two horizontal plates are separated by a distance h and the interstitial space is filled with water. the two plates are motionless. at time t, the top plate is moved with a velocity U=1 m/s. what is the velocity field u(y,t)
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
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PURPOSE OF THE EXERCISE:
This exercise tests your ability to solve an unsteady viscous flow problem by reducing the Navier–Stokes equations to a diffusion (heat‐equation type) problem and applying separation of variables to capture the transient evolution from rest to steady Couette flow. -
ENGINEERING CONTEXT:
Understanding the transient response of a suddenly started plate is crucial in designing lubrication systems and predicting start-up behavior in fluid film bearings. Once mastered, you can predict how quickly a fluid layer adjusts to boundary motion in devices such as journal bearings. -
PHYSICAL CONTEXT:
The dominant physical effect is viscous momentum diffusion across the gap between the plates. -
ASSUMPTIONS:
Assume the fluid is incompressible and Newtonian (water, with \rho=1000\,\mathrm{kg/m^3} and constant viscosity), the flow is one-dimensional (velocity in the x direction depends only on the vertical coordinate y), and body forces are negligible. These assumptions are reasonable due to the large horizontal extent of the plates and the small gap compared to in-plane dimensions. -
DRAWINGS:
A simple diagram shows two horizontal, infinite plates separated by a distance h, with the coordinate y running from 0 (bottom plate, no-slip condition: u(0,t)=0) to h (top plate, suddenly moved to u(h,t)=U=1\,\mathrm{m/s} for t>0). Indicate that at t=0 the fluid is at rest, i.e. u(y,0)=0. -
SOLUTION TECHNIQUE:
The governing equation is the one-dimensional diffusion equation for momentum,
\frac{\partial u}{\partial t} = \nu\,\frac{\partial^2 u}{\partial y^2}\,,
with \nu=\eta/\rho. The boundary conditions are
u(0,t)=0\quad\text{and}\quad u(h,t)=U\,,
and the initial condition is
u(y,0)=0\,.
To account for the nonhomogeneous boundary condition at y=h, decompose the solution as
u(y,t)= u_{\mathrm{steady}}(y) + \tilde{u}(y,t)\,,
where the steady part is the linear Couette profile,
u_{\mathrm{steady}}(y)= U\,\frac{y}{h}\,,
which satisfies u(0)=0 and u(h)=U, and \tilde{u}(y,t) represents the transient correction that satisfies homogeneous boundary conditions:
\tilde{u}(0,t)=0,\quad \tilde{u}(h,t)=0\,,
with initial condition
\tilde{u}(y,0)= -U\,\frac{y}{h}\,.
For \tilde{u}(y,t), apply separation of variables in the domain 0\le y\le h using eigenfunctions that satisfy
Y_n(0)=0\quad \text{and} \quad Y_n(h)=0\,.
The appropriate eigenfunctions are
Y_n(y)= \sin\left(\frac{n\pi y}{h}\right)\,,
with eigenvalues \lambda_n=\nu\left(\frac{n\pi}{h}\right)^2. Expanding the initial condition in this eigenfunction basis yields
\tilde{u}(y,t)= \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi y}{h}\right) \exp\Bigl[-\nu\Bigl(\frac{n\pi}{h}\Bigr)^2 t\Bigr]\,,
with coefficients determined by
A_n= \frac{2}{h} \int_0^h \left[-U\,\frac{y}{h}\right]\sin\left(\frac{n\pi y}{h}\right)dy\,.
A brief calculation shows that
A_n= -\frac{2U}{(n\pi)^2}\Bigl[1-(-1)^n\Bigr]\,.
Noting that only odd values of n contribute (since 1-(-1)^n=0 for even n), the complete solution is
u(y,t)= U\,\frac{y}{h} - \frac{4U}{\pi^2}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\sin\left(\frac{(2n+1)\pi y}{h}\right)e^{-\nu\Bigl(\frac{(2n+1)\pi}{h}\Bigr)^2t}\,.
- REFLECTION:
This series solution shows how the fluid velocity evolves from a quiescent state to the steady linear profile of Couette flow. The transient terms decay exponentially with time, governed by the viscous diffusion timescale \sim h^2/\nu. In practical engineering applications, understanding this transient behavior is key when predicting startup times in lubrication layers or thin film flows.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 11.2: Couette Flow Part 1 (HIGHLY RELEVANT)
- Chapter 10 Fundamentals: Solving the boundary layer equations via a similarity transform for the fla… (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
- Chapter 11.2: Couette Flow Part 2 (HIGHLY RELEVANT)
- Chapter 10 Fundamentals: The boundary-layer approximation for the steady isobaric laminar flat plate… (HIGHLY RELEVANT)
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