Couette flow equation

For extremely narrow openings (cracks) with deep flow paths (such as mortar joints and tight-fitting components) the flow is laminar and the flow rate, Q (mVs), can be described by the Couette flow equation - ... 

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

Cottrell equation Cottrell unit Couchman equation Couette flow Couette viscometers Cough drops Coughlozenges... 

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

The attractive feature of LADM Is that once the fluid structure Is known (e.g., by solution of the YBG equations given In the previous section or by a computer simulation) then theoretical or empirical formulas for the transport coefficients of homogeneous fluids can be used to predict flow and transport In Inhomogeneous fluid. For diffusion and Couette flow In planar pores LADM turns out to be a surprisingly good approximation, as will be shown In a later section. 

For the steady, planar Couette flow to be examined In a later section, the momentum balance equation yields... 

Couette Flow Simulation. MD typically simulate systems at thermodynamic equilibrium. For the simulation of systems undergoing flow various methods of nonequilibrium MD have been developed (Ifl iZ.). In all of these methods the viscosity Is calculated directly from the constitutive equation. 

However, several flow transition regimes have been identified between laminar and fully turbulent flow. The cessation of laminar Couette flow is marked by the appearance of Taylor vortices in the gap between the two cylinders. For the case of stationary outer cylinder, the critical angular velocity, C0crit> of inner cylinder at which these flow instabilities first appear can be estimated by using the following equations [102] ... 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

As fj = a I we can integrate this equation for a capillary and for the Couette flow... 

The field of transport phenomena is the basis of modeling in polymer processing. This chapter presents the derivation of the balance equations and combines them with constitutive models to allow modeling of polymer processes. The chapter also presents ways to simplify the complex equations in order to model basic systems such as flow in a tube or Hagen-Poiseulle flow, pressure flow between parallel plates, flow between two rotating concentric cylinders or Couette flow, and many more. These simple systems, or combinations of them, can be used to model actual systems in order to gain insight into the processes, and predict pressures, flow rates, rates of deformation, etc. 

Write a program to solve by means of RFM the equation of motion and using the velocity field, calculate viscous dissipation and solve for the energy equation. Neglect inertial and convective effects. Consider T0=200°C, Ti=150°C, /x=24000 Pa-s, k=0.267 W/mK, i o=0.1 m, i i=0.13 m, k=0.769, cc=0.496 rad/s. Compare the numerical results with the analytical solution. Hint The couette flow is constant along the angular direction, hence, it is no necessary to use the whole domain. 

In situations where separations comparable to the collector radius (h a) need to be considered, the particle no longer experiences Couette flow. However, Equation (4) correctly predicts that oa approaches se at large separations, whereas at small separations, where the distinction between Ob and b is most important, the conditions of its derivation are practically satisfied. 

Simple shear flow is the characteristic of Couette flow, and no pressure gradient exists in the direction of motion. Since the pressure gradient term is also zero, the governing equation reduces to... 

Shear stress The shear stress in the four mentioned geometries can be determined by measuring the moment M (in Nm), or pressure AP (in N/m2) during flow. For the Couette flow and the cone and plate flow the relationships for shear stress and shear rate are easy to handle in order to determine the viscosity. For the parallel plates flow and Poisseuile flow, however, more effort is needed to determine the shear stress at the edge of the plate, qR, in parallel plates flow or the shear rate at the wall, qw, in Poisseuile flow. In Table 15.1 equations for shear stresses and shear rates are shown. 

Figure 4.2 shows a Couette flow of a fluid of constant density p, viscosity p, and thermal conductivity k between parallel plates. The bottom plate is at rest, while the top plate is moving at a constant velocity Vj. The upper and lower plates are at uniform temperatures Tx and r2, respectively. The equation of motion for fully developed flow in the x-direction is... 

The simplest type of flow of a medium that yields itself to an analytical description within the framework of the precise hydrodynamical equations of viscous liquids (Navier-Stocks equations) is the Couette flow. This flow occurs under the impact of tangential stresses generated in a viscous liquid by a solid surface moving in it. The magnitude of the force that has to be applied to this surface to securse its movement in the viscous medium characterizes the tangential stresses and the velocity of its movement — the shear velocity. 

