Stokes Navier drag force

For the case of creeping flow, that is flow at very low velocities relative to the sphere, the drag force F on the particle was obtained in 1851 by Stokes(1) who solved the hydrodynamic equations of motion, the Navier-Stokes equations, to give ... 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

Exact results are known for the isothermal drag force in the two limits Kn. 0 and Kn. -> oo for Ma. <<1. For Kn. 0, Stokes law is readily derived from the Navier-Stokes equation with stick boundary conditions... 

Due to its great importance in reactor simulations, a brief survey of the main steps involved in Stokes solution of the Navier-Stokes equation for creeping motion about a smooth immersed rigid sphere is provided. The details of the derivation is not repeated in this book as this task is explained very well in many textbooks [14, 15, 88, 140]. Consider an incompressible creeping motion of a uniform flow of speed 1/ = v vj ej approaching a solid sphere of radius rp, as illustrated in Fig. 5.1. To describe the surface drag force in the direction of motion it is convenient... 

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

Dandy and Dwyer [30] computed numerically the three-dimensional flow around a sphere in shear flow from the continuity and Navier-Stokes equations. The sphere was not allowed to move or rotate. The drag, lift, and heat flux of the sphere was determined. The drag and lift forces were computed over the surface of the sphere from (5.28) and (5.33), respectively. They examined the two contributions to the lift force, the pressure contribution and the viscous contribution. While the viscous contribution always was positive, the pressure contribution would change sign over the surface of the sphere. The pressure... 

Employing singular perturbation methods to solve the Navier-Stokes equations, Saffman obtains, for the drag, torque about the sphere center, and lift force,... 

The term (-ur]lk) in Eq. 3.23 is the Darcy resistance term, and the term (rjW u) is the viscous resistance term the driving force is still considered to be the pressure gradient. When the permeability k is low, the Darcy resistance dominates the Navier-Stokes resistance, andEq. 3.23 reduces to Darcy s law. Therefore, the Brinkman equation has the advantage of considering both viscous drag along the walls and Darcy effects within the porous medium itself. In addition, because Brinkman s equation has second-order derivatives of u, it can satisfy no-slip conditions at solid surfaces bounding the porous material (e.g. the walls of a packed bed reactor), whereas Darcy s law cannot. In that sense, Brinkman s equation is more exact than Darcy s law. 

Suppose there are two canonical types of inertial flow around a particle with each its own fluid-particle interaction force, such as a steady-state drag and a history related force. Each of these forces is due to a particular fluid flow field around the particle in question. Unless the principle of separation of scales applies and/or inertial effects can safely be ignored, the Hnear addition of two flow fields each described by its own Navier—Stokes equation does not yield the total flow field, just because of the convective or inertial terms in the Navier—Stokes equation, e.g.. 

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