Use Bingham's model to solve the problem of anti-plane...

Use Bingham's model to solve the problem of anti-plane viscoplastic flow in a circular pipe aligned with the $z$-axis. Assume a velocity field of the form $\mathbf{v}=w(r) \mathbf{k}$, where $r=\sqrt{x^2+y^2}$ is the radius in a system of cylindrical polar coordinates. Suppose the material occupies the annular region $(0<) a \leq r \leq b$ and that $w(b)=0$ and $w(a)=W$. Assume that the axial pressure gradient vanishes and find the velocity field $w(r)$.

Key Concepts

Bingham Plastic Model

The Bingham plastic model describes materials that behave as rigid solids under low stress and flow as viscous fluids when the applied stress exceeds a critical yield stress. This concept is important in viscoplastic flow analysis because it provides the constitutive relation that distinguishes between regions that are flowing versus those that are unyielded, which in turn affects the velocity profile and stress distribution in the material.

Viscoplastic Flow

Viscoplastic flow refers to the behavior of materials that have both viscous and plastic characteristics. In such flows, material deformation does not occur until the applied stress surpasses a certain yield stress. This dual behavior is central in solving problems involving materials like Bingham plastics, as it necessitates identifying yield surfaces and determining the zones where flow occurs versus zones where the material remains rigid.

Anti-Plane Shear Flow

Anti-plane shear flow involves the deformation of a material where displacement occurs in one direction while spatial variations occur in the perpendicular plane. This simplification reduces a three-dimensional flow problem to a more manageable form, focusing on one velocity component. It is particularly useful in analyzing flows along a single axis, where the stress and strain fields can be expressed solely in terms of the radial coordinate in cylindrical coordinates.

Cylindrical Polar Coordinate System

The cylindrical polar coordinate system is used to describe problems with rotational or circular symmetry. It involves coordinates (r, ?, z), where r is the radial distance and ? is the angular coordinate, with z often aligned with the axis of symmetry. Utilizing this coordinate system simplifies the mathematical formulation of the problem by aligning the coordinate lines with the natural boundaries and symmetry of the geometry.

Boundary Conditions and Annular Geometry

In problems involving annular regions, boundary conditions are essential for determining the unique solution to the flow field. In this context, prescribed velocities at the inner and outer boundaries (r = a and r = b) are used to solve for the velocity profile across the gap. Annular geometry introduces additional complexity due to the curvature and finite thickness of the flow region, which is addressed by applying the appropriate physical constraints at each boundary.

Please give Ace some feedback