Momentum Equation for compressible 1D flow

In summary, the differential momentum equation is derived using the steady state flow equation and the force exerted by the solid boundary on the fluid passing through the control volume.
  • #1
Bohr1227
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Homework Statement


Derive the differential momentum equation: [tex]dp=-\rho udu[/tex].

Homework Equations


[tex]\oint_{ CS}^{ } \rho \vec{U}(\vec{U}\cdot \vec{n})dA=-\oint_{ CS}^{ } p\vec{n}dA[/tex] for steady state flow...

The Attempt at a Solution


I tried to solve it like in the attached picture, but can someone explain where the [tex]p+\frac{1}{2}dp[/tex] (marked with red line in the picture) comes from? Is it a linear interpolation or something?
Untitled1.png


Thanks!
 

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  • #2
That is a term for representing the force that the solid boundary exerts (in the x direction) on the fluid passing through the control volume.
 

FAQ: Momentum Equation for compressible 1D flow

What is the momentum equation for compressible 1D flow?

The momentum equation for compressible 1D flow is a fundamental equation in fluid mechanics that describes the relationship between the forces acting on a fluid and the resulting acceleration of the fluid. It is also known as the Euler equation or the Navier-Stokes equation.

What are the assumptions made in the momentum equation for compressible 1D flow?

The momentum equation for compressible 1D flow assumes that the flow is one-dimensional, meaning that the fluid properties vary only in one direction. It also assumes that the flow is steady, meaning that the fluid properties do not change over time. Additionally, the equation assumes that the fluid is compressible, meaning that its density can change with changes in pressure and temperature.

How is the momentum equation for compressible 1D flow derived?

The momentum equation for compressible 1D flow is derived from the Navier-Stokes equations, which are a set of equations that describe the motion of a fluid. The derivation involves applying the principles of conservation of mass, momentum, and energy to a control volume in the fluid.

What are the main applications of the momentum equation for compressible 1D flow?

The momentum equation for compressible 1D flow is used in a wide range of engineering applications, including aerodynamics, gas dynamics, and rocket propulsion. It is also used in the design of aircraft, cars, and other vehicles that experience compressible flow.

How can the momentum equation for compressible 1D flow be solved?

The momentum equation for compressible 1D flow can be solved using numerical methods or analytical solutions. In numerical methods, the equation is discretized and solved using numerical techniques. Analytical solutions involve making simplifying assumptions to the equation and solving it using mathematical methods.

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