Lines of Equal Potential

Suppose the mathematician has managed, by good fortune or guesswork, to write down the stream function for a steady flow of fluid under certain boundary conditions. By equating the stream function to a sequence of con­stants, a family of streamlines can be drawn and a picture of the flow can be exhibited. Now suppose that, guided by the streamlines, the mathematician draws another family of curves. These new curves are to be drawn so that they always cut across the streamlines at right angles. A network of orthogonal lines is built up. If the first set of lines were the streamlines of the flow, what are these new lines that have been drawn so that they are always at right angles to them?

They are called potential lines. They are in fact another way of implic­itly representing the velocity distribution of a flow. Their immediate interest is that the potential lines of a given flow can always be reinterpreted as the streamlines of a new flow, while the old streamlines become the potential lines of the new flow. Streamlines and potential lines can be interchanged, provided that appropriate changes are made to the boundary conditions of the flow. This possibility of interchange can be interpreted to mean that, just as there exists a stream function, so there must exist another, closely related function ready to perform the same role with regard to the lines of potential that у played with regard to streamlines. This function is called the potential function, and it is conventionally designated by the Greek letter phi, ф. The role of the potential function may be illustrated by the uniform flow along the x-axis, where the axis can be taken as a solid boundary. This flow is the one discussed earlier whose stream function is у = – Uy. The streamlines are
horizontal lines parallel with the x-axis, so the potential lines are vertical lines parallel with the y-axis. Now switch the potential lines and the streamlines, that is, switch the two families of curves given by у = constant and ф = con­stant. The streamlines are now vertical and parallel with the y-axis, which can be treated as a boundary to the new flow. The horizontal lines parallel with the x-axis are the new potential lines.

The intimate relationship between potential lines and the streamlines finds expression in the mathematics of irrotational flow. Because the two families of curves are orthogonal, it is possible to write the equations for the velocity components u and v of a given flow either in terms of the stream function that applies to the flow or in terms of the potential function that applies to it. The result gives rise to the following relationships between ф and у:

= dy = дф dx dy

It follows immediately from these equations that the potential function ф obeys Laplace’s equation, just as the stream function does when represent­ing an irrotational flow. One useful mathematical property of solutions to Laplace’s equation is that they are additive. If yj is a solution and y2 is a solution, then y3 = yj + У2 is also a solution. Stream functions can be added. Again, the point can be illustrated by reference to the simplest possible cases. The flow of speed U along the x-axis (yj = – Uy) can be combined with, say, a flow of the same speed U but along the y-axis (that is, the flow arrived at by switching the streamlines and the potential lines of the original flow so that y2 = Ux), and the result is another flow that moves diagonally and whose stream function is y3 = yj + У2. In this way complicated flows can be con­structed out of simple flows.