Question:medium
Pick the CORRECT statement(s) corresponding to the following equation
\[ \dfrac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = -\dfrac{1}{\rho}\nabla p + \mathbf{g} + \vartheta \nabla^{2}\mathbf{u} \]
where \(\mathbf{u}\) is the velocity vector, \(p\) is the pressure, \(\rho\) is the density, \(\mathbf{g}\) is the gravitational acceleration vector, \(t\) is time, and \(\vartheta\) is the kinematic viscosity.
Pick the CORRECT statement(s) corresponding to the following equation
\[ \dfrac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = -\dfrac{1}{\rho}\nabla p + \mathbf{g} + \vartheta \nabla^{2}\mathbf{u} \] where \(\mathbf{u}\) is the velocity vector, \(p\) is the pressure, \(\rho\) is the density, \(\mathbf{g}\) is the gravitational acceleration vector, \(t\) is time, and \(\vartheta\) is the kinematic viscosity.
\[ \dfrac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = -\dfrac{1}{\rho}\nabla p + \mathbf{g} + \vartheta \nabla^{2}\mathbf{u} \] where \(\mathbf{u}\) is the velocity vector, \(p\) is the pressure, \(\rho\) is the density, \(\mathbf{g}\) is the gravitational acceleration vector, \(t\) is time, and \(\vartheta\) is the kinematic viscosity.
Show Hint
Check what the convective term does to linearity, and what the single-viscosity term assumes about the fluid.
Updated On: Jul 28, 2026
- It represents the conservation of linear momentum.
- It is a nonlinear partial differential equation.
- It is valid for both Newtonian and non-Newtonian fluids.
- It is valid for incompressible flow.
Show Solution
The Correct Option is A, B, D
Solution and Explanation
The given equation is the Navier-Stokes momentum equation. Each statement can be checked by recalling what assumptions go into deriving this particular form, and what mathematical character its terms have.
- It represents the conservation of linear momentum: true. The left side is the acceleration of a fluid particle following the flow, and the right side is the sum of the forces per unit mass acting on it, which is Newton's second law for a fluid.
- It is a nonlinear partial differential equation: true. The convective term $\mathbf{u} \cdot \nabla \mathbf{u}$ has the unknown $\mathbf{u}$ multiplying its own derivative, which makes the whole equation nonlinear in the velocity field.
- It is valid for both Newtonian and non-Newtonian fluids: false. The last term uses a single constant kinematic viscosity $\vartheta$, which only describes a Newtonian fluid. A non-Newtonian fluid needs a different, shear-rate dependent stress model.
- It is valid for incompressible flow: true. The viscous term written as $\vartheta \nabla^2 \mathbf{u}$ is the simplified incompressible form of the viscous stress divergence, so this is the standard incompressible Navier-Stokes equation.
So the correct statements are the first, second and fourth, giving option A, B, D.
Was this answer helpful?
0
