/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 The cone and plate viscometer sh... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 67

The cone and plate viscometer shown is an instrument used frequently to characterize non-Newtonian fluids. It consists of a flat plate and a rotating cone with a very obtuse angle (typically \(\theta\) is less than 0.5 degrees). The apex of the cone just touches the plate surface and the liquid to be tested fills the narrow gap formed by the cone and plate. Derive an expression for the shear rate in the liquid that fills the gap in terms of the geometry of the system. Evaluate the torque on the driven cone in terms of the shear stress and geometry of the system.

Short Answer

Expert verified
The shear rate for the given system is derived as \(\theta \omega / \Delta y\). The torque on the driven cone is determined as \(\pi\tau{R}^{4} / 2\theta\), where R is the cone radius. These expressions provide the relationship between the system's geometry, the shear stress, and the shear rate.

Step by step solution

01

Understand System Geometry and Define Variables

Analyze the system. Let \(\Delta x\) be the gap width at any point \(\Delta r\) from the cone tip, \(\Delta y\) be the vertical height of the gap, and \(\theta\) be the cone angle. Follow the convention that the positive direction of \(x\) is downward along the cone, the positive direction of \(r\) is outward from the axis of cone rotation. Derive relevant equations in terms of these variables.
02

Derive Expression for Shear Rate

The shear rate can be defined by the change in the angular velocity with respect to the change in radius, symbolically represented as \(\omega / \Delta r\). For small angles (\(\theta\)), \(\Delta y / \Delta r = tan(\theta) 鈮� \theta\), such that \(\Delta r = \Delta y / \theta \). Substitute \(\Delta r\) contextually to get the shear rate as \(\omega / (\Delta y / \theta)\) or \(\theta \omega / \Delta y\).
03

Derive Expression for Torque

Derive the expression for the torque on the cone (\(T\)) by expressing it as the radial force \(\Delta F\) acting on the fluid (\(\Delta F = \tau \Delta A\)) times the radius of rotation. Then integrate over the full cone disc area. The force due to shear stress on annulus of fluid is \(\Delta F = \tau 2\pi\Delta r \Delta y\), and substituting \(\Delta y\) in terms of \(\Delta r\) from step 2, it becomes \(\Delta F = \tau 2\pi\Delta r^{2} / \theta\). The Torque (\(T\)) then becomes the integral of \( r \Delta F \), i.e., \( T = \int_{0}^{R} r \Delta F dr = \int_{0}^{R} 2\pi\tau{r}^{3} / \theta dr = \pi\tau{R}^{4} / 2\theta \) where R is the cone radius.

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