One can also show that all one dimensional time-dependent perturbations of a steady multifluid flow exist for all times, and stay bounded—as in the case of one fluid. Similar results can be obtained for axisymmetric Poiseuille flows of several fluids. A similar study is also made for plane Poiseuille or Couette flows of several fluids having a Phan-Thien-Tanner constitutive equation [50]. 

Following [47] we restrict now the study of stability to Oldroyd models (where di = 0). It is easy to check that the steady Couette flow, solution of the steady equations corresponding to system (16)-(17), is given by... 

The Couette flow analysis uses the constitutive equation as its basis. The total shear stress in the boundary layer is written as... 

Because the coefficients of the invariants are not specified, the use of the Reiner-Rivlin equation is rather arbitrary. On the other hand, the time-dependent characteristics of viscoelastic fluids are not described by this equation. Neither are normal stresses in Couette flow correctly described by Eq. (13.3). 

It is also possible to calculate the shear viscosities and the twist viscosities by applying the SLLOD equations of motion for planar Couette flow, Eq. (3.9). If we have a velocity field in the x-direction that varies linearly in the z-direction the velocity gradient becomes Vu=ye ej, see Fig. 3. Introducing a director based coordinate system (Cj, C2, 63) where the director points in the e3-direction and the angle between the director and the stream lines is equal to 0, gives the following expression for the strain rate in the director based coordinate system. 

The SLLOD equations of motion presented in Eqs. [123] are for the specific case of planar Couette flow. It is interesting to consider how one could write a version of Eqs. [123] for a general flow. One way to do this is introduce a general strain tensor, denoted by Vu. For the case of planar Couette flow, Vu = j iy in dyadic form, where 1 and j denote the unit vector in the x and y directions, respectively. The matrix representation is... 

However, it is clear that for a general tensor Vu, trajectory analysis based on the SLLOD dynamics in Eqs. [129] will yield incorrect results. Equation [132] has an extra term in the force, which is equivalent to saying that the momenta in Eqs. [129] are not peculiar with respect to a general flow (indeed, Eqs. [129] yield peculiar velocities for the case of planar Couette flow), and therefore the flow profile produced will not be q Vu as expected. Equations [129] also lead to problems when one is considering definitions of pressure... 

Following Eqs. [202], the equation of motion for the particle coordinate in a planar Couette flow can be written as follows ... 

As we have already demonstrated, the SLLOD equations have been highly successful for studying moderate shear rate systems. To review, the equations of motion for planar Couette flow, with Nose-Hoover thermostats, - " are ... 

Alternatively, one could use SLLOD equations to do direct simulations, such as shear a system under planar Couette flow and measure the shear stress. As we have already discussed, this approach has been used successfully to calculate a host of transport properties. It is important to remember, however, that direct simulation is often unable to simulate realistic materials at experimentally accessible shear rates. At low shear rates, the nonequilibrium response becomes small compared to the magnitude of the equilibrium fluctuations that naturally arise. The extremely small signal-to-noise ratio would demand prohibitively long simulations before any meaningful answers could be obtained. 

Equation (3-66) is the balance between centrilugal forces and the radial pressure gradient that is responsible for the fact that ur = 0. Thus, in the Couette flow problem the acceleration associated with curved fluid pathlines does not necessarily lead to a radial flow, but may be balanced by a radial pressure gradient. 

It should be noted that these equations are completely characterized by the two dimensionless parameters e and f22/f2i. In particular, the Reynolds number, Re = ucic/v = a2i2 /v, does not appear in spite of the fact that it would have appeared in the full equations, (3—56)—(3—58), if these had been nondimensionalized in the same way. From a mathematical point of view, this is because the viscous terms turned out to be identically equal to zero in (3-56), whereas the inertia and pressure terms were zero in (3-57) - compare (3-66) and (3-67). Thus the form of the velocity and pressure fields in the Couette flow problems is completely independent of Re. In this sense, the Couette flow problem is very similar to a unidirectional flow. 

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