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21-Mat-A2 Materials Transport Phenomena · December 2019

Question 1 of 5: Power-Law Fluid Between Parallel Plates

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2019 — 12-MTL-A2 Transport Phenomena in Materials Engineering. Three-hour, open-book exam (one textbook of the candidate's choice permitted, with margin notations, no loose notes); any non-communicating calculator permitted. Each of the five questions is worth 25 points, and any four constitute a complete paper — only the first four questions as they appear in the answer book are marked. All five are solved below for completeness. Candidates were told to state all assumptions clearly.

Reference texts: Bird, R. B., Stewart, W. E. & Lightfoot, E. N., Transport Phenomena — power-law non-Newtonian flow between parallel plates and annular fully-developed duct flow (Questions 1, 3), matching this paper's own Appendix A conservation-equation tables; Incropera, F. P. et al., Fundamentals of Heat and Mass Transfer — natural-convection correlations for a horizontal and a vertical flat plate (Question 2); Geiger, G. H. & Poirier, D. R., Transport Phenomena in Materials Processing — mould-resistance-controlled solidification of castings (Question 4); Shewmon, P. G., Diffusion in Solids — the Boltzmann–Matano graphical method for a concentration-dependent interdiffusion coefficient (Question 5).

Question 1: Power-Law Fluid Between Parallel Plates (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

QuantitySymbolValue
Constitutive law (power law / Ostwald–de Waele)$\tau_{yx}$$-\eta_0(dV_x/dy)^n$
Plate half-spacing (plates at $y=\pm B$)$B$symbolic
Plate width (into the page)$W$symbolic
Pressure drop over length $L$$\Delta P/L$symbolic, drives the flow

Find. The velocity profile $V_x(y)$ and the volumetric flow rate $Q$ for pressure-driven flow of the power-law fluid between two stationary parallel plates separated in the $y$-direction.

plate (y=+B)plate (y=−B)y=0 (centreline)yx (flow)Vx(y)
Fig. 1 — power-law fluid between stationary plates at $y=\pm B$; flow driven in the $x$-direction by a constant pressure gradient, symmetric velocity profile shown.

Approach. Write a differential momentum balance across the gap to get the (constitutive-law-independent) shear-stress profile, then combine it with the power-law relation and integrate twice, applying centreline symmetry and no-slip at both walls.

  1. Differential momentum balance. For steady, fully developed flow with no $y$- or $z$-momentum, a balance on a differential slab of thickness $dy$, width $W$, length $L$ gives a linear shear-stress profile that is independent of the fluid model, zero at the centreline by symmetry: $$\tau_{yx}=\frac{\Delta P}{L}\,y\qquad(-B\le y\le B)$$
  2. Combine with the power law. For the upper half ($0\le y\le B$) the velocity decreases outward from the centreline, so $dV_x/dy\le0$ there; substituting the constitutive law and solving for the slope: $$-\eta_0\left(\frac{dV_x}{dy}\right)^n=\frac{\Delta P}{L}\,y\;\;\Rightarrow\;\;\frac{dV_x}{dy}=-\left(\frac{\Delta P\,y}{\eta_0 L}\right)^{1/n}$$
  3. Integrate with no-slip at the wall. Integrating from $y$ to $B$ and applying $V_x(B)=0$: $$V_x(y)=\int_y^B\left(\frac{\Delta P}{\eta_0 L}\right)^{1/n}y'^{\,1/n}\,dy'=\frac{n}{n+1}\left(\frac{\Delta P}{\eta_0 L}\right)^{1/n}\left[B^{(n+1)/n}-y^{(n+1)/n}\right]$$ By centreline symmetry the same expression holds for $-B\le y\le 0$ with $y$ replaced by $|y|$: $$\boxed{V_x(y)=\frac{nB}{n+1}\left(\frac{\Delta P\,B}{\eta_0 L}\right)^{1/n}\left[1-\left(\frac{|y|}{B}\right)^{(n+1)/n}\right]}$$
  4. Volumetric flow rate. Integrating $V_x$ across the full gap (symmetric, so twice the $0\to B$ half) and width $W$: $$Q=2W\!\int_0^B\!V_x\,dy=2W\cdot\frac{nB}{n+1}\left(\frac{\Delta P\,B}{\eta_0 L}\right)^{1/n}\!\cdot B\cdot\frac{n+1}{2n+1}$$ $$\boxed{Q=\frac{2nWB^2}{2n+1}\left(\frac{\Delta P\,B}{\eta_0 L}\right)^{1/n}}$$ Newtonian check ($n=1$, $\eta_0=\mu$): the formula collapses to $Q=\tfrac{2}{3}WB^2\!\cdot\!\Delta P B/(\mu L)=\dfrac{2WB^3\Delta P}{3\mu L}$, which is exactly the classical slit (parallel-plate) Poiseuille-flow result for a full gap $2B$.
QuantityResult
Velocity profile, $V_x(y)$$\dfrac{nB}{n+1}\left(\dfrac{\Delta P\,B}{\eta_0 L}\right)^{1/n}\left[1-\left(\dfrac{|y|}{B}\right)^{(n+1)/n}\right]$
Centreline (max) velocity, $V_{x,\max}$$\dfrac{nB}{n+1}\left(\dfrac{\Delta P\,B}{\eta_0 L}\right)^{1/n}$
Volumetric flow rate, $Q$$\dfrac{2nWB^2}{2n+1}\left(\dfrac{\Delta P\,B}{\eta_0 L}\right)^{1/n}$
Newtonian limit ($n=1$) check$Q\to 2WB^3\Delta P/(3\mu L)$ — matches classical slit flow
Check
Assumes steady, fully developed, incompressible laminar flow with no slip at either wall, negligible edge/end effects (width $W$ large compared with $2B$), and that the power law's sign convention is applied consistently with $\tau_{yx}=0$ enforced at the centreline by symmetry rather than by extrapolating a fractional power of a negative slope.
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