the spherical coordinates have the form... 

The three Navier-Stokes equations can be put in very compact form by using the shorthand notation of vector calculus [6, p. 66 7 8, p. 80]. Furthermore, it is often convenient to use these equations in polar or spherical coordinates their transformations to those coordinate systems are shown in many texts [6, p. 66 8, p. 80]. The corresponding equations for fluids with variable density are also shown in numerous texts [6, p. 66 7 8, p. 80]. If we set /A = 0 in the Navier-Stokes equations, thus dropping the rightmost term, we find the Euler equation which is often used for three-dimensional flow where viscous effects are negligible. 

The Navier-Stokes equations are the differential momentum balances for a three-dimensional flow, subject to the assumptions that the flow is laminar and of a constant-density newtonian fluid and that the stress deformation behavior of such a fluid is analogous to the stress deformation behavior of a perfectly elastic isotropic solid. These equations are useful in setting up momentum balances for three-dimensional flows, particularly in cylindrical or spherical geometries. 

We consider Reynolds numbers not exceeding 400 this is the higher limit for which the flow is axisymmetric in the case of the rigid sphere and for which a bubble of air in water remains quasi-spherical. As the flow is considered axisymmetric, the Navier-Stokes equations can be... 

By measuring velocity of a spherical particle sinking in a liquid under gravity force the viscosity of the liquid can be found (the buoyancy effect should be taken into account). Note that in Section 7.3.3, using an electric field as an action force, the same Stokes law has been applied (with some precautions) to evaluation of velocity and mobility of spherical ions in isotropic liquids or nematic liquid crystals For large Reynolds numbers, Re = pv//ri>l the flow in no longer laminar and even becomes turbulent. Then, the convective term (vV)v should be added to the left part of the Navier-Stokes equation... 

For flow past a sphere the stream function ij/ can be used in the Navier-Stokes equation in spherical coordinates to obtain the equation for the stream function and the velocity distribution and the pressure distribution over the sphere. Then by integration over the whole sphere, the form drag, caused by the pressure distribution, and the skin friction or viscous drag, caused by the shear stress at the surface, can be summed to give the total drag. 

A more rigorous cell model, inasmuch as it involves a solution of the Navier-Stokes equation for creeping flow rather than a modification of that equation as in the case of Brinkman, is that of Happel (1958). In this case the basic cell is that of a single sphere surrounded by a concentric spherical envelope of fluid, the volume of which bears the same ratio to the volume of the cell as bed voidage does to unity. The crucial feature of this model is that the outer surface of the fluid envelope is frictionless (zero shear stress), so that it is often referred to as the free surface model. The solution is... 

When the bubble surface does not deform, it is relatively easy to solve the problem on a fixed grid. Balasubramanian and Lavery [41] used the fixed grid method to study the thermocapillary migration of a spherical bubble under microgravity for large Reynolds and Marangoni numbers. The problem they studied was a stationary bubble surrounded by the liquid with a steady state velocity field. The full Navier-Stokes equation was solved in a three-dimensional spherical (r,9,(j)) coordinate system. The origin of the coordinate is at the center of mass of the bubble. 

In order to calculate the friction coeflScient of a spherical colloid, the Navier-Stokes equations for liquids have to be solved, with the boundary condition that the liquid layer adjacent to the particle adheres to its surface, thus moving with the same velocity. The problem was solved long ago by Stokes. The result is the famous equation... 

The fluid velocity distribution given by Eqs. (93)-(96) are only valid for an isolated particle. However, there are a number of practically important situations, like the deep-bed filtration process, when the flow past an assembly of spheres (forming a porous mediiun) takes place. In this case, the flow field around a single sphere is influenced by the presence of other spheres. Various models that describe the flow field in the packed bed consisting of spheres are available. The sphere in cell models [81-83] assume that each sphere in the packed bed is surrounded by the spherical cavity filled with fluid. The size of the cavity is determined by the overall average porosity of the medium. The general solution of the Navier-Stokes equation for the stream function inside the cavity may be written as [7]... 

In order to formulate the flow equations for a fluid, for instance, for the gas in the cyclone or swirl tube, we must balance both mass and momentum. The mass balance leads to the equation of continuity the momentum balance to the Navier-Stokes equations for an incompressible Newtonian fluid. When balancing momentum, we have to balance the x-, y- and -momentum separately. The fluid viscosity plays the role of the diffusivity. Books on transport phenomena (e.g. Bird et ah, 2002 Slattery, 1999) will give the full flow equations both in Cartesian, cylindrical and spherical coordinates. 

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