Question

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.

• A. It represents the conservation of linear momentum.
• B. It is a nonlinear partial differential equation.
• C. It is valid for both Newtonian and non-Newtonian fluids.
• D. It is valid for incompressible flow.

Hint:

Check what the convective term does to linearity, and what the single-viscosity term assumes about the fluid.

Answer 1

The Correct Option is A, B, D

Step 1: Recognise the equation.
This is the Navier-Stokes equation written per unit mass. The left side is the material acceleration of a fluid particle. The right side collects the forces per unit mass acting on it: pressure gradient, gravity, and viscous diffusion.

Step 2: Check statement (A).
By Newton's second law, mass times acceleration equals net force. This equation is exactly that law applied to a moving fluid element, written as acceleration equals force per unit mass. So it states conservation of linear momentum, and (A) is TRUE.

Step 3: Check statement (B).
The convective term \(\mathbf{u}\cdot\nabla\mathbf{u}\) multiplies the unknown velocity by its own spatial derivative. This makes the equation nonlinear in \(\mathbf{u}\), unlike the other, linear terms. So (B) is TRUE.

Step 4: Check statement (C).
The viscous term \(\vartheta \nabla^{2}\mathbf{u}\) uses one constant kinematic viscosity \(\vartheta\), which comes from a linear Newtonian stress-strain rate relation. A non-Newtonian fluid has a viscosity that depends on the shear rate itself, so this exact form no longer applies. So (C) is FALSE.

Step 5: Check statement (D).
There is no density-variation term coupling into this momentum balance, and the viscous term used here is the simplified form that assumes \(\nabla \cdot \mathbf{u} = 0\). This is the standard incompressible Navier-Stokes momentum equation, so (D) is TRUE.

Final Answer:
The equation is Newton's second law for a Newtonian, incompressible fluid, and it is nonlinear from the convective acceleration term. \[ \boxed{\text{A, B, D}} \]