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Chapter 2: Problem 56

Fluids of viscosities \(\mu_{1}=0.1 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\) and \(\mu_{2}=0.15 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\) are contained between two plates (each plate is \(1 \mathrm{m}^{2}\) in area). The thicknesses are \(h_{1}=0.5 \mathrm{mm}\) and \(h_{2}=0.3 \mathrm{mm},\) respectively. Find the force \(F\) to make the upper plate move at a speed of \(1 \mathrm{m} / \mathrm{s}\). What is the fluid velocity at the interface between the two fluids?

Short Answer

Expert verified
The force required to move the upper plate at a speed of \(1 \mathrm{m} / \mathrm{s}\) is 700 N. The fluid velocity at the interface between the two fluids is 0.5 m/s.

Step by step solution

01

Compute the force due to each fluid

Using the formula F=\(\mu A (du/dh) \), force on the upper plate due to fluid 1, \(F_{1} = \mu_{1}A(du/dh_{1}) \), and due to fluid 2, \(F_{2} = \mu_{2}A(du/dh_{2})\). Given that \(\mu_{1}=0.1 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}, h_{1}=0.5 \mathrm{mm}, \mu_{2}=0.15 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}, h_{2}=0.3 \mathrm{mm}\), and the area A = 1 \( \mathrm{m}^{2}\), and \( du/dh\) represents the change in speed with respect to thickness of the fluid, which becomes \(1 \mathrm{m} / \mathrm{s}\) for each fluid thickness.
02

Calculate the forces

Substitute the known values into the force equations, \(F_{1} = (0.1 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2})(1 \mathrm{m}^{2})(1 \mathrm{m} / \mathrm{s}/0.5 \mathrm{mm}) = 200 N\), and \(F_{2} = (0.15 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2})(1 \mathrm{m}^{2})(1 \mathrm{m} / \mathrm{s}/0.3 \mathrm{mm}) = 500 N\). Note that the thickness must be converted from mm to meter.
03

Compute total force

The total force F required to move the upper plate at a speed of \(1 \mathrm{m} / \mathrm{s}\) is the sum of the forces due to each fluid, i.e., \(F = F_{1} + F_{2} = 200 N + 500 N = 700 N\).
04

Determine fluid velocity at the interface

The velocity at the interface \(v_{i}\) can be determined from the velocity profile of each of the fluids. For a Newtonian fluid such as in this problem, the velocity profile is linear. Thus, \(v_{i}\) for fluid 1 can be expressed as \((du/dh_{1})h_{1}\), and for fluid 2 as \((du/dh_{2})h_{2}\). Substituting the known values, we find: \(v_{i} = (1 \mathrm{m} / \mathrm{s}/0.5 \mathrm{mm})0.5 \mathrm{mm} = 0.5 m/s\).

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Most popular questions from this chapter

An insulation company is examining a new material for extruding into cavities. The experimental data is given below for the speed \(U\) of the upper plate, which is separated from a fixed lower plate by a 1 -mm-thick sample of the material, when a given shear stress is applied. Determine the type of material. If a replacement material with a minimum yield stress of \(250 \mathrm{Pa}\) is needed, what viscosity will the material need to have the same behavior as the current material at a shear stress of \(450 \mathrm{Pa} ?\) $$\begin{array}{ccccccccccc}\tau(P a) & 50 & 100 & 150 & 163 & 171 & 170 & 202 & 246 & 349 & 444 \\ U(\mathrm{m} / \mathrm{s}) & 0 & 0 & 0 & 0.005 & 0.01 & 0.025 & 0.05 & 0.1 & 0.2 & 0.3 \end{array}$$

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For the velocity fields given below, determine: a. whether the flow field is one- , two-, or three-dimensional, and why. b. whether the flow is steady or unsteady, and why. (The quantities \(a\) and \(b\) are constants.) (1) \(\vec{V}=\left[a y^{2} e^{-b t}\right]\) (2) \(\vec{V}=a x^{2} i+b x j+c k\) (3) \(\vec{V}=\)axyi\(-\)byt (4) \(\vec{V}=a x \hat{i}-b y j+c t \hat{k}\) (5) \(\bar{V}=\left[a e^{-\ln x}\right] i+b t^{2}\) (6) \(\vec{V}=a\left(x^{2}+y^{2}\right)^{1 / 2}\left(1 / z^{3}\right) \hat{k}\) (7) \(\vec{V}=(a x+t) i-b y^{2}\) (8) \(\vec{V}=a x^{2} \hat{i}+b x z j+c y k\)

The flow field for an atmospheric flow is given by \\[\vec{V}=-\frac{M y}{2 \pi} \hat{i}+\frac{M x}{2 \pi}\\] where \(M=1 \mathrm{s}^{-1}\), and the \(x\) and \(y\) coordinates are the parallel to the local latitude and longitude. Plot the velocity magnitude along the \(x\) axis, along the \(y\) axis, and along the line \(y=x,\) and discuss the velocity direction with respect to these three axes. For each plot use a range \(x\) or \(y=0 \mathrm{km}\) to \(1 \mathrm{km}\). Find the equation for the streamlines and sketch several of them. What does this flow field model?
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