Irrotational Flow and Laplace’s Equation

The motion of a fluid element involves three different kinds of change:

(1) translation, (2) strain, and (3) rotation. Translation involves change of position of the element, strain involves a deformation of the shape of the element, and rotation involves a change of angular orientation of the ele­ment. Rotation may seem to be an intuitively clear idea because the image that comes to mind is the rotation of a rigid body in which the fluid element is pictured as if it behaves like, say, a spinning ball. Sometimes fluid elements are indeed represented as spinning balls. Although shape is not really crucial, the picture of a sphere is sometimes invoked when explaining the striking result that a fluid element in an ideally inviscid fluid can never be made to rotate if it is not already rotating, nor can it be stopped from rotating if it is already in rotation. The rotation of an ideal fluid element can neither be created nor destroyed by, for example, the motion of a solid body that is immersed in, and surrounded by, a fluid. The argument is that, in a perfect fluid, neither the surrounding fluid nor such a moving body can exert any traction on the smooth surface of the element in order to change its exist­ing state of rotatory motion. It will be evident that, in light of this result, the origin of rotation becomes something of a mystery.19

Cowley and Levy, however, do not avail themselves of an intuitive pic­ture of fluid elements as rotating spheres of fluid. They opt for the more austere technical definition. Technically, the rotation of a fluid element (in two-dimensional flow) is defined as the average angular velocity of any two infinitesimal linear elements within the fluid element that are instantaneously perpendicular to one another. Mathematically this definition is expressed in the formula

1(dv du |

rotation = — ———- — .

2 ^dx dy )

The virtue of the technical definition is that commonsense comparisons tend to omit the possibility that the angular velocity of the two linear elements might cancel out so that, under some circumstances, rotation can be equal to zero by virtue of the deformation of the fluid element.20 A flow in which the quantity in the brackets in the previous formula is zero is called an ir – rotational flow.

Methodologically, the important point about the rotation of a fluid ele­ment is that by neglecting it, and restricting attention to irrotational flow, the mathematics is greatly simplified. Why is this? A glimpse into the reasons can be gained by taking another look at the stream function discussed in the pre­vious section. Consider the following expression involving the stream func­tion y. The expression is arrived at by differentiating у twice with respect to x and twice with respect to y and adding the result. Thus,

dy dy

dx2 + dy2′

It will be recalled that differentiating у once yields the velocity components of the flow and that the x and y components of the fluid velocity at a point are given by

dy

u =—— and

dy

= dy

dx

Substituting these definitions of the velocity components in the expression under consideration gives

dV+dV=+Af+d^VAf_—1

dx2 dy2 dx ^ dx J dy ^ dy J

dv du dx dy

The result of the substitution is precisely the expression that was used in the technical definition of the term “rotation.” It is in fact twice the rotation. If the rotation is zero, that is, if the flow is irrotational, then this term must be zero, and so, therefore, is the expression cited at the outset of the discussion. In other words, if the flow is irrotational, then the stream function у is gov­erned by the equation

dy+dy= 0

dx2 dy2

This equation is called Laplace’s equation. Although the equation itself may look far from simple, it is not difficult to appreciate that it is simpler than if the right-hand side were equated to some complicated function of x and y rather than to zero. Irrotational flow is thus a (relatively) simplified form of flow governed by Laplace’s equation.

Laplace’s equation is one of the most significant differential equations in the history of mathematical physics.21 The equation is often written as V2y = 0.22 The restriction to “irrotational” flow, which it signifies, not only simplified the mathematics, but it brought out the analogies between fluid flow and the results that had emerged or were emerging in other fields. Ir – rotational flow obeyed simple mathematical laws that were similar to those in areas such as the theory of gravitational force, the theory of heat, the theory of elasticity, and the theory of magnetism and electricity. Maxwell used the analogy, and Laplace’s equation, to shed light on the hydrodynamics of the flow of fluid through an orifice and the vena contracta, that is, the contraction shown by the jet of fluid a short distance from the orifice.23 Because of the electrical analogies, irrotational flows used to be called “electrical” flows. The interplay between hydrodynamics and the theory of electric phenomena was not only suggestive theoretically, but it was also exploited in the laboratory. In the interwar years it provided the basis of a laboratory-bench technique used by E. F. Relf at the National Physical Laboratory for graphically plotting the streamlines of the flow around objects with complicated shapes, such as aerofoils.24 The resulting representation was, of course, a representation of the flow as it would take place if the air were an ideal fluid.25