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Today we will explore the deformation law for Newtonian fluids. Can anyone summarize what we mean by a Newtonian fluid?
Isn't it a fluid whose viscosity remains constant regardless of the stress applied?
Exactly! That's a key characteristic. This section will explain how we can relate the stress and strain in these fluids using the Navier-Stokes equations. Let's define our fundamental terms.
What exactly does 'deformation law' mean in this context?
Great question. The deformation law describes how shear stress is related to the rate of strain in a fluid. The simplest assumption is a linear relationship, which we will derive using several postulates.
Stokes formulated three key postulates for Newtonian fluids. The first is that the stress tensor τ_ij is at most a linear function of the strain rates ϵ_ij. Can anyone explain the significance of that?
It means that the relationship between stress and strain is proportional, which helps us predict how a fluid behaves when forces are applied.
Exactly! The second postulate states that fluids are isotropic and the third asserts that at zero strain rates, the stress must equal the hydrostatic pressure condition, τ_ij = -p δ_ij. Remember that δ_ij is the Kronecker delta.
So in simpler terms, when the fluid is not flowing, it just applies pressure evenly in all directions?
Yes, that's correct! These principles set the stage for understanding how we can express the behavior of these fluids mathematically.
Now, let’s transform our understanding to arbitrary axes. Can anyone recall what the principal axes are?
They are the axes where shear stresses vanish.
Correct! We now introduce coordinates that might not eliminate shear stresses and show how they relate to the principal stresses and strain rates. We will use transformation rules.
What equations do we use for these transformations?
Good question! We use equations that relate τ_ij and ϵ_ij in the new coordinate systems—these develop from the principles of trigonometry as well. Let’s work through the transformations together.
Finally, let’s apply what we’ve learned. How does understanding the deformation law aid in hydraulic engineering?
It helps predict how fluids will flow through pipes and other systems, crucial for design!
Exactly! Additionally, knowing the relationships allows us to model flow characteristics in both laminar and turbulent conditions, helping engineers design more efficient systems.
How would we measure these physical quantities in practice?
In practice, we can use tools like viscometers and stress sensors to observe the relationship between stress and strain rates. This empirical data can confirm our models!
In fluids that flow, viscosity won't change; when forces apply, the stress stays the same!
Imagine a calm lake with even water pressure, where every drop feels the same weight above it, much like how fluids respond in hydrostatics.
Stokes=evaluate Stress in Three ways: Linear relationship, Isotropic nature, Zero strain at rest.
H.I.S = Hydrostatic, Isotropic, Shear stress - remember the key principles!
Term
Newtonian Fluid
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Term
Hydrostatic Pressure
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Term
Stokes' Postulate
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A fluid whose viscosity is constant and does not change with the applied stress.
The internal stress developed due to the viscosity of the fluid.
The rate at which deformation occurs in a material due to applied stress.
The pressure exerted by a fluid at equilibrium due to the weight of the fluid above.
A function that is one when its indices are equal and zero otherwise.
Having identical properties in all directions.
This section discusses the deformation law related to Newtonian fluids, deriving its relation to strain and stress as dictated by the Navier-Stokes equations.
In this section, the principles governing how Newtonian fluids deform under shear stress are examined, focusing on the relationship between viscous stress and strain rates. Stokes' postulates of isotropy and linearity are highlighted, along with the implications of hydrostatic pressure in a fluid at rest.
In this section, we delve into the deformation law for Newtonian fluids, which provides a foundation for understanding viscous fluid flow described by the Navier-Stokes equations. We begin with the relationship between the stress tensor τ_ij and strain rates ϵ_ij, introducing Stokes' three fundamental postulates. He hypothesizes that the stress tensor is a linear function of strain rates, that fluids are isotropic in their properties, and that under static conditions, stress corresponds to hydrostatic pressure. We also relate shear and normal stresses to the strain rates through transformation laws, leading to the general deformation equations that connect stress and strain in arbitrary coordinate systems. This culminates in a comprehensive understanding of the flow characteristics of Newtonian fluids, significantly influencing hydraulic engineering and fluid mechanics.