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Homework Statement
ρ= density, vi = i-th velocity component, gi=i-th component of gravity vector, p=pressure, μ= viscosity, D/Dt = material derivative
Homework Equations
Continuity equation: div v = 0
Energy conservation for a Newtonian fluid?
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In summary, the derivation of the Continuity equation is found in Transport Phenomena by Bird, Stewart, and Lightfoot. It involves dotting the equation of motion with the velocity vector, also known as the Mechanical Energy Balance Equation. This can be combined with the Overall Energy Balance equation to yield the Thermal Energy Balance equation.
ρ= density, vi = i-th velocity component, gi=i-th component of gravity vector, p=pressure, μ= viscosity, D/Dt = material derivative
Continuity equation: div v = 0
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The derivation you are looking for is in Transport Phenomena by Bird, Stewart, and Lightfoot. It involves dotting the equation of motion (momentum balance equation) with the velocity vector. Incidentally, this is called the Mechanical Energy Balance Equation. It can be combined with the Overall Energy Balance equation to yield the Thermal Energy Balance equation.Display_Name said:
Related to Energy conservation for a Newtonian fluid?
1. What is a Newtonian fluid?
A Newtonian fluid is a type of fluid that follows Newton's laws of motion, meaning that its viscosity (resistance to flow) remains constant regardless of the applied shear stress. This means that the fluid's behavior is predictable and can be described by a linear relationship between the shear stress and the rate of shear strain.
2. Why is energy conservation important for Newtonian fluids?
Energy conservation is important for Newtonian fluids because it allows us to accurately predict the behavior of the fluid and its flow patterns. By conserving energy, we can ensure that the fluid follows Newton's laws of motion and that the forces acting on the fluid are balanced. This is crucial for understanding and predicting the behavior of fluids in various applications, such as in pipelines or industrial processes.
3. How is energy conserved in a Newtonian fluid?
In a Newtonian fluid, energy is conserved through the balance of forces acting on the fluid. This means that the energy input into the fluid (such as through pumping or stirring) must be equal to the energy output (such as through flow or heat dissipation). Additionally, the fluid must follow Newton's laws of motion, which ensures that energy is not lost due to friction or other factors.
4. What are some practical applications of energy conservation for Newtonian fluids?
Energy conservation for Newtonian fluids has many practical applications. Some examples include optimizing the design of pipelines and pumps to minimize energy consumption, predicting and controlling flow patterns in industrial processes to improve efficiency, and understanding the behavior of fluids in medical procedures such as blood flow in arteries.
5. Are there any limitations to energy conservation for Newtonian fluids?
While energy conservation is a fundamental principle for Newtonian fluids, there are some limitations to its application. In real-world situations, there may be factors such as turbulence, non-Newtonian behavior, or external forces that can affect the conservation of energy. Additionally, the accuracy of predictions based on energy conservation may also be affected by experimental error or uncertainties in fluid properties.
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