Velocity planar flow

Illustration Common flow types. Experimental studies of drop breakup have been mainly confined to linear, planar flows. All linear flows in 2D are encapsulated by the general velocity field equations... 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

This expression is known as the fiber density continuity equation. It states that a fiber, which moves out of one angular position must move into a neighboring one, conserving the total number of fibers. If the initial distribution function, fio, is known, an expression for the angular velocity of the particle, , must be found to solve for eqn. (8.152) and determine how the distribution function varies in time. The motion of the fibers can often be described by the motion of a rigid single rod in a planar flow. 

Equation (4.4) is an unwitting statement that the velocity (l/t) of a sol s planar flow is inversely proportional to t. A capillary viscometer is designed to maintain r, l, V, and t (1 atm) constant, so that ti, is direcdy proportional to tt. The generalized equation for a single measurement (single-point viscometry) is... 

Problem 1.3 (Worked Example) Consider a planar flow with the following velocity gradient tensor in the x y plane ... 

MIXED FLOW. Other flows with extensional components also have coil-stretch transitions. The smaller the extensional component is relative to the overall strain rate, the higher the overall strain rate at which the transition takes place (Giesekus 1962, 1966) A steady planar flow, for example, can be considered to be a mixture of a shearing and an extensional flow in such a mixed flow, the velocity gradient tensor, Vv, can be expressed as (Fuller and Leal 1980, 1981)... 

Figure 3.17 Birefringence as a function of the eigenvalue of the velocity gradient tensor, G, for planar flows generated in a four-roll mill, for dilute solutions of polystyrenes of three different molecular weights in polychlorinated biphenyl solvent. Here G is the strain rate and a the flow type parameter. For planar extension, a — 1 and G = is the extension rate for simple shear, a = 0 and G = y is the shear rate. The different symbols correspond to a values of 1.0 (0)> 0.8 (A), 0.5 (-1-), and 0.25 (diamonds). The curves are theoretical predictions from the FENE dumbbell model, including conformation-dependent drag (discussed in Section 3.6.2.2.2). (From Fuller and Leal 1980, reprinted with permission from Steinkopff Publishers.)...

Hence, both velocity components for two-dimensional planar flow in rectangular coordinates are related to the stream function in the following manner. Either... 

For two-dimensional planar flow with =0 and no dependence of and Vy on spatial coordinate z, the velocity vector is... 

Potential Flow Transverse to a Long Cylinder Via the Scalar Velocity Potential. The same methodology from earlier sections is employed here when a long cylindrical object of radius R is placed within the flow field of an incompressible ideal fluid. The presence of the cylinder induces Vr and vg within its vicinity, but there is no axis of symmetry. The scalar velocity potential for this two-dimensional planar flow problem in cylindrical coordinates must satisfy Laplace s equation in the following form ... 

Potential Flow Transverse to a Long Cylinder Via the Stream Function. For two-dimensional planar flow in cylindrical coordinates, the radial and polar velocity components are related to the stream function rfr via the following expressions ... 

In summary, Laplace s equation must be satisfied by the scalar velocity potential and the stream function for all two-dimensional planar flows that lack an axis of symmetry. The Laplacian operator is replaced by the operator to calculate the stream function for two-dimensional axisymmetric flows. For potential flow transverse to a long cylinder, vector algebra is required to determine the functional form of the stream function far from the submerged object. This is accomplished from a consideration of Vr and vg via equation (8-255) ... 

The characteristic scale of velocity variation (11.81) is equal to R. It means that for any region whose size is much smaller than R, the flow can be approximated as a quasi-planar flow in the meridian plane, formed by superposition of uniform flow and simple shear flow. The uniform flow induces the force Fs on the drop S2, while the shear flow produces torque Tj. At a sufficient distance from the drop S2 we have... 

Here a = — 9 Woo /9y is the potential velocity gradient in the oxidizer stream, 7 = 0 for planar flow, andy = 1 for axially symmetric flow. Because x (Z) is the... 

Abstract. In this paper, the permeability of ordered fibrous porous media for normal flows is predicted theoretically and numerically. Moreover, microscopic velocity profiles in the unit cell are investigated in detail for normal flows. Porous material is represented by a unit cell which is assumed to be repeated throughout the media and ID fibers are modeled. Fibers are presented as cylinders with the same radii. Planar flow that perpendicular to the axes of cylinders is considered in this paper. All numerical calculations are performed using Gerris program [6]. The quantitative comparison of numerical and theoretical results of computation of the permeability of ordered fibrous media is reasonably good and is about 10-15%. 

We discuss streamline traeing and volume flow rate computations in planar flow liquids and gases, however, are both allowed, and a heterogeneous, anisotropic formation is permitted. Let us consider the Darcy velocity... 

So far we have treated two-dimensional planar flows. However, many three-dimensional problems are also amenable to analytical solution. To proceed, we introduce the notion of the point spherical source. Actually, the concept is best taught through global mass conservation considerations. We consider two-dimensional flows first. First, the radial Darcy velocity is proportional to dp/dr. This, times the area 2cr in planar problems, must be constant hence, in such flows, dp/dr goes like 1/r, which on integration leads to the expected logarithmic pressure. In three dimensions, dp/dr x 4 7t r must remain constant thus, dp/dr goes like l/i, so that p(r) varies like 1/r. This describes the point spherical source. We could also have started more formally with the spherically symmetric form of Laplace s equation,... 

Here g is the gravity vector and tu is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integr on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors Vi and V9, the momentum equation becomes... 

A useful simphfication of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m through a single planar entrance and a single planar exit (Fig. 6-4), to whi(m the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tu is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + pgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. 

A situation which is frequently encountered in tire production of microelectronic devices is when vapour deposition must be made into a re-entrant cavity in an otherwise planar surface. Clearly, the gas velocity of the major transporting gas must be reduced in the gas phase entering the cavity, and transport down tire cavity will be mainly by diffusion. If the mainstream gas velocity is high, there exists the possibility of turbulent flow at tire mouth of tire cavity, but since this is rare in vapour deposition processes, the assumption that the gas widrin dre cavity is stagnant is a good approximation. The appropriate solution of dre diffusion equation for the steady-state transport of material tlrrough the stagnant layer in dre cavity is... 

To permit a more general discussion, we can replace the snowplow with a piston, and replace the snow with any fluid (Fig. 2,3), We consider the example shown in a reference frame in which the undisturbed fluid has zero velocity. When the piston moves, it applies a planar stress, a, to the fluid. For a non-viscous, hydrodynamic fluid, the stress is numerically equal to the pressure, P, The pressure induces a shock discontinuity, denoted by which propagates through the fluid with velocity U. The velocity u of the piston, and the shocked material carried with it (with respect to the stationary frame of reference), is called the particle velocity, since that would be the velocity of a particle caught up in the flow, or of a particle of the fluid. 

Forced-flow development enables the mobile phase velocity to be optimized without regard to the deficiencies of a capillary controlled flow system [34,35). In rotational planar chromatography, centrifugal force, generated by spinning the sorbent layer about a central axis, is used to drive the solvent... 

Simple shear (also known as planar Couette flow) is achieved when fluid is contained between two plane parallel plates in relative in-plane motion. If the velocity direction is taken to be x, one has x = y, all other xa 3 zero and... 

---