Stream function conditions

The boundary conditions for the stream-function-vorticity system requires specifying the stream function on all the boundaries. This is usually straightforward for known inflow and outflow conditions and solid walls. The vorticity boundary conditions comes from evaluating Eq. 3.281 on the boundary. Along the boundary, which usually corresponds with one of the coordinate directions, one of the terms in Eq. 3.281 (i.e., the one in which the derivatives align with the boundary) can be evaluated explicitly since the stream function is already specified. Thus the boundary conditions becomes a relationship between the boundary vorticity and a boundary-normal second derivative of stream function. For example, consider the natural convection in a long horizontal tube. Here, since there is no inflow or outflow, the stream function is simply zero all around the tube wall. Thus the vorticity boundary conditions are... 

The boundary conditions provide a tight coupling between the vorticity and stream-function fields. Also velocities still appear in the convective terms. Given the stream-function field, velocity is evaluated from the definition of stream function. That is, velocity is computed from stream-function derivatives. 

Consider the condition, which determines the velocity of the curved flame front propagation in the channel. Inside the stagnation zone filled by combustion products the pressure is constant and is equal to the value at infinity (when x = oo). Because of Bernoulli s integral along the streamline restricting the stagnation zone, the gas motion velocity remains unchanged. Since at x = oo the flow is plane-parallel (ptJO = const, v — 0), distributions of velocity u and of the stream function are associated with the vorticity distribution ... 

R is the radius of the conduits n the velocity at the outlet of the conduit, i.e., the impinging velocity. The velocity components in terms of stream function were given by Eq. (1.26) while the conditions of irrotational flow were determined by Eq. (1.21). 

In the 1960s, the start of application of computers to the practice of marine research gave a pulse to the development of numerical diagnostic hydrodynamic models [33]. In them, the SLE (or the integral stream function) field is calculated from the three-dimensional density field in the equation of potential vorticity balance over the entire water column from the surface to the bottom. The iterative computational procedure is repeated until a stationary condition of the SLE (or the integral stream function) is reached at the specified fixed density field. Then, from equations of momentum balance, horizontal components of the current vector are obtained, while the continuity equation provides the calculations of the vertical component. The advantage of this approach is related to the absence of the problem of the choice of the zero surface and to the account for the coupled effect of the baroclinicity of... 

In incompressible flow with constant properties and no body forces, the dynamics are independent of the thermodynamics. Once the kinematic flow field is described by the stream function vj/, any number of temperature distributions may be solved with different thermal boundaiy conditions. 

Assuming die flow remains laminar and has a boundary layer-like characteristic, write down the governing equations together with the boundary and initial conditions. If the y coordinate is replaced by the stream function derived by ... 

In terms of the stream function, the boundary conditions on velocity are ... 

At the inflow boundary and on top of the computational domain, analytic solution for the disturbance velocity was used in accordance with Eqn. (2.7.1) and (2.7.2). On the flat plate, the no-slip condition simultaneously provides a Dirichlet boundary condition for the stream function and the wall vorticity at every instant of time. 

Fig. 10. Some monthly averaged observed distributions of atmospheric methane from the SAMS satellite, Jones and Pyle, near solstice conditions, compared to the model distribution for July by Solomon and Garcia. Light dashed arrows indicate the residual Eulerian stream function, showing the advection pattern.

The second firee surface condition (59b) expresses also that Zg is a stream fimction in a planar flow, the stream function

[Pg.314]

A solution exists when the differential equation (3.181) and its associated boundary conditions can be fulfilled with this statement. In order to show that this is applicable, we will form a stream function... 

In place of the partial differential equations (3.313) to (3.315), two ordinary non-linear differential equations appear. The continuity equation is no longer required because it is fulfilled by the stream function. The solution has to satisfy the following boundary conditions ... 

By definition (4.27), the stream function / satisfies the first equation in equation (4.26). The boundary conditions for / are... 

Under the long wavelength and quasistationary approximations and with the use of the linearized forms of the hydrodynamic and thermodynamic boundary conditions, first, we solve the Orr-Sommerfeld equation for the amplitude of perturbed part of the stream function from the Navier-Stokes equations. Second, we solve the equation for the amplitude of perturbed part of the temperature in the liquid film. The dispersion relation for the fluctuation of the solid-liquid interface is determined by the use of these solutions. From the real and imaginary part of this dispersion relation, we obtain the amplification rate cr and the phase velocity =-(7jk as follows ... 

The corresponding boundary conditions are no mass flux through the coast and a bounded stream function far offshore. Equation (2.38) forms together with the boundary conditions an eigenvalue problem that can be solved analytically only for a special analytical form of the shelf topography and must be solved numerically for naturally shaped shelf topography. 

Here, we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition,... 

By passing from the fluid velocity components Vr, V to the stream function according to formulas (2.1.3), we arrive at Eq. (2.1.4). It follows from the remote boundary conditions (2.2.2) that in the general solution (2.1.5) it suffices to retain only the first term (corresponding to the case n = 2). The no-slip conditions (2.2.1) allow us to find the unknown constants A2, B2, C2, and D2. The resulting expression for the stream function,... 

To solve the corresponding hydrodynamic problem, it is convenient to use the spherical coordinates and introduce the stream function according to (2.1.3). Condition (2.5.2) acquires the form... 

Drops and bubbles. Axisymmetric shear flow past a drop was studied in [474,475], We denote the dynamic viscosities of the fluid outside and inside the drop by p and p.2- Far from the drop, the stream function satisfies (2.5.3) just as in the case of a solid particle. Therefore, we must retain only the terms with n = 3 in the general solution (2.1.5). We find the unknown constants from the boundary conditions (2.2.6)-(2.2.10) and obtain... 

Following [270], we first consider steady-state diffusion to the surface of a solid spherical particle in a translational Stokes flow (Re - 0) at high Peclet numbers. In the dimensionless variables, the mathematical statement of the corresponding problem for the concentration distribution is given by Eq. (4.4.3) with the boundary conditions (4.4.4) and (4.4.5), where the stream function is determined by (4.4.2). 

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