Scalar velocity potential

For an irrotational, incompressible, and frictionless fluid flow there exists a scalar velocity potential 4> such that the velocity vector V is... 

Hence, temperature profiles in pure isotropic solids, the scalar velocity potential for ideal fluid flow, and dynamic pressure profiles for flow through porous media are all based on the solution of Laplace s equation. Whenever the divergence of a vector vanishes and the vector is expressed as the gradient of a scalar, Laplace s equation is required to calculate the scalar profile. 

Potential Flow around a Gas Bubble Via the Scalar Velocity Potential, An incompressible fluid with constant approach velocity (i.e., S Vapproach) flows upward past a stationary nondeformable gas bubble of radius R. This two-dimensional flow is axisymmetric about the spherical coordinates because this coordinate system provides the best match with the macroscopic boundary at r = / . The appropriate partial differential equation for is... 

The boundary condition at large r is employed to calculate the radial and tangential velocity components, as well as the functional form of the scalar velocity potential. Since the velocity vector far from the bubble is... 

Now, this expression for the scalar velocity potential far from the bubble is compared with the tangential velocity component ... 

This indicates that df/d9 = 0, or / = Ci (i.e., constant). It is acceptable to set Cl to zero because any constant will satisfy Laplace s equation. Furthermore, the value of Ci does not affect the velocity profile because one calculates the components of the velocity vector from the gradient of d>, and the gradient of Cl vanishes. At most, Ci will affect the magnitude of along an equipotential. Analysis of this problem far from the bubble yields the following expression for the scalar velocity potential ... 

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. 

This motion of the bubble induces axisymmetric two-dimensional flow in the liquid phase. In the potential flow regime, one calculates the scalar velocity potential tb(r, 0) via Laplace s equation. The general solution in spherical coordinates 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 ... 

The fact that cos 0 is a good function for the angular part of the scalar velocity potential is obvious because both terms in Laplace s equation reveal the same angular dependence. Once again, F(r) r" is appropriate ... 

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 ... 

There are no dimensionless numbers in this potential flow equation because convective forces per unit volume and dynamic pressure forces per unit volume both scale as pV /L. Furthermore, potential flow theory provides the formalism to calculate p and the dimensionless scalar velocity potential such that the vorticity vector vanishes and overall fluid mass is conserved for an incompressible fluid. Hence,... 

Two generic planar wavemaker configurations are shown in Figs. 2.1(a) and 2.1(b). The fluid motion may be obtained from the negative gradient of a dimensional scalar velocity potential T(x, z,t) according to... 

The scalar velocity potential must be a solution to the Laplace equation... 

---