Streamfunction

This corresponds to a Hamiltonian system which is characterized by a weak oscillatory perturbation of the SHV streamfunction T r, ) —> Tfr, Q + HP, (r, ( ) x sin(fEt). The equations of fluid motion (4.4.4) are used to compute the inertial and viscous forces on particles placed in the flow. Newton s law of motion is then... 

For steady-state (no time variation) two-dimensional flows, the notion of a streamfunction has great utility. The stream function is derived so as to satisfy the continuity equation exactly. In cylindrical coordinates, there are two two-dimensional situations that are worthwhile to investigate the r-z plane, called axisymmetric coordinates, and the r-0 plane, called polar coordinates. 

The physical meaning of the stream function is that fluid flows along streamlines, which are lines of constant stream function. Since, by definition, flow cannot cross streamlines, the mass flow rate between any two streamlines must be constant. Furthermore the magnitude of the flow rate between two streamlines is determined by the difference in the values of the streamfunction on the two streamlines. 

Fig. 3.1 Illustration of the mass flow and streamfunction on a two-dimensional axisymmetric area element.

It is seen that the relationship between streamfunction and vorticity is described by a Poisson equation. Depending on the particular coordinates, the operator on the right-hand side may reduce to a Laplacian. In this case of axisymmetric flow, the operator is not a Lapla-cian. 

Deriving the axisymmetric stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.60, and 3.60), but considering flow only in the z-r plane. In general, there may be a circumferential velocity component ui, but there cannot be variations of any variable in the circumferential direction 0. The derivation depends on two principal conjectures. First, the velocity field is presumed to be described in terms of a streamfunction that has the separable form... 

We note, for future reference, that the function f plays a role in the present problem at 0(a/R) that is analogous (in the plane) to the streamfunction for a general 2D flow. The concept of a streamfunction will be discussed in detail in Chap. 7. 

The scalar function f that appears in (7-38) is known as the streamfunction. The physical significance of if is best seen through the relationship (7-37). In particular, if we substitute (7-38) into (7-37), we obtain... 

To demonstrate that such a sequence of eddies does exist, we can show that there is an infinite sequence of dividing streamlines, [Pg.457]

As demonstrated by Moffatt, this streamfunction has infinitely many zeroes as r approaches zero, namely... 

We saw in Section C that the creeping-motion and continuity equations for axisymmetric, incompressible flow can be reduced to the single fourth-order PDE for the streamfunction,... 

The nonzero-velocity components, expressed in terms of the streamfunction, are... 

One extremely useful result can be proven that applies to any problem in which the streamfunction is expressed in the general form (7-131). This result can be formalized in the form of a theorem ... 

The asymptotic form of the streamfunction for r oo can be obtained from the uniform... 

Thus, regardless of the details of the body geometry, or the form of the boundary conditions at the body surface, the streamfunction must exhibit the asymptotic form (7 147) if there is a uniform streaming flow at large distances from the body. Comparing (7 147) and (7-131), we see that many of the coefficients A and B in the general axisymmetric solution must be zero for this case. In particular, the asymptotic condition (7-147) requires that... 

Now, we have expressed the general streamfunction, (7-149), and the disturbance flow contribution in (7-150) and (7-151), in terms of spherical coordinates. However, we have not yet specified a body shape. Thus the linear decrease of the disturbance flow with distance from the body must clearly represent a property of creeping-flows that has nothing to do with specific coordinate systems. Indeed, this is the case, and the velocity field (7-151) plays a very special and fundamental role in creeping-flow theory. It is commonly known as the Stokeslet velocity field and represents the motion induced in a fluid at Re = 0 by a point force at the origin (expressed here in spherical coordinates).17 We shall see later that the Stokeslet solution plays an important role in many aspects of creeping-flow theory. 

Figure 7-12. The streamlines and contours of constant vorticity for uniform streaming flow past a solid sphere (Stokes problem). The streamfunction and vorticity values are calculated from Eqs. (7-158) and (7-162). Contour values plotted for the streamfunction are in increments of 1/16, starting from zero at the sphere surface, whereas the vorticity is plotted at equal increments equal to 0.04125.

To do this, we must first determine the form of streamfunction at large distances from the sphere when the velocity field has the asymptotic form (7 176). Specifically, if we transform (7—176) to spherical coordinates by using the general relationships... 

From the form (7-181) and the definitions (7-102) of the streamfunction in terms of the axisymmetric velocity components ur and ug, we can easily obtain the asymptotic form for xj/. The result is... 

In summary then, the leading-order problem is just the translation of a spherical drop through a quiescent fluid. The solution of this problem is straightforward and can again be approached by means of the eigenfunction expansion for the Stokes equations in spherical coordinates that was used in section F to solve Stokes problem. Because the flow both inside and outside the drop will be axisymmetric, we can employ the equations of motion and continuity, (7-198) and (7-199), in terms of the streamfunctions f<(>> and < l)), that is,... 

Because the problem is axisymmetric about the z axis, we can solve the creeping-flow equation in terms of the streamfunction Hence, after applying the asymptotic condition... 

Several brief comments should be made regarding the solution just outlined. First, the derivation assumes, implicitly, that / / 0 remembering the far-field condition (7-238) and the form of the streamfunction (7-244), the reader may well wonder about the efficacy of trying to use this solution to deduce conditions when U = 0. The simplest response is... 

To calculate this angle, we first express the right-hand side of (7-290) in terms of the streamfunction that was previously calculated ... 

Thus we begin by considering the full Navier-Stokes equation expressed in terms of the streamfunction characteristic velocity and the sphere radius a as a characteristic length scale. Using spherical coordinates, with ij = cos 9, this equation is... 

Equation (9-203) can be solved quite easily. The physical significance of (9-203) is that 6 o must be constant along lines (or surfaces) parallel to u. In other words, the projection of V6>o in the direction of u is zero. We may also note that the streamfunction [Pg.645]

Streamline implies the existence of a streamfunction, which we have seen to he true only for axisymmetric and 2D flows. In three dimensions, recirculating flows are associated with regions of closed stream surfaces, or pathlines. 

A heated circular cylinder is suspended in a fluid that is undergoing a simple shear flow. Assume that the cylinder does not rotate so that the streamfunction representing flow in the (inner) region near the cylinder is... 

The condition (10 16) is just the kinematic condition (10 13) expressed in terms of the streamfunction, whereas (10 15) requires that the velocity field approach a uniform streaming motion at large distances from the cylinder. Equation (10 14), subject to (10-15) and (10 16), is solved easily by means of separation of variables, or other standard transform methods, with the resulting solution... 

The boundary-layer equations are third order with respect to 7 and first order with respect to x [this can be seen clearly if (10-40) and (10 41) are expressed in terms of the streamfunction]. Hence, to specify completely the velocity profiles in the boundary layer, we require one additional boundary condition in 7 and an initial profile at the leading edge of the boundary layer (x is usually defined so that this point corresponds to x = 0). In addition, the potential-flow equations are second order and thus require at least one boundary condition in addition to (10 12). In Section A, it was suggested that an appropriate condition was... 

Problem 10-4. Boundary Layer on a Flat Plate in an Accelerating Flow, Uoo = Ax. Consider flow past a flat plate in the throat of a 2D channel as depicted in the figure. If the free-stream velocity is given by = ax, where x is the distance from the leading edge and X is a constant, show that the flow in the boundary layer on the plate is governed by an ODE. Solve for the streamfunction numerically. How does the boundary-layer thickness grow with x How does the shear stress vary with x ... 

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