18 Questions
In the Navier-Stokes Equation, the term involving the pressure gradient is responsible for the acceleration of the fluid.
False
The Navier-Stokes Equation applies to both compressible and incompressible fluids.
True
In Couette flow, parallel plates move at different velocities creating net acceleration in the flow.
False
Microfluidic Systems typically deal with fluid flow at macroscopic scales.
False
Newtonian Law of Viscosity states that the rate of deformation is directly proportional to the shear stress applied.
True
The Navier-Stokes Equation includes terms for viscous forces, pressure forces, and body forces.
True
In the context of microfluidic systems, for laminar flow, the gradient of velocity is zero.
True
According to the Navier-Stokes Equation in the provided text, the steady-state condition implies that the partial derivative of velocity with respect to time is not zero.
False
The 2nd Newtonian Law of Viscosity mentioned in the text involves the change of velocity with respect to time and coordination system.
True
In microfluidic systems, the force element with velocity 'v' determines the floating velocity in the fluid.
False
According to the provided content, pulsatile flow implies that the gradient of velocity is not zero.
True
For inertial forces to be acting in fluid flow as per the text, the gradient of velocity must be zero.
False
The Laplacian operator on a scalar is defined as the sum of the partial derivatives of the scalar function with respect to each spatial variable squared.
True
The Navier-Stokes Equation involves the Laplacian operator applied to a vector.
False
In fluid mechanics, the assumption of incompressible fluid implies that the density of the fluid can vary throughout the flow.
False
According to the conservation of mass, if there are no sources and sinks in the flow, then the density multiplied by the velocity field divergence must be zero.
True
The Navier-Stokes Equation describes how the velocity field of a fluid evolves over time.
True
Newtonian Law of Viscosity states that the shear stress in a fluid is directly proportional to the rate of strain in the fluid.
True
Study Notes
Navier-Stokes Equation
- The Navier-Stokes equation is a non-linear partial differential equation that describes the velocity distribution of a fluid in dependency of position and time.
- It is a conservation of momentum equation applied to a fluid element.
Navier-Stokes Equation for Incompressible Newtonian Fluids
- The final Navier-Stokes equation for incompressible Newtonian fluids is: ∂ρv/∂t + ρv∇v = -∇p + η∇²v + f_volume
- It consists of pressure, friction, and volume forces.
Couette Flow Profile
- Couette flow is a flow between two parallel plates, where the top plate moves with velocity uH and the bottom plate moves with velocity uL.
- The flow has no acceleration, no net pressure force, and no net convective transport of momentum.
- The flow profile is calculated using the Navier-Stokes equation without body force.
Microfluidic Channel
- The velocity in the channel is determined by the force (pressure) element with velocity v, and the velocity gradient is added.
- In a steady-state, ∂v/∂t = 0, and for laminar flow, ∇v = 0.
- For pulsatile flow, ∂v/∂t ≠ 0 (e.g., aortic vessel).
2nd Newtonian Law
- The 2nd Newtonian law states that the change of velocity is equal to the sum of the forces acting on the fluid element.
- It is described by the equation: dv/dt = Σf_j
- The law is applied by changing the coordination system, using the chain rule.
Laplacian Operator
- The Laplacian operator is a mathematical operator used to describe the distribution of a scalar or vector field.
- It is defined as: ∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z²
- It is used to describe the distribution of pressure and velocity in a fluid.
Flow Assumptions
- The first assumption is that the fluid is incompressible with constant density ρ.
- The second assumption is that the conservation of mass is maintained, meaning no sources and sinks.
- These assumptions lead to the continuity equation: ∇⋅v = 0.
Test your knowledge on the Navier-Stokes Equation, a non-linear partial differential equation that specifies the velocity distribution of a fluid in relation to position and time. Learn about the Conservation of Momentum applied to fluid elements in the context of Microfluidic Systems.
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