--- title: "Navier-Stokes equation and its terms" subject: "Applied Computational Fluid Dynamics" chapter: 2 key: "semester_7/applied_computational_fluid_dynamics:navier-stokes-terms-meaning" source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/navier-stokes-terms-meaning site: BCT Engineering --- # Navier-Stokes equation and its terms Applied Computational Fluid Dynamics · Chapter 2: Governing Equations and Numerical Foundations - Asked in: year not given, 8 marks - Video: [Description and Derivation of the Navier-Stokes Equations](https://www.youtube.com/watch?v=NjoMoH51UZc) — LearnMechE - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/navier-stokes-terms-meaning - Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-2 ## Question Write the Navier-Stokes momentum equation for an incompressible Newtonian fluid with constant properties. Explain the physical meaning of each term and state the assumptions used. Why is the equation difficult to solve? ## Short answer The incompressible Navier–Stokes equation is ρ(∂u/∂t + u·∇u) = −∇p + μ∇²u + ρg. Terms: local acceleration, nonlinear convective acceleration, pressure gradient, viscous diffusion, body force. It is difficult because the convective term is nonlinear, pressure lacks an explicit equation, and turbulence at high Re demands very fine grids or modelling. ## Answer The **Navier-Stokes (N-S) equations** express Newton's second law ($\text{mass}\times\text{acceleration}=\text{force}$) for a fluid element. #### Equation (vector form) $$ \rho\left(\frac{\partial\vec V}{\partial t}+(\vec V\cdot\nabla)\vec V\right)=-\nabla p+\mu\nabla^2\vec V+\rho\vec g $$ In Cartesian form, x-direction: $$ \rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)+\rho g_x $$ with similar equations for $y$ and $z$. It is used together with $\nabla\cdot\vec V=0$. #### Meaning of terms | Term | Name | Physical meaning | |---|---|---| | $\rho\,\partial\vec V/\partial t$ | Local (unsteady) inertia | Rate of change of momentum at a fixed point | | $\rho(\vec V\cdot\nabla)\vec V$ | Convective inertia | Momentum carried by the flow; non-linear term | | $-\nabla p$ | Pressure force | Force per unit volume from the pressure gradient | | $\mu\nabla^2\vec V$ | Viscous (diffusion) force | Momentum diffusion by molecular friction | | $\rho\vec g$ | Body force | Gravity (buoyancy etc.) per unit volume | The left side is mass times acceleration per unit volume; the right side is the sum of forces. #### Assumptions - Continuum, Newtonian fluid (stress proportional to strain rate). - Constant density and viscosity (incompressible, isothermal). - Stokes' hypothesis for bulk viscosity is not needed when $\nabla\cdot\vec V=0$. #### Why difficult to solve - The convective term $u\,\partial u/\partial x$ is **non-linear**, so superposition fails and no general analytical solution exists. - Four unknowns ($u,v,w,p$) are coupled with four equations, and pressure has no equation of its own (needs pressure-velocity coupling). - At high $Re$ the solution is turbulent, with a wide range of scales, so very fine meshes are needed (DNS) or modelling (RANS/LES). - Analytical solutions exist only for simple cases such as Couette and Poiseuille flows, which are used to test CFD codes. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The Navier-Stokes equation expresses: A. Newton's second law for a fluid element B. Conservation of mass only C. The first law of thermodynamics D. Bernoulli's principle only Answer: A. The left side is mass times acceleration per unit volume and the right side is the sum of forces. (Key point: Physical basis) 2. Which term makes the equation non-linear? A. Pressure gradient -grad p B. Viscous term mu del^2 V C. Convective inertia rho (V.grad)V D. Body force rho g Answer: C. The convective term such as u du/dx is non-linear, so superposition fails. (Key point: Convective term) 3. The term mu del^2 V represents: A. Gravity force B. Momentum diffusion by molecular friction C. Local unsteady inertia D. Pressure force Answer: B. It is the viscous (diffusion) force per unit volume. (Key point: Viscous term) 4. Which assumption is used in this form of the equation? A. Inviscid fluid B. Compressible flow C. Non-Newtonian fluid D. Newtonian fluid with constant density and viscosity Answer: D. The form given is for a Newtonian, incompressible fluid with constant properties. (Key point: Assumptions) 5. Why is pressure hard to handle in the incompressible equations? A. Pressure is constant B. It has no equation of its own, so pressure-velocity coupling is needed C. It appears only in the energy equation D. It is not a variable Answer: B. Four unknowns (u, v, w, p) are linked to momentum and continuity, but pressure has no separate transport equation. (Key point: Difficulty of solving) ## Related in Applied Computational Fluid Dynamics - [Plane Poiseuille flow from Navier-Stokes](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aplane-poiseuille-flow-numerical) — Applied Computational Fluid Dynamics, chapter 2 - [Enthalpy formulation and heat transfer equation](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aenthalpy-energy-equation-heat-transfer) — Applied Computational Fluid Dynamics, chapter 2 - [Stokes flow and settling velocity](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Astokes-flow-settling-particle) — Applied Computational Fluid Dynamics, chapter 2 ## Related in other subjects - [Navier-Stokes equations for Newtonian fluid](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Anavier-stokes-newtonian-fluid) — Fluid Mechanics, chapter 4 - [Navier-Stokes and Bernoulli equations with terms](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Anavier-stokes-bernoulli-terms) — Fluid Mechanics, chapter 5 - [Navier-Stokes simplification, velocity distribution](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Anavier-stokes-simplified-couette) — Fluid Mechanics, chapter 5 - [Derive Navier-Stokes equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aderive-navier-stokes-equation) — Mathematics III, chapter 2, asked 3×