Reconsider the case of shear flow (see Chapter 2). If the...

Reconsider the case of shear flow (see Chapter 2). If the top $(y=1)$ and bottom $(y=0)$ plates are a unit distance apart and the top moves with unit velocity, the pressure is constant and velocity is given by $$ u(x, y, z)=(y, 0,0) . $$ It is reasonable that the force this flow exerts pushing up on the top plate will be different from the force it exerts in the direction the plate is moving. This exercise is to verify this intuition by calculating these force directly from the explicit solution for the velocity and pressure. (a) Find the deformation tensor and stress tensor. Find the stress tensor at the top wall. In particular, find the force the flow exerts at the top wall in the normal direction to the wall (pushing up), and verify that in this case it is the pressure. Find the force exerted at the top wall in the direction the wall is moving. Sketch the geometry, velocity, and Cauchy stress vectors. (b) Verify that forces depend on orientation by checking that the Cauchy stress vector at mid-channel is different on the $x-y, y-z$ and $x-z$ planes.

Reconsider the case of shear flow (see Chapter 2). If the top $(y=1)$ and bottom $(y=0)$ plates are a unit distance apart and the top moves with unit velocity, the pressure is constant and velocity is given by
$$
u(x, y, z)=(y, 0,0) .
$$
It is reasonable that the force this flow exerts pushing up on the top plate will be different from the force it exerts in the direction the plate is moving. This exercise is to verify this intuition by calculating these force directly from the explicit solution for the velocity and pressure.
(a) Find the deformation tensor and stress tensor. Find the stress tensor at the top wall. In particular, find the force the flow exerts at the top wall in the normal direction to the wall (pushing up), and verify that in this case it is the pressure. Find the force exerted at the top wall in the direction the wall is moving. Sketch the geometry, velocity, and Cauchy stress vectors.
(b) Verify that forces depend on orientation by checking that the Cauchy stress vector at mid-channel is different on the $x-y, y-z$ and $x-z$ planes.

Key Concepts

Orientation Dependence of Stress

Orientation dependence of stress refers to the phenomenon where the magnitude and direction of the force exerted by a fluid on a surface depend on the orientation of that surface. This is a direct consequence of the tensorial nature of the stress tensor; when it is projected onto surfaces with different normal vectors, the resulting Cauchy stress vectors (tractions) can vary significantly. This concept is crucial for analyzing forces in complex geometries and understanding how stress distributions change with surface orientation.

Cauchy Stress Vector (Traction Vector)

The Cauchy stress vector, or traction vector, represents the force per unit area acting on an oriented surface within a continuum. It is obtained by contracting the stress tensor with the unit normal vector to the surface. This concept is key to understanding how different components of stress (normal and shear) act on variously oriented surfaces, leading to different physical behaviors such as pressure and viscous shear forces.

Shear Flow

Shear flow refers to a type of fluid motion in which layers of the fluid move parallel to each other with different velocities. This results in a velocity gradient that is oriented perpendicular to the flow direction, giving rise to shear stresses. In continuum mechanics, shear flows are often used as basic models to understand the behavior of viscous fluids and the distribution of stresses in simple geometries.

Deformation Tensor

The deformation tensor, often referred to as the strain rate tensor in fluid mechanics, is a measure of the rate at which a fluid element deforms. It is derived from the velocity gradient by taking its symmetric part. This tensor quantifies extensional and shear deformations and is fundamental in relating the kinematics of fluid motion to the internal stresses, especially in viscous fluids.

Stress Tensor

The stress tensor is a fundamental concept in continuum mechanics that describes the distribution of internal forces within a material. In fluids, it typically comprises an isotropic part associated with pressure and a deviatoric part associated with viscous effects. The stress tensor provides essential information on how forces are transmitted across different surfaces within the fluid, and it is used to compute forces acting on boundaries.

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