Stream function potential flow

Stream function for flow past cylinder Dimensionless radial component of electric potential in circular capillary, zFip/RT... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

Outside the jet and away from the boundaries of the workbench the flow will behave as if it is inviscid and hence potential flow is appropriate. Further, in the central region of the workbench we expect the airflow to be approximately two-dimensional, which has been confirmed by the above experimental investigations. In practice it is expected that the worker will be releasing contaminant in this region and hence the assumption of two-dimensional flow" appears to be sound. Under these assumptions the nondimensional stream function F satisfies Laplace s equation, i.e.. 

Example 2.7. To show what form the energy equation takes for a two-phase system, consider the CSTR process shown in Fig. 2.6. Both a liquid product stream f and a vapor product stream F (volumetric flow) are withdrawn from the vessel. The pressure in the reactor is P. Vapor and liquid volumes are and V. The density and temperature of the vapor phase are and L. The mole fraction of A in the vapor is y. If the phases are in thermal equilibrium, the vapor and liquid temperatures are equal (T = T ). If the phases are in phase equilihrium, the liquid and vapor compositions are related by Raoult s law, a relative volatility relationship or some other vapor-liquid equilibrium relationship (see Sec. 2.2.6). The enthalpy of the vapor phase H (Btu/lb or cal/g) is a function of composition y, temperature T , and pressure P. Neglecting kinetic-energy and potential-energy terms and the work term,... 

For CO 0, Eq. (11-7) reduces to the stream function for steady creeping flow past a rigid sphere, i.e., Eq. (3-7) with k = co. The parameter 3 may be regarded as a characteristic length scale for diffusion of vorticity generated at the particle surface into the surrounding fluid. When co is very large, 3 is small, and the flow can be considered irrotational except in the immediate vicinity of the particle. In the limit co go, Eq. (11-7) reduces to Eq. (1-29), the result for potential flow past a stationary sphere. 

The motion of a sphere moving through a stagnant incompressible fluid is equivalent to the uniform fluid flow about a fixed sphere. For axisymmetrical potential flow the velocity components of the fluid can be obtained by use of the Stokes stream function as explained by [26, 170] ... 

The potential-flow solution for streaming motion past a circular cylinder was obtained earlier and given in terms ofthe streamfunctionin(10-17). To calculate the pressure gradient in the boundary layer, we first determine the tangential velocity function, ue, as defined in (10-37) ... 

Complex numbers owe their origin to the quest for the square root of a negative number. Thus the so-called imaginary number i = is a fundamental element of complex numbers, written as z = X + iy, in which x is the real part and y is the imaginary part. Although real numbers quantify physical quantities, complex numbers provide very convenient representations of many physical phenomena. In quantum mechanics, the wave function is a complex function. Two-dimensional, incompressible, irrotational flows are represented by a complex flow potential, w = 9 h- t /, with 9, the velocity potential, as the real part, and /, the stream function, as the imaginary part. 

Let and be the potential and the stream function of the potential flow of a fluid. Since and are determined up to additive constants, we can assume... 

The velocity potential and the stream function for a noncirculatory flow of ideal fluid past an elliptic cylinder has the form [26]... 

Dialysis is a diffusion-based separation process that uses a semipermeable membrane to separate species by vittue of their different mobilities in the membrane. A feed solution, containing the solutes to he separated, flows ou one side of the membrane while a solvent stream, die dialysate, flows on die other side (Fig. 21. -1). Solute transport across the membrane occurs by diffusion driven by the difference in solme chemical potential between the two membrane-solution interfaces. In practical dialysis devices, no obligatory transmembrane hydraulic pressure may add an additional component of convective transport. Convective transport also may occur if one stream, usually the feed, is highly concentrated, thus giving rise to a transmembrane osmotic gradient down which solvent will flow. In such circumstances, the description of solute transport becomes more complex since it must incorporate some function of die trans-membrane fluid velocity. 

We consider a close-separation, ideal cascade Wiose external streams have molar flow rates Xif (positive if a product, negative if a feed), and compositions X/t expressed as mole fraction. Let us look for a function of composition x ), to be called the separation potential, with the property that the sum over all external streams, to be called the separative capacity D,... 

Particles can be deposited on the fs.fz. and on the stagnant cup. Let us estimate that part of collision efficiency connected with the deposition on the fs.fz. It means that the coordinates n - 4, 3 + 3p) must be substituted into the equation of the stream function for the potential flow. Note that the substitution of the coordinates (n/ 2, aj, + a ) yields Sutherland s equation (10.11). The substitution of the coordinates (tc - P, a, + 3p) leads to the equation. 

