Derivation of Navier-Stokes Equation in Cartesian Coordinates
Problems raised in the differenential momentum equation can be solved by introducing relationships between stress components and equations between stress components and velocity gradients . Since the velocity gradient in flow field has the meaning of strain rate, the relationship between velocity gradient and stress is the same as the relationship between strain and stress in solid mechanics . In solid mechanics, this relationship is called Hooke's law , and according to it, stress is proportional to the strain of an elastic material. Without these laws, problems in solid mechanics cannot be solved. In electrical engineering, there is a similar law, i.e. Ohm's law, and in heat transfer, Fourier's law plays a similar role. These relationships between solid and fluid mechanics contain knowledge about how an object will behave after being acted on by a force. Let us derive equations between the velocity gradient and stress components discovered by Stokes in 1845. First, t