Transient Viscous Diffusion of Momentum (COMSOL) This problem is one of the few instances in this book in which we investigate transient or unsteady-state fluid flow. It illustrates how momentum is transferred under the diffusive influence of viscosity. As shown in Fig. E6.4.1, consider a liquid of

Question

Transient Viscous Diffusion of Momentum (COMSOL)

This problem is one of the few instances in this book in which we investigate transient or unsteady-state fluid flow. It illustrates how momentum is transferred under the diffusive influence of viscosity.

As shown in Fig. E6.4.1, consider a liquid of density [latex]\rho=1 \mathrm{~g} / \mathrm{cm}^{3}[/latex] and viscosity [latex]\mu[/latex] (whose values will be given later), contained between two parallel plates AB and CD of length [latex]L=2 \mathrm{~cm}[/latex] and separation [latex]H=1 \mathrm{~cm}[/latex]. The lower plate CD is stationary, and the upper plate AB can undergo two types of movement:

At time [latex]t=0[/latex], it is suddenly set in motion with a constant [latex]x[/latex] velocity [latex]v_{x}=v_{x 0}[/latex]. In our problem, [latex]v_{x 0}=1 \mathrm{~cm} / \mathrm{s}[/latex] and [latex]\mu=0.5 \mathrm{~g} / \mathrm{cm} \mathrm{s}=0.5 \mathrm{P}[/latex].
At time [latex]t=0[/latex], it is oscillated to the right and left in such a way that its [latex]x[/latex] velocity varies with time according to [latex]v_{x 0}=a \sin \omega t[/latex]. In our problem, the amplitude [latex]a=1 \mathrm{~cm} / \mathrm{s}[/latex], the angular velocity [latex]\omega=2 \pi \mathrm{s}^{-1}[/latex], and the viscosity may be either [latex]\mu=0.1[/latex] or [latex]0.5 \mathrm{P}[/latex].

Very similar types of motion occur in concentric-cylinder viscometers, as in Figs. [latex]1.1[/latex] and 11.13(b), although with smaller moving-surface separations.

Use COMSOL to investigate how [latex]v_{x}[/latex] varies between the midpoints of the lower and upper plates at different times, and interpret the results. If needed, further details of the implementation of COMSOL are given in Chapter [latex]14 .[/latex] All mouseclicks are left-clicks (L-click, the same as Select) unless specifically denoted as right-click [latex](\mathrm{R})[/latex].

Note that COMSOL will employ the full Navier-Stokes and continuity equations in its solution. However, because the only nonzero velocity component is [latex]v_{x}[/latex], the following familiar diffusion-type equation is effectively being solved:

[latex]\rho \frac{\partial v_{x}}{\partial t}=\mu \frac{\partial^{2} v_{x}}{\partial y^{2}}[/latex] (E6.4.1)

Accepted Answer

Case 1 (Constant Upper-Plate Velocity)

Select the Physics

Open COMSOL and L-click Model Wizard, 2D, Fluid Flow (the little rotating triangle is called a "glyph"), Single-Phase Flow, Laminar Flow, Add. Note the symbols used by COMSOL--- [latex]u, v, w[/latex] for velocity components and [latex]p[/latex] for pressure.
L-click Study, Time Dependent, Done.

Define the Parameters

R-click Global Definitions and select Parameters. In the Settings window to the right of the Model Builder window, enter the parameter [latex]v x 0[/latex] in the Name column and its value and units, [latex]1[\mathrm{~cm} / \mathrm{s}][/latex], in the Expression column. Similarly, define [latex]L=2[\mathrm{~cm}], H=1[\mathrm{~cm}][/latex], rho [latex]1=1\left[\mathrm{~g} / \mathrm{cm}^{3} ight][/latex], and [latex]m u 1=0.5[\mathrm{~g} / \mathrm{cm} / \mathrm{s}][/latex].

Establish the Geometry

R-click Geometry and select Rectangle. Note the default is based on the lower-left corner being at [latex](0,0)[/latex]. Enter the values for Width and Height as [latex]L[/latex] and [latex]H[/latex]. L-click Build Selected.

Define the Fluid Properties

R-click the Materials node within Component 1 and select Blank Material. Note that the new material will have the required material parameters based on the physics of the problem. Enter the parameters [latex]r h o 1[/latex] and [latex]m u 1[/latex] as the values for Density and Dynamic viscosity.

