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Chapter 6: Problem 80

Two immiscible, incompressible, viscous fluids having the same densities but different viscosities are contained between two infinite, horizontal, parallel plates (Fig. \(P 6.80\) ). The bottom plate is fixed, and the upper plate moves with a constant velocity \(U\) Determine the velocity at the interface. Express your answer in terms of \(U, \mu_{1},\) and \(\mu_{2}\). The mo:ion of the fluid is caused entirely by the movement of the upper plate; that is, there is no pressure gradient in the \(x\) direction. The fluid velocity and shearing stress are continuous across the interface between the two fluids. Assume laminar flow:

Short Answer

Expert verified
The velocity at the interface is \( \frac{U \mu_{2}}{\mu_{1} + \mu_{2}} \)

Step by step solution

01

Understand the Dynamics

First, we understand that at the bottom plate, the velocity is going to be zero because it is stationary, and at the top, the velocity will be \(U\) due to the motion of the plate. As the fluids are viscous, there would be a velocity gradient on moving from the bottom to the top.
02

Apply the Newton's Law of Viscosity

Since there is no pressure gradient in the x-direction, we can express the shear stress \( \tau \) in the two layers as: \(\tau_{1} = \mu_{1} \frac{du_{1}}{dz}, \quad \tau_{2} = \mu_{2} \frac{du_{2}}{dz}\) where z is the direction perpendicular to the plates, \( \mu_{1} \) and \( \mu_{2} \) are the dynamic viscosities of the two fluids, and \(u_{1}\) and \(u_{2}\) are the velocities of the points in the two layers.
03

Apply The Principle of Shear Stress Continuity

The given problem also mentions that the shear stress is continuous across the interface, so we can equate the shear stresses in the two layers: \(\mu_{1} \frac{du_{1}}{dz} = \mu_{2} \frac{du_{2}}{dz}\)
04

Find the Velocities in terms of Shear Stress

Integrating the above expressions of shear stresses from bottom to the interface (for 1st layer) and from top to the interface (for 2nd layer), we get: \(u_{1} = \frac{\tau_{1} z}{\mu_{1}} + C_{1}, \quad u_{2} = U - \frac{\tau_{2}(H-z)}{\mu_{2}}\)
05

Solve for Velocity at Interface

At the interface, the velocities of the two layers are equal i.e., \( u_{1} = u_{2}\). Substituting the values obtained in the previous step, we get: \(\frac{\tau_{1} z}{\mu_{1}} + C_{1} = U - \frac{\tau_{2}(H-z)}{\mu_{2}} \). Solving this, we get the velocity at the interface \( u_{i} \) as follows: \( u_{i} = \frac{U \mu_{2}}{\mu_{1} + \mu_{2}} \)

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Most popular questions from this chapter

Air at \(25^{\circ} \mathrm{C}\) flows normal to the axis of an infinitely long cylinder of 1.0 -m radius. The approach velocity is \(130 \mathrm{km} / \mathrm{hr}\). Find the rotational velocity of the cylinder so that a single stagnation point occurs on the lower-most part of the cylinder. Assume potential flow.

The three components of velocity in a flow field are given by $$\begin{aligned}u &=x^{2}+y^{2}+z^{2} \\\v &=x y+y z+z^{2} \\\w &=-3 x z-z^{2} / 2+4\end{aligned}$$ (a) Determine the volumetric dilatation rate and interpret the results. (b) Determine an expression for the rotation vector. Is this an irrotational flow field?

For a certain incompressible, two-dimensional flow field the velocity component in the \(y\) direction is given by the equation $$v=3 x y+x^{2} y$$ Determine the velocity component in the \(x\) direction so that the volumetric dilatation rate is zero.

The flow in the impeller of a centrifugal pump is modeled by the superposition of a source and a free vortex. The impeller has an outer diameter of \(0.5 \mathrm{m}\) and an inner diameter of \(0.3 \mathrm{m}\). At the outlet from the impeller, the flowing water has the following velocity componeats, relative to the impeller: radial component \(2 \mathrm{m} / \mathrm{s}\) and tangential comporent \(7 \mathrm{m} / \mathrm{s}\). (a) Find the strength of the source and the vortex required to model this flow. (b) Assume that the impeller blades are shaped like the streamlines and plot an impeller blade shape. (c) Find the radial and tangential components of velocity at the inlet to the impeller.

The two-dimensional velocity field for an incompressible Newtonian fluid is described by the relationship $$\mathbf{V}=\left(12 x y^{2}-6 x^{3}\right) \hat{\mathbf{i}}+\left(18 x^{2} y-4 y^{3} \hat{\mathbf{j}}\right.$$ where the velocity has units of \(\mathrm{m} / \mathrm{s}\) when \(x\) and \(y\) are in meters. Determine the stresses \(\sigma_{x x}, \sigma_{y y},\) and \(\tau_{x y}\) at the point \(x=0.5 \mathrm{m}\) \(y=1.0 \mathrm{m}\) if pressure at this point is \(6 \mathrm{kPa}\) and the fluid is glycerin at \(20^{\circ} \mathrm{C}\). Shew these stresses on a sketch.
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