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GATE ME Conservation Laws — practice questions

Explore 4 free Conservation Laws questions in GATE ME Mechanical Engineering. Open each question or exercise to read its content, reference answers and available explanation.

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Conservation Laws questions and exercises with answers

Showing questions 1–4 of 4.

The CONSERVATION OF MASS equation for an incompressible fluid ($\rho$ = const) is:Read the answer and worked solution →The Navier-Stokes equations express:Read the answer and worked solution →For an INCOMPRESSIBLE Newtonian fluid with constant μ, the Navier-Stokes equation reduces to:Read the answer and worked solution →Conservation of THERMAL ENERGY in a flowing fluid involves which key terms?Read the answer and worked solution →

Worked practice questions

The CONSERVATION OF MASS equation for an incompressible fluid ($\rho$ = const) is:

  1. $\nabla \cdot \mathbf{u} = 0$ (divergence-free velocity field)
  2. $\rho \mathbf{u} = 0$
  3. $\nabla \rho = 0$
  4. $\partial u/\partial t = 0$
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Correct answer: A. $\nabla \cdot \mathbf{u} = 0$ (divergence-free velocity field)

**General mass conservation:** $\partial \rho/\partial t + \nabla \cdot (\rho \mathbf{u}) = 0$. For incompressible flow ρ = const: $\nabla \cdot \mathbf{u} = 0$. The velocity field has zero divergence — fluid cannot accumulate.

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The Navier-Stokes equations express:

  1. Conservation of mass only
  2. Conservation of MOMENTUM for a Newtonian viscous fluid
  3. Conservation of energy
  4. Conservation of species
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Correct answer: B. Conservation of MOMENTUM for a Newtonian viscous fluid

N-S is the momentum equation for a Newtonian (linear stress-strain-rate) fluid. **Force-per-volume balance:** time rate of momentum change + advection = pressure gradient + viscous stress + body force.

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For an INCOMPRESSIBLE Newtonian fluid with constant μ, the Navier-Stokes equation reduces to:

  1. $\rho D\mathbf{u}/Dt = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{f}$
  2. $\rho \mathbf{u} = 0$
  3. $\partial u/\partial t = 0$
  4. $\nabla p = 0$
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Correct answer: A. $\rho D\mathbf{u}/Dt = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{f}$

Incompressible + constant μ: $\rho(\partial \mathbf{u}/\partial t + \mathbf{u}\cdot\nabla \mathbf{u}) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{f}$. LHS = inertial; RHS = pressure + viscous + body force. The famous N-S equation.

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