Define the Boundary Conditions

L-click Laminar Flow, Wall 1, and note the default No Slip boundary conditions that are highlighted.
R-click the Laminar Flow node and select Inlet to define an inlet boundary condition, and L-click the left boundary [latex](\# 1)[/latex] to select it as the inlet. Note that clicking on a boundary will change its color from red to blue. Within the Settings window L-click the dropdown box in the Boundary Condition node to select Pressure (equal to zero), deselect Suppress backflow, and leave Normal flow checked. R-click the Laminar Flow node and select Outlet, and L-click the right boundary [latex](\# 4)[/latex] to select it as the outlet, with zero pressure, Normal flow checked, and Suppress backflow unchecked. R-click the Laminar Flow node and L-click Wall. Select the top boundary (again changing from red to blue), and within the Settings window change the Boundary Condition from No Slip to Sliding wall. Enter vx0 as the value for the tangential velocity [latex]U w[/latex].

Create the Mesh and Solve the Problem

L-click the Mesh node and change the Element Size in the Settings window to Fine. L-click Build All to construct the mesh. R-click Mesh 1, and Statistics will show that the mesh contains approximately 4,000 elements.
L-click the Study 1 little triangular glyph, and then select the Step 1 Time Dependent node and define the output range segments using the range function. For the requested output times, enter the three ranges as: range [latex](0.0,0.01,0.1)[/latex] [latex]\operatorname{range}(0.2,0.1,0.5) \operatorname{range}(1.0,0.5,2.0)[/latex].
R-click the Study node and select Compute, which gives (after about 15 seconds) a surface plot of the magnitudes of the velocities --- from low near the bottom (blue), medium at the middle (turquoise), and high near the top (red).
Save your results occasionally --- for example, in the file Ex6.4-11.mph.

Display the Results (Case 1)

Create a data set at the midpoint in the [latex]y[/latex]-direction. R-click the Data Sets node within the Results tree and select Cut Line 2D. In the Settings window enter the values [latex](\mathrm{L} / 2,0)[/latex] and [latex](\mathrm{L} / 2, \mathrm{H})[/latex] for Points 1 and [latex]2 .[/latex] L-click Plot near the top of the Settings window to display the cut line.
R-click Results and L-click 1D Plot Group. In the Settings window, Leftclick the dropdown dialog and change the data set to Cut Line 2D 1, which was created above. Change the Time selection to "From list" and select the desired individual plot times by holding down the CTRL key (Command on a Mac) and then L-clicking on the individual values [latex]0.01,0.02,0.05,0.1,0.2,0.5[/latex], and [latex]2.0[/latex].
R-click the 1D Plot Group 3 and select Line Graph. Within the Settings window enter the variable for the [latex]x[/latex]-component of velocity, [latex]u[/latex], in the Expression field.
Expand the Coloring and Style section by L-clicking the triangular glyph to the left. Change the Color option to Black by L-clicking the dropdown dialog box.
L-click Plot in the Settings window to update the plot. Save in Ex6.416.mph or similar.

Discussion of Case 1 (Constant Upper-Plate Velocity)

Fig. E6.4.2 shows how the velocity [latex]v_{x}[/latex] varies with distance [latex]y[/latex] from the lower plate at various times. The result is a classical case of diffusion---in this instance, of [latex]x[/latex] momentum (generated by the motion of the upper plate, which moves at [latex]1 \mathrm{~cm} / \mathrm{s})[/latex], into the liquid below it. For the smallest time plotted [latex](t=0.01 \mathrm{~s})[/latex], most of the liquid, from [latex]y=0[/latex] to [latex]0.7 \mathrm{~cm}[/latex], is essentially undisturbed, followed by a rising [latex]v_{x}[/latex] between [latex]y=0.7[/latex] and [latex]1 \mathrm{~cm}[/latex]. For early times (or for an infinitely deep liquid), before the influence of the lower plate is felt, there is an analytical solution:

[latex]\frac{v_{x}}{v_{x 0}}=\operatorname{erfc} \frac{z}{2 \sqrt{ u t}},[/latex] (E6.4.2)

in which [latex]z=H-y[/latex] is the downward distance from the upper plate, [latex] u=\mu / ho[/latex] is the kinematic viscosity, and [latex]\operatorname{erfc}[/latex] denotes the complementary error function.

