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βπ =βπ +βππh=βπ +ππβh=βπ +ππ πhhh Now, N-S equations can be reiterated in fewer terms: π ( π π£ π₯ + ( βπ£β β . π» ) π£ ) = β π π π + π π» 2 π£ ππ π π₯ ππ₯ π₯ π(ππ£π¦ +(βπ£ββ.π»)π£ )=βπππ +ππ»2π£ ππ π π¦ ππ¦ π¦ { π ( π π£ π§ + ( βπ£β β . π» ) π£ ) = β π π π + π π» 2 π£ ππ π π§ ππ§ π§ (3.3) (3.4) Moving forward, the modified pressure is called pressure in the body of thesis. Other definitions of pressure will be mentioned if needed. 3.1.2- Darcyβs law for porous media The fluid dynamics in porous media is dominated by viscous forces at relatively low flow rates. Darcyβs experimental law is an established equation to predict the pressure gradient for laminar flow passing through porous media: π»π =βππ£ (3.5) πΎ Where K is an experimental term called the permeability of porous media. Darcyβs law predicts the dominancy of viscous forces. Additionally, it shows the pressure gradient depends on the flow average velocity, viscosity, and permeability. However, when the fluid flow rate increases, microscopic inertial forces grow rapidly. Therefore, the pressure gradient deviates from what Darcyβs law predicts. Many experimental terms are suggested to account for this deviation. [88] The most well-known ones are Forchheimer and Brinkman correction terms in the literature. Forchheimer presented a general correction term to Darcyβs law for fluids at high flow rates in his article at 1901: [89] π 20PDF Image | Analysis of Fluid Flow in Redox Flow Batteries
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