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Questions and Answers
What is the relationship between shear stress and velocity gradient for Newtonian fluids?
What is the unit of dynamic or absolute viscosity?
Which of the following fluids is considered a Newtonian fluid?
What does it mean when a fluid is described as 'inviscid'?
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What does the kinematic viscosity, ν, represent?
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How does viscosity generally vary for a fluid?
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What does the negative sign in the equation 𝜏𝑤 = −𝜇 𝜕𝑈𝑥 𝜕𝑦 indicate?
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What equation is used to calculate the thickness of the film (viscous sub-layer) in Newton's law of viscosity?
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In the Blasius relation for the friction factor in a smooth tube, what does 'Cf' represent?
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When dealing with fully rough pipes, what happens to the friction factor 'Cf'?
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What does the Moody chart depict?
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In which zone on the Moody chart are the pipes considered hydraulically fully rough?
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What property of a fluid determines its resistance to deformation in shear?
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Which condition, due to the viscosity of a fluid, is observed when a thin layer next to a wall sticks to it and does not move?
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In the context of fluid motion near a solid boundary, what is the term used for the layers of fluid that are slowed down by the layer below them?
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What determines whether a fluid layer experiences no slip near a solid boundary?
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Which property of a fluid is crucial when there is motion between one layer of fluid and another?
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What kind of motion causes the layers of fluid to form a boundary layer near a solid surface?
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Study Notes
Newton's Law of Viscosity
- Newton's law of viscosity can be applied to the layer near the wall: 𝜏𝑤 = −𝜇 𝜕𝑈𝑥 𝜕𝑦
- The minus sign indicates that 𝜏𝑤 is in the opposite direction to x
Integrating the Equation
- The equation can be integrated across the film: 𝛿 𝑈 𝛿 𝑚 ∫0 𝜏𝑤 𝑑𝑦 = −𝜇 ∫0 𝑚 𝑑𝑈𝑥
- On integration and rearrangement, it gives 𝜏𝑤 = -m 𝑈𝑚 𝑑
Thickness of the Film (Viscous Sub-Layer)
- To obtain the thickness of the film (viscous sub-layer), 𝛿, we need to know 𝜏𝑤
- 𝜏𝑤 can be calculated from the empirical (experimental) relation of Blasius for the friction factor in a smooth tube/pipe: 𝐶𝑓 = 0.079𝑅𝑒𝐷−0.25
Friction Factor and Relative Roughness
- The relationship between 𝐶𝑓, 𝑅𝑒𝐷, and the dimensionless form of 𝑒 (𝑒/𝐷, also known as the relative roughness) is embodied in the Moody chart
- The zone of pipes that are hydraulically fully rough is to the right of the dashed line (marked “Complete turbulence, rough pipes”)
Wall Shear Stress
- The wall feels a friction force in the direction of the fluid motion as the fluid tries to drag it along
- The friction force per unit area is the shear stress because it is parallel to the area on which it acts
- Shear Stress ∝ Velocity gradient
Newtonian Fluids
- For Newtonian fluids, the relationship is given by: 𝜏 = 𝜇 𝑑𝑈 𝑑𝑦 (N/m²)
- 𝜇 is the dynamic or absolute viscosity, with units Pa s (Pascal seconds) or Poise (1g/cms)
- Shear Stress = Viscosity x Velocity gradient
Non-Newtonian Fluids
- Not all fluids behave in this simple way
- Examples of non-Newtonian fluids include blood, custard, and some paints
Viscosity
- Viscosity is not constant for a fluid but varies with pressure and temperature
- The main variations are with temperature
- Kinematic viscosity (ν) is sometimes used, which is the viscosity divided by the density: 𝜈 = 𝜇 / 𝜌 (m²/s)
Ideal Fluid
- In an ideal fluid, the effects of viscosity are non-existent, 𝜇 = 0, and the fluid is said to be “inviscid”
- However, all real fluids have viscosity
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Description
Explore the concept of shear stress in fluid mechanics and its relationship with viscosity. Learn how the friction force per unit area is related to shear stress, and how it depends on velocity gradient in Newtonian fluids.