As time increases, more and more of the liquid is set into motion by the diffusion of [latex]x[/latex] momentum from the upper plate. Eventually --- theoretically at [latex]t=\infty[/latex] but effectively at a time of about 2 s---a steady state is reached, in which the shear stress exerted by the upper plate is counterbalanced by the shear stress in the opposite direction from the stationary lower plate. Since the velocity profile is linear, [latex]d v_{x} / d y[/latex] is constant; thus, these shear stresses are equal (and opposite) because they are both given by [latex] au_{y x}=\mu d v_{x} / d y[/latex].

Case 2 (Oscillating Upper-Plate Velocity)

Now solve the oscillatory surface time-dependent problem by first saving the model under a new name, such as Ex6.4-17.mph (or a file name of your choice).
Follow Step 3 above (Parameters under Global Definitions) to add the parameters for [latex]a[/latex] and omega with values [latex]1.0[\mathrm{~cm} / \mathrm{s}][/latex] and [latex]2^{*} \mathrm{pi}[1 / \mathrm{s}] .[/latex] Also update the viscosity by changing the value for [latex]m u 1[/latex] to [latex]0.1[\mathrm{~g} / \mathrm{cm} / \mathrm{s}][/latex].
Update the top wall boundary condition, Wall 2. L-click Laminar Flow, Wall 2 and enter the tangential velocity value a* [latex]\sin ([/latex] omega*t [latex])[/latex].
Update the requested output times. L-click the Study 1, Step 1 Time Dependent node and enter the ranges as: range [latex](0,2,18)[/latex] range [latex](19,0.25,20)[/latex].
R-click the Study 1 node and select Compute. The solution took about 100 seconds on the author's computer. The result is a surface plot showing velocity magnitudes, not needed here.
Update the requested times by selecting [latex]1 \mathrm{D}[/latex] Plot Group 3 . In the Settings window, change the Time selections to [latex]t=19,19.25,19.5,19.75[/latex], and [latex]20 \mathrm{~s}[/latex] by L-clicking the first value 19 and concluding the selection with Shift + L-click on the value 20 , giving five values altogether. L-click Plot.
Save the model, in Ex6.4-23.mph for example.
Use Step 3 to update the value of [latex]m u 1[/latex] to [latex]0.5[\mathrm{~g} / \mathrm{cm} / \mathrm{s}][/latex].
R-click the Study node and select Compute.
L-click on the 1D Plot Group 3, observe the changes to the solution, and make a final file save.

Discussion of Case 2 (Oscillating Upper-Plate Velocity)

The results for the two values of the viscosity are shown in Figs. E6.4.3 and E6.4.4. Since [latex]\omega=2 \pi[/latex], the period of the oscillations is [latex]T=1 \mathrm{~s}[/latex]. Thus, velocity profiles separated by [latex]0.25[/latex] s will occur over one complete cycle of times [latex]t=0[/latex], [latex]0.25 T, 0.5 T, 0.75 T[/latex], and [latex]T[/latex]. Note two main effects:

(a) Penetration depth. For the lower viscosity, [latex]\mu=0.1 \mathrm{P}[/latex], the surface oscillations do not penetrate as far into the liquid as those for the higher viscosity of [latex]\mu=0.5 \mathrm{P}[/latex]. In other words, the lower viscosity does not permit as rapid a diffusion of [latex]x[/latex] momentum as the higher viscosity.

(b) Phase change. Note that for [latex]t=T[/latex], when the upper plate has a zero velocity, much of the liquid has a negative velocity, largely due to the diffusional effects of the negative surface velocity at an earlier time such as [latex]t=0.75 T[/latex]. Similar observations can be made for other values of time.

Both effects have applications in the testing of non-Newtonian fluids, for which the phase difference between stress and strain is illustrated in terms of displacements in Fig. 11.11. Since the velocity [latex]v_{x}[/latex] is the derivative of the distance moved, the displacement [latex]D[/latex] of the upper plate is:

[latex]D=-\frac{a}{\omega} \cos \omega t.[/latex] (E6.4.3)

For an infinitely deep liquid, there is again an exact solution:

[latex]v_{x}=a e^{-\beta z} \sin (\omega t-\beta z),[/latex] (E6.4.4)

in which [latex]z[/latex] is the depth and:

[latex]\beta^{2}=\frac{\omega}{2 u} .[/latex] (E6.4.5)

The attenuation and phase change with depth [latex]z[/latex] are very apparent from Eqn. (E6.4.4).

Question 6.4: Transient Viscous Diffusion of Momentum (COMSOL) This proble...

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