Conservation law form of Navier Stokes Equation

• aerograce
In summary, the Navier-Stokes Equation can be written in conservation form by integrating by parts and using the continuity equation. This results in the density term with the pressure gradient dropping out and the term du^2/dx being equal to udu/dx.
aerograce
I am pretty confused about how to write Navier-Stokes Equation into conservation form, it seems that from my notes,
first, the density term with the pressure gradient dropped out.
and second, du^2/dx seems to be equal to udu/dx.
Why is it so? I attached my notes here for your reference.

aerograce said:
I am pretty confused about how to write Navier-Stokes Equation into conservation form, it seems that from my notes,
first, the density term with the pressure gradient dropped out.
and second, du^2/dx seems to be equal to udu/dx.
Why is it so? I attached my notes here for your reference.
View attachment 109122
View attachment 109121
This is pretty straightforward to do. Integrating by parts, $$u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=u\frac{\partial u}{\partial x}+\frac{\partial (uv)}{\partial y}-u\frac{\partial v}{\partial y}$$Next, adding and subtracting ##u(\partial u/\partial x)## gives:
$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=2u\frac{\partial u}{\partial x}+\frac{\partial (uv)}{\partial y}-u\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)$$
But, from the continuity equation, the last term in parenthesis in this equation is zero. So$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=2u\frac{\partial u}{\partial x}+\frac{\partial (uv)}{\partial y}=\frac{\partial u^2}{\partial x}+\frac{\partial (uv)}{\partial y}$$
The rest of the derivation is straightforward.

1. What is the conservation law form of Navier Stokes Equation?

The conservation law form of Navier Stokes Equation is a set of partial differential equations that describe the motion of a fluid in a given space. It is derived from the principles of conservation of mass, momentum, and energy.

2. How is the conservation law form of Navier Stokes Equation different from the original Navier Stokes Equation?

The original Navier Stokes Equation is a single equation that combines the conservation of mass, momentum, and energy into one expression. The conservation law form separates these principles into individual equations, making it easier to apply different boundary conditions and simplifying the solution process.

3. What are the applications of the conservation law form of Navier Stokes Equation?

The conservation law form of Navier Stokes Equation is used in various fields such as fluid dynamics, aerodynamics, and weather forecasting. It is also used in the design and analysis of aircraft, automobiles, and other fluid systems.

4. How is the conservation law form of Navier Stokes Equation solved?

The conservation law form of Navier Stokes Equation is a system of partial differential equations, which can be solved numerically using computational methods such as finite difference, finite element, and finite volume methods. Analytical solutions are only possible for simple and ideal cases.

5. What are the limitations of the conservation law form of Navier Stokes Equation?

The conservation law form of Navier Stokes Equation assumes that fluids are incompressible, inviscid, and Newtonian. It also does not take into account turbulence effects and other complex phenomena such as chemical reactions and phase changes. These limitations make it difficult to accurately model real-world fluid systems.

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