Answer :
To apply Blasius's similarity solution for flow over a flat plate and convert the momentum and energy equations into ODEs, one must assume laminar flow, linearize the Navier-Stokes and continuity equations, and utilize the symmetry of the problem. This simplification process transforms complex partial differential equations into more solvable ODEs, aiding in the analysis of the flow characteristics near the flat plate.
To apply Blasius's similarity solution for flow over a flat plate and convert the momentum and energy equations into ordinary differential equations (ODEs), we start by assuming a laminar, vorticity-free fluid flow. By exploiting the symmetry of the problem, we work in a coordinate frame where the flow velocity can be considered as a function of just one spatial coordinate, v = n2v(y), implying the problem is effectively one-dimensional. Consequently, the Navier-Stokes equations are simplified significantly.
For the stationary flow (dv/dt = 0), the y-component of the Navier-Stokes equation becomes the static Pascal equation, indicating that the pressure is evenly distributed and not influenced by the movement of the plate or fluid. For the z-component, which represents the only Cartesian component of the fluid's velocity, we arrive at the 1D Laplace equation, V2v.
Moreover, the momentum continuity equation can also be linearized and reduced to an ODE. The linearized equations encapsulate basic fluid dynamic principles like conservation of mass through the continuity equation, and momentum conservation, which takes the form of non-linear partial differential equations but can be simplified through assumptions into solvable ODEs.
In summary, the process of linearizing the hydrodynamic equations and applying Blasius's similarity solution allows us to transform the complex partial differential equations of fluid motion into more manageable ODEs, which are easier to analyze and solve for various flow characteristics such as velocity profiles and thermal gradients near the flat plate.
