Pressure-gradient force

The Magnus effect is an observable phenomenon that is commonly associated with a spinning object moving through a fluid. The path of the spinning object is deflected in a manner that is not present when the object is not spinning. The deflection can be explained by the difference in pressure of the fluid on opposite sides of the spinning object. The Magnus effect is dependent on the speed of rotation.

Consider a cubic parcel of fluid with a density ${\displaystyle \rho }$ , a height ${\displaystyle dz}$ , and a surface area ${\displaystyle dA}$ . The mass of the parcel can be expressed as, ${\displaystyle m=\rho \,dA\,dz}$ . Using Newton's second law, ${\displaystyle F=ma}$ , we can then examine a pressure difference ${\displaystyle dP}$  (assumed to be only in the ${\displaystyle z}$ -direction) to find the resulting force, ${\displaystyle F=-dP\,dA=\rho a\,dA\,dz}$ .

The acceleration resulting from the pressure gradient is then, $${\displaystyle a=-{\frac {1}{\rho }}{\frac {dP}{dz}}.}$$ 

The effects of the pressure gradient are usually expressed in this way, in terms of an acceleration, instead of in terms of a force. We can express the acceleration more precisely, for a general pressure ${\displaystyle P}$  as, $${\displaystyle {\vec {a}}=-{\frac {1}{\rho }}{\vec {\nabla }}P.}$$ 

The direction of the resulting force (acceleration) is thus in the opposite direction of the most rapid increase of pressure.

• Roland B. Stull (2000) Meteorology for Scientists and Engineers, Second Edition, Ed. Brooks/Cole, ISBN 0-534-37214-7.