/*! 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 59 A concentric cylinder viscometer... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 59

A concentric cylinder viscometer may be formed by rotating the inner member of a pair of closely fitting cylinders. For small clearances, a linear velocity profile may be assumed in the liquid filling the annular clearance gap. A viscometer has an inner cylinder of \(75 \mathrm{mm}\) diameter and 150 mm height, with a clearance gap width of \(0.02 \mathrm{mm}\) A torque of \(0.021 \mathrm{N} \cdot \mathrm{m}\) is required to turn the inner cylinder at \(100 \mathrm{rpm} .\) Determine the viscosity of the liquid in the clearance gap of the viscometer.

Short Answer

Expert verified
After calculation, the viscosity of the liquid in the clearance gap of the viscometer can be determined.

Step by step solution

01

Understand the viscosity

In this problem, the torque required to turn the inner cylinder at a certain speed is known, and the task is to calculate the viscosity of the liquid in the cylinder's clearance gap. This is a direct application of the concept of viscosity, which is the measure of a liquid's resistance to shear or flow.
02

Use the torque and shear stress relationship

We know that torque is given by \( T = r \cdot F \), where \( r \) is the radius and \( F \) is the force. In our case, \( F \) is the tangential force due to the shear stress \( \tau \). The shear stress in cylindrical coordinates is given by \( \tau = \frac{F}{A} = \frac{r \cdot T}{2 \cdot \pi \cdot r \cdot h} = \frac{T}{2 \cdot \pi \cdot h} \), where \( h \) is the height of the cylinder. Now, we calculate the shear stress using the given values.
03

Calculate the viscosity

Knowing the shear stress, we can calculate the viscosity. The viscosity \( \mu \) is given by the relationship \( \tau = \mu \cdot \frac{dv}{dr} \), where \( dv \) is the velocity difference (in our case, this is the rotational speed of the inner cylinder) and \( dr \) is the clearance gap width. We rearrange for \( \mu \) and obtain \( \mu = \frac{\tau \cdot dr}{dv} \) and insert the previously calculated shear stress and the given values into this formula.

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Most popular questions from this chapter

The velocity for a steady, incompressible flow in the \(x y\) plane is given by \(\vec{V}=i A / x+i A y / x^{2},\) where \(A=2 \mathrm{m}^{2} / \mathrm{s},\) and the coordinates are measured in meters. Obtain an equation for the streamline that passes through the point \((x, y)=\) (1,3). Calculate the time required for a fluid particle to move from \(x=1 \mathrm{m}\) to \(x=2 \mathrm{m}\) in this flow field.

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.

A block of mass \(M\) slides on a thin film of oil. The film thickness is \(h\) and the area of the block is \(A\). When released, mass \(m\) exerts tension on the cord, causing the block to accelerate. Neglect friction in the pulley and air resistance. Develop an algebraic expression for the viscous force that acts on the block when it moves at speed \(V\). Derive a differential equation for the block speed as a function of time. Obtain an expression for the block speed as a function of time. The mass \(M=5 \mathrm{kg}, m=1 \mathrm{kg}, A=25 \mathrm{cm}^{2},\) and \(h=0.5 \mathrm{mm} .\) If it takes \(1 \mathrm{s}\) for the speed to reach \(1 \mathrm{m} / \mathrm{s}\), find the oil viscosity \(\mu .\) Plot the curve for \(V(t)\).

A concentric-cylinder viscometer is shown. Viscous torque is produced by the annular gap around the inner cylinder. Additional viscous torque is produced by the flat bottom of the inner cylinder as it rotates above the flat bottom of the stationary outer cylinder. Obtain an algebraic expression for the viscous torque due to flow in the annular gap of width \(a\). Obtain an algebraic expression for the viscous torque due to flow in the bottom clearance gap of height \(b\). Prepare a plot showing the ratio, \(b / a,\) required to hold the bottom torque to 1 percent or less of the annulus torque, versus the other geometric variables. What are the design implications? What modifications to the design can you recommend?

The velocity distribution for laminar flow between parallel plates is given by \\[\frac{u}{u_{\max }}=1-\left(\frac{2 y}{h}\right)^{2}\\] where \(h\) is the distance separating the plates and the origin is placed midway between the plates. Consider a flow of water at \(15^{\circ} \mathrm{C},\) with \(u_{\max }=0.10 \mathrm{m} / \mathrm{s}\) and \(h=0.1 \mathrm{mm} .\) Calculate the shear stress on the upper plate and give its direction. Sketch the variation of shear stress across the channel.
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