Question

The continuity equation of two- dimensional steady incompressible flow is_______

The question was posed to me in an interview for internship.

Question is from Turbulent Boundary Layer on a Flat Plate topic in portion Boundary Layer Flow & Forces on Sub-merged Bodies of Fluid Mechanics

Answer 1

The continuity equation for a two-dimensional steady incompressible flow is given by:

∂u∂x+∂v∂y=0\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0∂x∂u​+∂y∂v​=0

---

Explanation:

1. Two-Dimensional Flow:

   • The flow has two velocity components: $u$ (velocity in the $x$-direction) and $v$ (velocity in the $y$-direction).

2. Steady Flow:

   • The flow properties do not change with time, i.e., ∂∂t=0\frac{\partial}{\partial t} = 0∂t∂​=0.

3. Incompressible Flow:

   • The density of the fluid remains constant, so mass conservation simplifies to the divergence of the velocity field being zero.

---

Derivation:

From the principle of conservation of mass:

∂ρ∂t+∇⋅(ρV⃗)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0∂t∂ρ​+∇⋅(ρV)=0

For steady ($\partial \rho /\partial t=0$) and incompressible flow ($\rho =\text{constant}$):

∇⋅V⃗=0\nabla \cdot \vec{V} = 0∇⋅V=0

Expanding in two dimensions (V⃗=ui^+vj^\vec{V} = u \hat{i} + v \hat{j}V=ui^+vj^​):

∂u∂x+∂v∂y=0\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0∂x∂u​+∂y∂v​=0

---

This equation ensures that the fluid entering a region equals the fluid leaving it, maintaining mass conservation.