Streamline unstable

If a turbulent fluid passes into a pipe so that the Reynolds number there is less than 2000, the flow pattern will change and the fluid will become streamline at some distance from the point of entry. On the other hand, if tire fluid is initially streamline (Re < 2000), the diameter of the pipe can be gradually increased so that the Reynolds number exceeds 2000 and yet streamline flow will persist in the absence of any disturbance. Unstable streamline flow has been obtained in this manner at Reynolds numbers as high as 40,000. The initiation of turbulence requires a small force at right angles to the flow to promote the formation of eddies. 

As the fluid flows over the forward part of the sphere, the velocity increases because the available flow area decreases, and the pressure decreases as a result of the conservation of energy. Conversely, as the fluid flows around the back side of the body, the velocity decreases and the pressure increases. This is not unlike the flow in a diffuser or a converging-diverging duct. The flow behind the sphere into an adverse pressure gradient is inherently unstable, so as the velocity (and lVRe) increase it becomes more difficult for the streamlines to follow the contour of the body, and they eventually break away from the surface. This condition is called separation, although it is the smooth streamline that is separating from the surface, not the fluid itself. When separation occurs eddies or vortices form behind the body as illustrated in Fig. 11-1 and form a wake behind the sphere. 

When a fluid is passed downwards through a bed of solids, no relative movement between the particles takes place, unless the initial orientation of the particles is unstable, and where the flow is streamline, the pressure drop across the bed is directly proportional to the rate of flow, although at higher rates the pressure drop rises more rapidly. The pressure drop under these conditions may be obtained using the equations in Chapter 4. 

In the case of real eigenvalues, Ai 2 = A, x is called a hyperbolic fixed point, corresponding to a saddle point of the streamfunction, and it is of unstable character. It is located at the intersection of two special streamlines, called separatrices, which are the stable and unstable manifolds of the hyperbolic fixed point. These lines are defined as the set of points that approach the fixed point in the limits t —> Too, respectively. The motion around a hyperbolic point can be obtained as a solution of (2.40)... 

A chaotic flow produces either transverse homocHnic or transverse heterocHnic intersections, and/or is able to stretch and fold material in such a way that it produces what is called a horseshoe map, and/or has positive Liapunov exponents. These definitions are not equivalent to each other, and their interrelations have been discussed by Doherty and Ottino [63]. The time-periodic perturbation of homoclinic and heteroclinic orbits can create chaotic flows. In bounded fluid flows, which are encountered in mixing tanks, the homoclinic and heteroclinic orbits are separate streamlines in an unperturbed system. These streamhnes prevent fluid flux from one region of the domain to the other, thereby severely limiting mixing. These separate streamlines generate stable and unstable manifolds upon perturbation, which in turn dictate the mass and energy transports in the system [64-66]. 

Many microfluidic geometries are not as simple as those described above, and the mixed flow type within complex geometries can lead to other interesting flow behaviors, especially at elevated De values. For example, contraction flows of highly elastic non-Newtonian fluids can be unstable due to the interaction of elastic stresses with streamline curvature in the vortices upstream of the contraction. The upstream vortices grow relative to the New-... 

The finite volume method, which returns to the balance equation form of the equations, where one level of spatial derivatives are removed is the method of choice always for the pressure equation and nearly always for the saturation equation. Commercial reservoir simulators are, with the exception of streamline simulators, entirely based on the finite volume method. See [11] for some background on the finite volume method, and [26] for an introduction to the streamline method. The robustness of the finite volume method, as used in oil reservoir simulation, is partly due to the diffusive nature of the numerical error, known as numerical diffusion, that arises from upwind difference methods. An interesting research problem would be to analyse the essential role that numerical diffusion might play in the actual physical modelling process particularly in situations with unstable flow. In the natural formulation, where the character of the problem is not clear, and special methods applicable to hyperbolic, or near hyperbolic problems are not applicable, the finite volume method, in the opinion of the author, is the most trustworthy approach. 

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