The Science of Soccer
D. Indumathi, The Institute of Mathematical Sciences, Chennai

Many of you may have watched on TV the FIFA football matches held in Brazil in June of this year. A football tournament is currently on in India, and in January 2015, the 2015 AFC Asian Cup international football tournament organised by the Asian Football Confederation (AFC) will be held in Australia.
What makes the game special and riveting to watch are some of the incredible goals shot by some master players. Even as the ball seems to be headed in a different direction, it suddenly turns and goes through the goal posts! An example is sketched in the back inside cover of this issue and represents Roberto Carlos' famous free kick in the game between Brazil and France in 1997. France's goalkeeper, Barthez, did not even move until it was too late and a ball-boy standing several meters from the goal ducked his head. Both keeper and ball-boy thought the ball was going far outside the post but it was a goal!
What makes the ball swerve so much? Such swerves are also seen in cricket (swing bowling), table tennis, golf, etc. All of them are due to the Magnus effect, named after German physicist Heinrich Gustav Magnus who first described this effect about 150 years ago.
To understand this effect we must first understand some properties of air. Air is a fluid, and is viscous, although its visosity is much less than say honey or even water. This viscosity opposes the relative** motion of different layers of the fluid; in effect, one layer drags the adjacent layer along with it.

Laminar flow
Consider a ball moving through the air. If the ball moves slowly enough (towards the left in the figure), the airflow is said to be laminar -- where the air flows smoothly around and over the ball in smooth lines called streamlines. The layers that separate out at point A when encountering the obstacle (ball) neatly rejoin on the other side at C.
There must be change in the speed and pressure of air so that it can change its flowing direction. Specifically, air slows down in the region where it bends away from the ball's surface (near points A and C) and speeds up where it bends toward the ball's surface (near points B and D). At the same time, due to Bernoulli's principle, the pressure is larger where the speed is less, and vice versa. So A and C are high pressure points.
The high pressure in front of the ball is balanced by the high pressure at the back of the ball. Thus the pressure forces cancel out completely and so the ball experiences no net pressure force. As a result, the only force due to the passing air acting on the ball is the viscous drag --- the friction-like downstream force due to the sliding of the viscous air across the ball's surface.

Turbulent airflow for a fast-moving ball
When the ball moves faster, the airflow become turbulent. Unlike in laminar flow, the air pressure is not distributed symmetrically around the ball in the turbulent flow. Thus the pressure forces cannot balance with each other and the ball experiences pressure drag --- downstream force exerted by unbalanced pressures in the moving air. In other words, it is the imbalanced pressure which slows down the motion of the ball in turbulent airflow.  The air in contact with the surface remains at rest and also slows down the nearby air to form a boundary layer. In the boundary layer, the air moves more slowly and has less total energy that the freely flowing air farther away from the ball.  Outside the boundary layer, the viscosity of the air can be neglected.
When the air flows toward the back of the ball, it travels from a low pressure region to a high pressure region, which is difficult. The air far enough outside the ball has enough energy to reach the back of the ball despite the pressure gradient; however, the air in the boundary layer does not. Thus the air flow becomes asymmetrical about the front and back of the ball. See the figure.
As the speed of the ball increases further, eventually, the boundary layer stops and reverses direction. The freely flowing air separates from the surface and the separated airflow produces eddies behind the ball. These eddies is then confined together to form a turbulent wake similar to that left behind a ship moving through water. The more the speed, the earlier the separation of the air layer and the eddies are formed closer to the ball.
Because of this turbulent wake, the air behind the ball no longer slows down and its pressure no longer rises. Thus the air pressure in front of the ball is higher than that behind the ball. The ball experiences pressure drag, thus slowing it down.
But this is slowing down the ball. What about curving the path of the ball in air?
Indeed, air drag is not the only possible force exerted by air sweeping past a moving ball. It is also possible for the passing air to exert lift forces on a moving ball. Unlike drag force, lift force pushes the ball sideways to either upward or downward directions which causes the ball to curve in flight. This happens when a ball, apart from its high speed, also has spin.
Suppose a ball is kicked to rotate clockwise rapidly about an axis perpendicular to its moving direction during its motion to the left.
The Magnus force is due to the interaction between the spinning ball and the viscous air. For a spinning ball, the passing air is moving in the same direction as the surface of contact on one side of the ball while it is moving in an opposite direction as the surface of contact on the other side --see figure. Therefore, the relative speed of the air compared to that of the ball is smaller on the side where it flows along with the rotating surface of the ball (at the top, in the figure). Thus the viscous air will separate from the ball's surface at a more downstream position on this side.
In contrast, on the other side of the ball (below), the point of separation is much earlier. The airflow pattern around the spinning ball is no longer symmetric.
According to the Newton's 3rd law of motion, there is a reaction force so that the ball is lifted up in the opposite direction. This is the Magnus effect. Another way to understand this is to see that the air speed is higher above than below the ball since the point of separation is much later above. So the pressure is lower above than below, and so there is a "push" from below on the ball due to this pressure gradient.
The net result is that the ball lifts up in a direction perpendicular to its direction of motion. If the spin is counterclockwise, then the Magnus force is downwards. This can be seen in the figure on the back cover.
So a vertical* spin causes lift. Similarly, if the ball was sent spinning in a horizontal direction (so-called side spin), the ball will swerve to the side (right or left, depending on the spin being clockwise or anticlockwise). This was achieved by football players like Beckham and Maradona by kicking the football with the inside* of their boots.
The technique behind a swerving free kick used by Lionel Messi, Ronaldo, Zlatan Ibrahimovic, Ronaldinho, Beckham, is illustrated in the figure.  The direction of the Magnus force is perpendicular to the direction the ball is moving in (in this diagram it is being kicked straight through the screen) and the axis around which it is spinning. The pure sidespin in the top-left figure creates a strong horizontal force, producing the swerving free kick invented by the Brazilian footballers. The pure backspin in the bottom-left figure produces the vertical lift seen for a golf ball.
Sources:
The Science of Soccer: http://www.physics.hku.hk (also for the figures)
Some beautiful curve goals of the 2014 FIFA World Cup can be seen in
http://www.deskeng.com/virtual_desktop/?p=8860