Consider an infinitesimal plane surface element, dS = ndS, located at point r. Here, dS is the area of the element, and n its unit normal. The fluid which lies on that side of the element toward which n points is said to lie on its positive side, and vice versa. The net flux of x-momentum across the element (in the direction of n) is πx · dS, which implies (from Newton’s second law of motion) that the fluid on the positive side of the surface element experiences a force πx ·dS in the x-direction due to short-range interaction with the fluid on the negative side.
It's hard to get past all this unusual notation, with the symbol n being used for two different things (momentum flux and normal), but I think I know what they are trying to say. Let [itex]\vec{N_x}[/itex] represent the net momentum flux (vector) through a surface of constant x. This momentum flux vector is not necessarily normal to the surface, since the fluid velocity is not necessarily normal to the surface. We can represent this momentum flux vector by:Niles said:Homework Statement
I am reading an introductory book on fluid mechanics (freely available on the WWW) and I came across this on page 10:
I don't understand the underlined part. We have a force-vector πx, which we decompose to find the overlap with some surface element dS. Shouldn't the force entering dS be directed along dS, and not πx as the author claims?
Thanks in advance.
Vector decomposition in fluid mechanics is the process of breaking down a vector quantity into its component parts in order to better understand and analyze its behavior. This is particularly important in fluid mechanics, where multiple forces and velocities can act on a fluid particle at the same time.
Vector decomposition is used in fluid mechanics to analyze the forces and velocities acting on a fluid particle in a specific direction. By breaking down the vector into its component parts, scientists can better understand the impact of individual forces and how they contribute to the overall behavior of the fluid.
The steps for vector decomposition in fluid mechanics include identifying the vector quantity, determining its direction and magnitude, and breaking it down into its component parts using trigonometric functions. The resulting components can then be analyzed separately to gain a better understanding of the overall behavior of the fluid.
Yes, vector decomposition can be used for non-uniform fluid motion. In this case, the vector quantities would have varying magnitudes and directions at different points in the fluid, and the process of decomposition would be used to analyze and understand their behavior at each point.
One limitation of vector decomposition in fluid mechanics is that it assumes the fluid is in a steady state and does not take into account the effects of turbulence or unsteady flow. Additionally, vector decomposition may not accurately represent the behavior of fluids with highly complex or chaotic motion.