Fluid Dynamics: Differential Conservation Laws and Equations

This lecture focuses on the differential approach to fluid dynamics, specifically addressing the conservation laws governing fluid flow. The instructor begins by recalling the principles of mass conservation and the diffusion equation, emphasizing the importance of understanding fluid behavior in detail. The lecture progresses to derive the fundamental equations of fluid flow using differential conservation laws, contrasting them with integral conservation laws. The continuity equation is introduced in differential form, highlighting its significance for incompressible fluids. The instructor then transitions to Newton's second law, discussing the forces acting on fluid elements, including body and surface forces. The concept of the Cauchy stress tensor is introduced, which combines pressure and viscous stresses. The lecture culminates in the formulation of the Cauchy equation, representing Newton's second law in differential form for incompressible flow. This comprehensive approach provides a solid foundation for understanding fluid dynamics and prepares students for further exploration of shear stress tensors and their implications in fluid mechanics.