If we know the potential function which describes some flow, we can compute the stream function, and conversely. The calculation is based on the following general integration property of partial derivatiyes. It A = A B, C), where A, B, and C are any mathematical functions, then... 

An interesting property of the stream function and the velocity potential is that if for a given flow y) and if/ = f ix, y), then there is another... 

The discussion has all been for the application of the stream function to potential flows. But sometimes it can be used to advantage for flows which are not irrotational and for which a potential function cannot exist (see Probs. 10.10 and 11.4). This is possible because the stream function is used in these cases simply as a way of combining the mass balance with the other pertinent equations. 

The velocity-potential stream-function methods shown in the preceding sections allow us to calculate the flow velocity and direction at any point in a two-dimensional, perfect-fluid, irrotational flow. Sometimes this is all the... 

Map out the following potential flows. In each case verify that Laplace s equation is satisfied, calculate the equation of the stream function, and indicate to what physical situation these flows might correspond. 

Although thej Stream function is used most often with the-potential function, it can be used for viscous flows for which no potential function exists. For example, the laminar flow of a viscous fluid along a sloping flat plate is given by the equation =7 [(pgd cos /3)/(2/l)][1 - (y/4) ], as described by Bird et al. [3, p. 39]. See Fig. 110.22. Calculate the stream function for this flow. Show that this flow cannot be represented by any potential function. 

Potential Flow around a Gas Bubble Via the Stream Function. The same axisymmetric flow problem in spherical coordinates is solved in terms of the stream function All potential flow solutions yield an intricate network of equipotentials and streamlines that intersect at right angles. For two-dimensional ideal flow around a bubble, the velocity profile in the preceding section was calculated from the gradient of the scalar velocity potential to ensure no vorticity ... 

Hence, two-dimensional axisymmetric potential flow in spherical coordinates is described by = 0 for the scalar velocity potential and = 0 for the stream function. Recall that two-dimensional axisymmetric creeping viscous flow in spherical coordinates is described by E E ir) = 0. This implies that potential flow solutions represent a subset of creeping viscous flow solutions for two-dimensional axisymmetric problems in spherical coordinates. Also, recall from the boundary condition far from submerged objects that sin 0 is the appropriate Legendre polynomial for the E operator in spherical coordinates. The methodology presented on pages 186 through 188 is employed to postulate the functional form for xlr. 

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) ... 

Except for the difference between sin 6 and cos0, notice the similarity between this form of Laplace s equation and (8-262) for the scalar velocity potential 4>. In fact, the general solution for the radial part of the stream function is exactly the same as that for from the preceding section. This is expected because and f satisfy the same equation for two-dimensional ideal flows that lack an axis of symmetry. The general solution for is... 

Axisymmetric irrotational (i.e., potential) flow of an incompressible ideal fluid past a stationary gas bubble exhibits no vorticity. Hence, V x v = 0. This problem can be solved using the stream fnnction approach rather than the scalar velocity potential method. Develop the appropriate equation that governs the solution to the stream function f for two-dimensional axisymmetric potential flow in spherical coordinates. Which Legendre polynomial describes the angular dependence of the stream function ... 

Three cases will be considered in this section inviscid flow, potential flow, and creeping flow. The fourth case, for boundary-layer flow, will be considered in Section 3.10. The solution of these equations may be simplified by using a stream function ip x, y) and/or a velocity potential (p x, y) rather than the terms of the velocity components v, Vy, and v. ... 

This potential exists only for a flow with zero angular velocity, or irrotationality. This type of flow of an.ideal or inviscid fluid (p = constant, p. = 0) is called potential flow. Additionally, the velocity potential exists for three-dimensional flows, whereas the stream function does not. 

In potential flow, the stream function and the potential function are used to represent the flow in the main body of the fluid. These ideal fluid solutions do not satisfy the condition that = Vy = 0 on the wall surface. Near the wall we have viscous drag and we use boundary-layer theory where we obtain approximate solutions for the velocity profiles in this thin. boundary layer taking into account viscosity. This is discussed in Section 3.10. Then we splice this solution onto the ideal flow solution that describes flow outside the boundary layer. 

Stream Function and Potential Function. A liquid is flowing parallel to the x... 

Instead of the potential < ), it is possible to introduce the flow-stream function i i for a two-dimensional flow. The stream lines (v / = constant) and the equipotential lines are perpendicular to each other. To express this, the following Cauchy-Rieman differential equations can be used ... 

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