Which of the following velocity fields satisfies continuity equation?

Continuity Equation Analysis in Fluid Dynamics

Physical Significance

The continuity equation (∇·V = 0) represents:

Conservation of mass in fluid flow
Balance between incoming and outgoing flow rates
Requirement for incompressible flows
Divergence-free velocity fields

A satisfying solution means the flow could physically exist without mass accumulation/depletion. Violation indicates either:

Compressible flow (density changes)
Non-physical velocity field
Transient flow conditions

Problem Statement

Identify which 2D/3D velocity fields satisfy the continuity equation for steady, incompressible flow:

a) u = 4xy + y², v = 6xy + 3x

Mathematical Analysis:
∂u/∂x = 4y
∂v/∂y = 6x
Divergence: 4y + 6x ≠ 0
Physical Interpretation:
Positive divergence indicates:

Net outflow from control volume
Density decreasing with time
Non-conservative mass flow

b) u = 2x² + y², v = -4xy

Mathematical Analysis:
∂u/∂x = 4x
∂v/∂y = -4x
Divergence: 4x – 4x = 0
Physical Interpretation:
Zero divergence means:

Mass inflow = Mass outflow
Density remains constant
Possible rotational flow pattern

c) 3D Field: u = 2x² – xy + z², v = x² – 4xy + y², w = -2xy – yz + y²

Mathematical Analysis:
∂u/∂x = 4x – y
∂v/∂y = -4x + 2y
∂w/∂z = -y
Divergence: (4x – y) + (-4x + 2y) + (-y) = 0
Physical Interpretation:
Three-dimensional balance shows:

Compensation through z-component
Complex flow path conservation
Valid for 3D incompressible flow

d) u = -Ky/(x²+y²), v = Kx/(x²+y²)

Mathematical Analysis:
∂u/∂x = (2Kxy)/(x²+y²)²
∂v/∂y = (-2Kxy)/(x²+y²)²
Divergence: 2Kxy/(x²+y²)² – 2Kxy/(x²+y²)² = 0
Physical Interpretation:
Represents potential flow:

Cylindrical coordinate system flow
Circular streamlines
Vortex-like motion without sources/sinks

Engineering Significance

Valid continuity solutions (b, c, d) are crucial for:

Computational Fluid Dynamics (CFD) validation
Hydraulic system design
Weather pattern modeling
Aerodynamic flow simulations

Non-physical solutions (a) would create mass imbalance in real-world applications.