Stream function boundary conditions

In the present eonfiguration, air inlet boundaries are assumed to be Pressure Inlet while outflow boundaries are assumed Pressure Outlet . Pressure inlet boundary conditions were used to define the total pressure and other scalar quantities at flow inlets. Pressure outlet boundary conditions were used to define the static pressure at flow outlets. At the nozzle inlet, the air pressure was varied. At the nozzle outlet, the pressure was supposed to be the external pressure (one atmosphere). At the wall of the nozzle standard wall function boundary condition was applied. Although the high velocity of air stream was a heat source that will increase the temperature in the nozzle, the nozzle length was very short and the process oecurs in a very short time. For simplification, it was assumed that the process is adiabatic i.e. no heat transfer occurred through walls. The flow model used was viscous, compressible airflow [1, 6-10]. The following series of equations were used to solve a compressible turbulent flow for airflow simulation [1,6-12] ... 

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. 

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... 

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. 

The three nondimensionalized boundary layer equations (Eqs. 6-64, 6-65, and 6 66) involve three unknown functions /<, i/, and T. two independent variables x and y, and two parameters Rej. and Pr. The pressure P (x ) depends on the geometry involved (it is constant for a flat plate), and it has the same value inside and outside the boundary layer at a specified x. Therefore, it can be,delennined separately from the free stream conditions, and dP ld.x in Eq. 6-65 can he treated as a known function of.v. Note that the boundary condition do not introduce any new parameters. 

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. 

According to (5-29), pressure //0) depends only on 6, and the problem reduces to the solution of (5-27) and (5-28) subject to boundary conditions (5-20). Equations (5-27) and (5-28) are known as the lubrication (thin-film) equations. We see that they resemble the equations for unidirectional flow. However, in this case, the boundaries are not required to be parallel. Thus uf can be a function of the stream wise variable 0, and will not be zero in general. Furthermore, because uf is a function of 9, so is dp(i))/<)(). Finally, whereas the unidirectional flow equations are exact, the lubrication equations are only the leading-order approximation to the exact equations of motion and continuity in the asymptotic, thin-gap limit, e 0. 

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,... 

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). 

The concentration part of the problem is described by the convective diffusion equation (4.4.3) (the subscript 1 is introduced for the stream function) and by the boundary conditions (4.4.4) and (4.4.5) specifying the concentration on the interface and remote from the drop. The diffusion Peclet number Pe is assumed to be small. 

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