6.1 Review of microscale hydrodynamics
6.1.5 Modified Stokes’ 2nd problem
The previous solution was only for the purely viscous fluid which follows Newton’s Law of Viscosity. If we consider a fluid which obeys a different constitutive law, the equa-
tion of motion will differ as will the resulting solution for the velocity of the fluid. In
addition, changing the boundary conditions for any fluid type will alter the solution to the differential equation. Adding a stationary upper boundary at y = h is called the Modi-
fied Stokes’ 2nd problem, and for the viscous fluid, the presence of the upper boundary means we cannot ignore the positive exponential term in Equation 6.24. Thus, for a plate oscillating atu(y,t)= Usin(ωt), we have a hyperbolic sine function, and the fluid velocity
becomes (Mitran et al., 2009)
u(y,t)= � Ueiωt sinh[δ0 A(h−y)] sinh[δ0 Ah] (6.33)
where �refers to taking the imaginary part, A = U/ω, andδ0 = (1+i)√ω/(2ν). Each
plot in figure 6.3 shows the flow profile for a given kinematic viscosities (ν = 1,2, or 4).
Each line within a plot is a moment in time after a steady state flow has been reached. As with Figure 6.2, red lines indicate the plane’s passage to the right, green lines indicate the plane’s passage to the left, and blue lines indicate the plane’s passage to the right again. The penetration depth is related to the inverse of�(δ0). When comparing the flow profiles
of Stokes’ 2nd problem and Modified Stokes’ 2nd problem, the effects of the oscillating
plane in Stokes’ 2nd problem penetrate farther into the fluid as in Modified Stokes’ 2nd problem, the fluid velocity must be zero at the upper boundary. Notice again the decrease
in velocity as a function of height for the lower viscosity fluid as compared to the higher viscosity fluid. The effects of the oscillating boundary penetrate farther into the fluid as ν
increases.
direction of plane
Figure 6.3: A simulation of Modified Stokes’ 2nd problem for a purely viscous fluid il- lustrates the effects of an increasing viscosity on the propagation of stress throughout the
fluid. The frequency is assumed to beω = 1 Hz. Each plot represents a different viscos-
ity (ν = 1,2,4), and within a plot, a line indicates a single time step after a steady state
has been reached. Red lines correspond to earlier times and blue lines correspond to later times. The three plots demonstrate that asνincreases, the coupling between the fluid and
the boundaries increases.
For viscoelastic fluids, the original differential equation for which I seek a solution will
not include Newton’s Law of viscosity, but will instead be a constitutive relation for the the viscous stress tensorτwhich, depending on the fluid, may or may not lead to an analytical
solution of the Modified Stokes’ 2nd problem. For two viscoelastic fluids, the linear vis- coelastic Maxwell model and the nonlinear upper convected Maxwell and Giesekus mod- els, modified Stokes’ 2nd problem has been solved analytically. Both Maxwell models are listed in Table 6.2.
The Maxwell model was first proposed by Maxwell in 1867 in differential form for
small changes in stress and strain (Macosko, 1994),
τ+λ∂τ
∂t =−ηγ˙ (6.34)
whereηis the fluid viscosity, andλis the material’s relaxation time. This model actually
falls out of the Generalized linear viscoelastic constitutive relation (in Table 6.2) if we as- sume only one single relaxation time within the fluid. Describing a fluid with a Generalized linear viscoelastic model implies a linear relationship between the change in stress (dτ) and
strain (γ) with a proportionality constant equal to the change in the relaxation modulusG,
ordτ= dGγ. These fluids do have memory as the generalized model integrates over all pre-
vious elapsed time. Graphically, Maxwell models are represented as a spring and dashpot combination, as shown in Figure 6.4, with the spring representing the elastic contribution and the dashpot representing the viscous contribution of the fluid. This concept that the application of slow stresses versus rapidly changing stresses affects a material’s response
τ
G η
Figure 6.4: The spring and dashpot represent the linear Maxwell model with a single re- laxation time.τis the stress applied to the fluid.
It is much simpler to consider the Navier-Stokes equation in its non-dimensional form where the position variables in the equation have been divided by the appropriate charac- teristic length scale, the velocity has been divided by the characteristic velocity, and the time has been divided by the ratio of the length scale over the velocity scale. The non- dimensional Navier-Stokes equation is then
∂u
∂t +∇ ·(�u�u)=
1
Re∇ ·(−p+τ) (6.35)
whereReis the Reynolds number. Assuming we still have no pressure gradient, a driving plane oscillation of u(t) = Usin(ωt) such that the fluid is only in the x direction, and
the viscous stress represented by the Generalized linear constitutive relation (where I will include G specific to the Maxwell model), the non-dimensional Navier-Stokes equation becomes (Mitran et al., 2009)
∂u ∂t = 1 Re ∂ ∂y �� t −∞ G(t−t�)∂u(y,t�) ∂y dt �� (6.36)
whereη0 is the zero-shear viscosity and for a Maxwell model,G(t) = (ηp/λ)e−t/λwith ηp
∂u ∂t = ηp λRe � t −∞ e−(t−t�)/λ∂2u(y,t�) ∂y2 dt �. (6.37)
This equation simplifies to a damped wave equation (Mitran et al., 2009),
∂2u ∂t2 + 1 λ ∂u ∂t = ηp λRe ∂2u ∂y2. (6.38)
The solution to Equation 6.38 has a very similar appearance to the solution for a purely viscous fluid (Equation 6.33) and has been written down by Mitran et al. (2009) in its non-dimensionalized form: u(y,t)=� � eitsinh[δ(H−y)] sinh(δH) � (6.39) whereδ = (1+i)�Re/(2η∗), in dimensional variablesRe= U2/(ων), andtis measured in
units ofω. The variableη∗is the complex viscosity of the fluid, and equal toη�−iη��where η�is the viscosity andη��is the elasticity of the fluid. Figure 6.5 illustrates Modified Stokes’
2nd problem for a viscoelastic fluid as a function of changing viscosity and elasticity. Three cases are shown in the figure: (i) η� = 0.5η��, (ii)η� = η��, (iii)η� = 2η��. For each case,
a series of curves is displayed with each curve illustrating a moment in time after a steady state flow has been achieved.
Initially in the figure the elasticity of the fluid dominates the fluid’s viscous response, and the coupling between the fluid and the boundaries is strong, as we see the effects of
the oscillating plate continue to affect fluid elements near the upper boundary. In addition,
channel. This increase is likely due to the constructive interference of propagating and reflected shear waves (off the stationary upper boundary). This increased velocity is not
present when the viscosity is on the same order or larger than the elasticity. In these cases
(η� ∼ η�� andη� > η��), the velocity profile begins to look more like a purely viscous fluid
in that the effect of the oscillating boundary is damped by the viscosity of the fluid more
direction of plane
Figure 6.5: Flow velocity as a function of height in a viscoelastic fluid with kinematic viscosityν = 1 as obtained using the analytical solution to Modified Stokes’ 2nd problem.
Each plot illustrates a different relationship between the fluid’s dynamic viscosity η� and
its elasticity�. Within a single plot, each curve represents a moment in time after steady
state has been reached. Red lines indicate earlier times and blue lines correspond to later times. As viscosityη�increases with respect to the elasticityη��, the effect of the oscillating
plane appears to die down more quickly; the velocity decays as height above the oscillating plane increases.
As I explore the flows produced by my biomimetic cilia in both viscous and viscoelastic fluids, these hydrodynamics solutions will be applied as a way to understand the underlying fluid dynamics in my systems, including the relationship among cilia beat frequency, flow velocity, and fluid viscosity and elasticity.