In Cartesian coordinate system, the velocity vector field

\(\mathrm {\left( \overrightarrow{\mathrm {v}}\right) }\) has three velocity components

\(\mathrm {u}\),

\(\mathrm {v}\), and

\(\mathrm {w}\) along the three mutually perpendicular directions:

\(\mathrm {x}\),

\(\mathrm {y}\), and

\(\mathrm {z}\), respectively. To calculate four different variables (three velocity components and the pressure), we actually need four different equations. The classical approach in doing so is to solve the Navier–Stokes equations, which is a vector equation and comprises of three different scalar equations, and the conservation of mass equation. This approach, though, looks simple is a cumbersome task in reality primarily because the Navier–Stokes equations are nonlinear. The approach, however, can be made easier if we place a large number of constraints on the system (such as assuming a low Reynolds number flow) and transform the nonlinear Navier–Stokes equations into relatively simple and linear Stokes equation, which is comparatively easy to solve for the velocity field and the pressure. Moreover, in some cases the solutions to these Stokes equation can still be complicated and thus we want even more simpler equations to solve. It is achieved by applying an additional kinematic constraint on the flow (i.e., assuming an irrotational flow field), which leads to a much simpler equation that can be solved directly to obtain the velocity. Such type of flow is known as the potential flow and resulting equation is called the Laplace’s equation.

$$\begin{aligned} \mathrm {\nabla ^{2}\phi }&\,\mathrm {=0} \end{aligned}$$

In general, there exists four fundamental solutions (two-dimensional elementary flows) of the Laplace’s equation, combining which all other steady flow conditions can be modeled. They are, uniform potential flow, source (or sink) flow, doublet potential, and vortex flow.

The source is a potential flow field in which flow emanating from a point spreads radially outwards. In contrast, a sink is the potential flow field in which the flow is directed toward a point from all the directions. The velocity potential for a two-dimensional source of strength

\(\mathrm {q}\) is given as

$$\begin{aligned} \mathrm {\phi }&\,\mathrm {=\frac{q}{2\pi }\ln r} \end{aligned}$$

Similarly, the stream function for a source flow is calculated as

$$\begin{aligned} {\mathrm {\psi }}&\,\mathrm {=\frac{q}{2\pi }\theta } \end{aligned}$$

A doublet flow is the potential flow field formed when a source and a sink of equal strengths are placed close to each other in such a way that the product of their strength and the distance between them remain constant. The velocity potential for a doublet flow is given by

$$\begin{aligned} {\mathrm {\phi }}&\,\mathrm {=\frac{\kappa }{2\pi }\frac{\cos \theta }{r}} \end{aligned}$$

Likewise, the stream function for a doublet flow is obtained as

$$\begin{aligned} {\mathrm {\psi }}&\,\mathrm {=-\frac{\kappa }{2\pi }\frac{\sin \theta }{r}} \end{aligned}$$

A line vortex is a two-dimensional steady flow, which circulates about a point. The velocity potential for a line vortex is given as

$$\begin{aligned} \mathrm {\phi }&\,\mathrm {=\frac{\Gamma }{2\pi }\theta } \end{aligned}$$

In a similar manner as above, the stream function for a line vortex is calculated as

$$\begin{aligned} \mathrm {\psi }&\,\mathrm {=-\frac{\Gamma }{2\pi }\ln r} \end{aligned}$$

The Laplace’s equation is a second-order linear partial differential equation; its linearity allows solutions to be constructed from the superposition of simpler and elementary solutions. For a two-dimensional incompressible and irrotational flow, if

\(\mathrm {\psi _{1}}\) and

\(\mathrm {\psi _{2}}\) are the solutions (stream functions) of the Laplace’s equation

\(\mathrm {\left( \nabla ^{2}\psi =0\right) }\) then their linear combination

\(\mathrm {\psi _{1}+\psi _{2}}\) will also be a solution.

The stream function for the flow due to the combination of a source of strength

\(\mathrm {q}\) at the origin, in an uniform flow of velocity

\(\mathrm {U_{a}}\) in

\(\mathrm {x}\)-direction will be

$$\begin{aligned} \mathrm {\psi }&\,\mathrm {=U_{a}r\sin \theta +\frac{q}{2\pi }\theta } \end{aligned}$$

The streamline passing through the stagnation point

\(\mathrm {S}\) is termed as stagnation streamline forming a shape of semi-infinite body which is also known as Rankine’s half-body.

The Kutta–Joukowski theorem states that the lift per unit span on a two-dimensional body is directly proportional to the circulation around the body. To apply this theorem, the curve around the body can be of any shape and size, but it must be enclosing the body completely. Also note that, the lift on the airfoil is of course produced due to pressure and shear stress distributions over the airfoil surface only and the circulation theory is not at all different from these distributions. Rather, the circulation in Kutta–Joukowski theorem is obtained from the same pressure distribution.