Momentum and mass balance in fluid mechanics

In summary, the conversation discusses the equations used to describe a Newtonian and incompressible fluid in fluid mechanics, specifically the dimensionless mass and momentum balance equations. The equations involve the Cauchy stress tensor and the Reynolds number, which represents the ratio of inertial forces to viscous forces in the fluid. The conservation laws of mass and momentum are used to derive these equations using the Material Derivative and the Reynolds Transport Theorem.
  • #1
Chuck88
37
0
When I am reading the paper in fluid mechanics, I found a paragraph and two equations:

"Since the fluid is considered Newtonian and incompress-
ible of density ρ and constant viscosity μ, the dimensionless
mass and momentum balance equations are:"

[tex]
\nabla \cdot \mathbf{v} = 0
[/tex]

[tex]
Re[\frac{d\mathbf{v}}{dt}+(\mathbf{v} - \mathbf{x}^t) \cdot \nabla \mathbf{v}] = \nabla \cdot \mathbf{T}
[/tex]

"Where ##\mathbf{T} = -p/Ca \mathbf{I} + (\nabla \mathbf{v} + \nabla \mathbf{v}^T)## is the Cauchy stress tensor"

Can someone provide me with the information about the "mass and momentum balance?" Also, what does ##Re## mean in the formulae?
 
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  • #2
Mass and momentum balance simply refer to two of the three conservation laws used to describe continuum mechanics: the conservation of mass and the conservation of momentum.

In this context, [itex]\mathrm{Re} = \frac{\rho u L}{\mu}[/itex] is the Reynolds number - a nondimensional number representing the ratio of inertial forces to viscous forces in the fluid.
 
  • #3
If you're new to continuum mechanics, look up the Material Derivative. Applied to density, you can derive that first equation. You might use the Material Derivative to derive Reynolds Transport Theorem. Applying that to momentum, you can find the second equation, which is basically Newton's second law for a continuum flow.
 

1. What is momentum in fluid mechanics?

Momentum in fluid mechanics is a measure of the motion of a fluid. It is a vector quantity that takes into account both the mass and velocity of the fluid. In simpler terms, it is the product of mass and velocity and represents the amount of force the fluid can exert.

2. How is momentum conserved in fluid mechanics?

Momentum is conserved in fluid mechanics through the principle of conservation of momentum. This principle states that the total momentum of a system remains constant unless acted upon by an external force. This means that in a closed system, the initial momentum will be equal to the final momentum.

3. What is mass balance in fluid mechanics?

Mass balance in fluid mechanics is the application of the law of conservation of mass. This law states that the mass of a fluid entering a control volume must be equal to the mass leaving that control volume, taking into account any changes in density or volume. It is important in analyzing and understanding the behavior of fluids in various systems.

4. How is mass balance calculated in fluid mechanics?

Mass balance is calculated by equating the mass entering a control volume to the mass leaving the control volume, taking into account any changes in density or volume. This can be represented mathematically as: Σmin = Σmout + Σmstored, where m represents mass and Σ represents the sum of all the values.

5. How are momentum and mass balance related in fluid mechanics?

Momentum and mass balance are closely related in fluid mechanics. The law of conservation of momentum can be derived from the law of conservation of mass by considering the relationship between mass, velocity, and momentum. In order for momentum to be conserved, mass must also be conserved. Therefore, the principles of momentum and mass balance are intertwined in the study of fluid mechanics.

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