nonlinear in the pressure. One example of the nonlinear nature of the Navier-Stokes equations can be seen in the case of a fluid flowing around a circular obstacle. In this case, the velocity of the fluid near the obstacle will be higher than the velocity of the fluid farther away from the obstacle. This results in a pressure gradient, with higher pressure near the obstacle and lower pressure farther away. To see this more explicitly, consider the case of a circular obstacle of radius placed in a uniform flow with velocity and density . Let be the velocity of the fluid at position and time , and let be the pressure at the same position and time. The Navier-Stokes equations in this case are: where is the kinematic viscosity of the fluid. Assuming that the flow is steady (meaning that the velocity and pressure do not vary with time), we can set the time derivative terms equal to zero: We can now consider the flow near the circular obstacle. In this region, the velocity of the fluid will be higher than the uniform flow velocity due to the presence of the obstacle. This results in a nonlinear term in the Navier-Stokes equations that is proportional to the velocity of the fluid. At the same time, the presence of the obstacle will also result in a pressure gradient, with higher pressure near the obstacle and lower pressure farther away. This can be seen by considering the continuity equation, which states that the mass flow rate through any surface must be constant. Since the velocity is higher near the obstacle, the mass flow rate through a surface near the obstacle will be higher than the mass flow rate through a surface farther away from the obstacle. This can be compensated for by a pressure gradient, with higher pressure near the obstacle and lower pressure farther away. As a result of these nonlinear effects, the Navier-Stokes equations in this case become difficult to solve, and approximations or numerical methods must be used to find the velocity and pressure fields in the flow. Consider the case of a two-dimensional fluid flow in a rectangular domain, with a velocity field and a pressure field . We can use a finite element method to solve the Navier-Stokes equation for the velocity field: To do this, we divide the domain into a series of smaller elements, and