--- title: "Navier-Stokes simplification, velocity distribution" subject: "Fluid Mechanics" chapter: 5 key: "semester_3/fluid_mechanics:navier-stokes-simplified-couette" source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/navier-stokes-simplified-couette site: BCT Engineering --- # Navier-Stokes simplification, velocity distribution Fluid Mechanics · Chapter 5: Hydrodynamics - Asked in: 2076 Chaitra, 1+2+1 marks - Video: [Fluid Mechanics Lesson 11A: Exact Solutions of the Navier-Stokes Equation](https://www.youtube.com/watch?v=XUEvY2Whoyw) — John Cimbala - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/navier-stokes-simplified-couette - Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-5 ## Question Write Navier-Stoke's equation in three dimensional form (derivation not required). If the flow is steady and incompressible; no flow or property variation in z-direction, fully developed flow (no property variation in x direction), model the above written Navier Stokes equation in simplified form using the assumptions. Can you develop simplified velocity distribution equation from the simplified model? ## Short answer Simplified equation: μ d²u/dy² = dp/dx. Integrating twice gives u = (1/(2μ))(dp/dx)(y²−hy) + (U/h)y; for fixed plates it's a parabolic Poiseuille profile, for zero pressure gradient it's linear Couette flow. ## Answer #### Navier-Stokes equation (3-D, incompressible, constant $\mu$) $$ \begin{aligned} \rho\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right) &= \rho g_x - \frac{\partial p}{\partial x} + \mu\nabla^2 u\\ \rho\left(\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}\right) &= \rho g_y - \frac{\partial p}{\partial y} + \mu\nabla^2 v\\ \rho\left(\frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}\right) &= \rho g_z - \frac{\partial p}{\partial z} + \mu\nabla^2 w \end{aligned} $$ with $\nabla^2 = \partial^2/\partial x^2 + \partial^2/\partial y^2 + \partial^2/\partial z^2$ and continuity $\partial u/\partial x + \partial v/\partial y + \partial w/\partial z = 0$. #### Simplification Assumptions: steady ($\partial/\partial t = 0$); no z-variation and $w = 0$; fully developed ($\partial/\partial x = 0$ for velocity). Continuity then gives $\partial v/\partial y = 0$, and with $v = 0$ at a wall, $v = 0$ everywhere. Take $x$ along the flow and gravity along $-y$ ($g_x = 0$, $g_y = -g$). - Left-hand side: all inertia terms vanish ($u\,\partial u/\partial x = 0$, $v = 0$). - x-momentum: $$ 0 = -\frac{\partial p}{\partial x} + \mu\frac{d^2u}{dy^2} $$ - y-momentum: $$ 0 = -\rho g - \frac{\partial p}{\partial y} $$ - z-momentum: $0 = -\partial p/\partial z$. So the pressure is hydrostatic across the flow and $dp/dx$ is constant, giving the simplified model: $$ \mu\frac{d^2u}{dy^2} = \frac{dp}{dx} $$ #### Velocity distribution Yes. Integrating twice: $$ \frac{du}{dy} = \frac{1}{\mu}\frac{dp}{dx}y + C_1,\qquad u = \frac{1}{2\mu}\frac{dp}{dx}y^2 + C_1y + C_2 $$ For Couette-type flow between plates at $y = 0$ (fixed) and $y = h$ (moving at $U$), with $u(0) = 0$, $u(h) = U$: $$ u = \frac{U y}{h} + \frac{1}{2\mu}\frac{dp}{dx}\left(y^2 - hy\right) $$ If $dp/dx = 0$, this reduces to the linear profile $u = Uy/h$ (simple Couette flow). If both plates are fixed ($U=0$), it gives the parabolic Poiseuille profile $u = \dfrac{1}{2\mu}\dfrac{dp}{dx}(y^2 - hy)$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is v = 0 everywhere? A. Because gravity acts along y B. Because pressure is constant C. du/dx = 0 and w = 0 give dv/dy = 0, and v = 0 at a wall D. Because flow is steady Answer: C. Continuity then gives dv/dy = 0 and the wall condition makes v = 0. (Key point: Continuity result) 2. What is the simplified x-momentum equation? A. rho u du/dx = -dp/dx B. d2u/dx2 = 0 C. dp/dx = rho g D. mu d2u/dy2 = dp/dx Answer: D. Inertia terms vanish, leaving 0 = -dp/dx + mu d2u/dy2. (Key point: Simplified model) 3. What does the y-momentum equation imply? A. Hydrostatic pressure variation across the flow B. Pressure constant C. Velocity linear D. Zero gravity Answer: A. 0 = -rho g - dp/dy. (Key point: y-momentum) 4. Velocity between a fixed plate and a plate moving at U? A. u = Uy/h only B. u = Uy/h + (1/2mu)(dp/dx)(y^2 - hy) C. u = (1/2mu)(dp/dx) y^2 D. u = U Answer: B. Integrating twice with u(0)=0, u(h)=U. 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