Laminar flow fixed parallel plates

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SUMMARY

The discussion centers on the derivation of equations 6.132 and the preceding equation in the context of laminar flow between fixed parallel plates. The equations are derived from the integration of partial differential equations, specifically focusing on the pressure distribution. The integration of equation 123 reveals that pressure is a function of both y and x, with the pressure gradient dp/dx treated as a constant, which is linked to viscosity and spatial properties. This understanding clarifies the relationship between pressure and flow characteristics in fluid mechanics.

PREREQUISITES
  • Understanding of partial differential equations in fluid mechanics
  • Familiarity with laminar flow concepts
  • Knowledge of pressure gradients and their implications in fluid dynamics
  • Basic calculus for integration techniques
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  • Study the derivation of the Navier-Stokes equations for fluid flow
  • Learn about boundary layer theory and its applications in laminar flow
  • Explore the relationship between viscosity and pressure gradients in fluid mechanics
  • Investigate numerical methods for solving partial differential equations in fluid dynamics
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Students and professionals in mechanical engineering, fluid dynamics researchers, and anyone studying laminar flow and its mathematical modeling.

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This all goes back to eqns. 122 and 123, and how you go about solving partial differential equations like these for the pressure. Eqn. 123 can be integrated with respect to y to show that the pressure will be equal to ρgy plus a function of x, f(x) (analogous to the constant of integration in solving an ordinary differential equation). You then substitute that result into equation 122, and find that f(x) is equal to x times the partial derivative of p with respect to x, plus a constant.
 
Ok, I get it now. I didn't see the step of bringing that equation back into the PDE's. I was also confused on how the dp/dx was treated as constant wrt x but on the following page it shows that dp/dx is in fact constant and related to the viscosity and spatial properties.
Thank you.
 

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