Question 1 of 5: Continuity Test of a Given Velocity Field
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams / EGBC — May 2013 — 04-CHEM-B1 Transport Phenomena. Three-hour open-book examination; any non-communicating calculator permitted. Five questions, of unequal weight, and all five must be answered (Q1 15, Q2 30, Q3 25, Q4 15, Q5 15 marks). The conservation equations (continuity, Navier–Stokes, shear-stress/velocity-gradient, energy and species) are supplied as Tables 1–5 appended to the paper. All five are solved below: four are pure derivations (Q1, Q2, Q3, Q5) and one (Q4) closes with a numerical mass-transfer rate.
Reference texts: R. S. Brodkey & H. C. Hershey, Transport Phenomena — A Unified Approach (McGraw-Hill) — the paper’s own reference, source of the appended Tables 1–5; R. B. Bird, W. E. Stewart & E. N. Lightfoot, Transport Phenomena (2nd ed., Wiley) — the equations of change, differential momentum/energy balances and diffusion with reaction; J. R. Welty, C. E. Wicks, R. E. Wilson & G. L. Rorrer, Fundamentals of Momentum, Heat and Mass Transfer (Wiley) — non-equimolar diffusion with a heterogeneous surface reaction; F. P. Incropera & D. P. DeWitt, Fundamentals of Heat and Mass Transfer (Wiley) — steady conduction in a spherical wall.
Question 1: Continuity Test of a Given Velocity Field (15 marks)
Given. A three-dimensional velocity field with components $u_x=4t+4x+4y$, $u_y=t-2y-2z$ and $u_z=t+x-2z$, for a fluid taken to be incompressible ($\rho=\text{const}$).
Find. Whether the field satisfies the incompressible continuity equation $\nabla\cdot\vec{u}=0$ and is therefore a kinematically admissible flow.
Approach. For a constant-density fluid the continuity equation reduces to a divergence-free (solenoidal) condition on the velocity, so evaluate the three spatial partial derivatives and test whether they sum to zero.
Reduce continuity for an incompressible fluid. The general continuity equation $\partial\rho/\partial t + \nabla\cdot(\rho\vec{u})=0$ with $\rho=\text{const}$ collapses to $\nabla\cdot\vec{u} = \dfrac{\partial u_x}{\partial x} + \dfrac{\partial u_y}{\partial y} + \dfrac{\partial u_z}{\partial z} = 0$. This is a constraint on the spatial divergence at every instant, so the explicit $t$-dependence of the field is admissible and does not by itself violate the condition.
Evaluate the three partial derivatives. Differentiating each component with respect to its own coordinate, $\dfrac{\partial u_x}{\partial x}=4$, $\dfrac{\partial u_y}{\partial y}=-2$ and $\dfrac{\partial u_z}{\partial z}=-2$.
Sum and test. Adding the three, $\nabla\cdot\vec{u} = 4 + (-2) + (-2) = \boxed{0}$. The divergence vanishes identically (at all $x,y,z,t$), so the field conserves mass for an incompressible fluid and is a physically possible flow.
Why the $y$-coefficient matters. Had the $u_y$ gradient been anything other than $-2$ (say $\partial u_y/\partial y = a$), the divergence would be $2+a\neq0$ and the flow impossible; the printed value $-2$ is exactly what incompressibility demands.
