Fluid Dynamics Notes

Basics About Fluids

  • Fluids are substances that continuously deform under the action of shearing forces (even very small ones).

Viscosity

  • Dynamic Viscosity:

    • Defined as the resistance of a fluid to deformation.
    • Shear stress ($\tau$) is considered when one layer of fluid slides over another.
    • Formula: $\tau = \text{Force per unit area}$.
    • Viscosity ($\eta$) is proportional to the velocity gradient or shear rate ($\frac{du}{dy}$).

  • Kinematic Viscosity:

    • A relevant ratio of the dynamic viscosity to fluid density; defined as $\nu = \frac{\eta}{\rho}$ where $\eta$ is dynamic viscosity and $\rho$ is density.

Newtonian vs. Non-Newtonian Fluids

  • Newtonian Fluids:

    • Deform according to constant viscosity irrespective of the velocity gradient.

  • Non-Newtonian Fluids:

    • Viscometry is dependent on several factors (temperature, shear rate, etc.).

Pressure and Head

  • Pressure:

    • Defined as the force exerted per unit area on a boundary or plane within the fluid.
    • Variation in pressure can be assessed using hydrostatics formula $p = \rho gh + p_{atm}$, where:
    • $p$ = pressure
    • $\rho$ = fluid density
    • $g$ = acceleration due to gravity
    • $h$ = height of the fluid column

  • Head:

    • Reference point commonly employed to simplify pressure calculation. Derived from the pressure formula when atmospheric pressure is considered zero, resulting in $p = \rho gh$.

Types of Flow

  • Uniform Flow:

    • Velocity is the same in magnitude and direction at every point in the fluid.

  • Non-Uniform Flow:

    • Velocity changes from point to point.

  • Steady Flow:

    • Conditions (velocity, pressure) remain constant with time, though they may vary spatially.

  • Unsteady Flow:

    • Conditions vary with time at given points.

Summary of Flow Types

  • Steady Uniform Flow:
    • Constant velocity and cross-section.
    • e.g., liquid flow through a uniform pipe.
  • Steady Non-uniform Flow:
    • Varying velocity at different points in the cross-section but constant over time.
  • Unsteady Uniform Flow:
    • Constant velocity across all points but changing over time.
  • Unsteady Non-Uniform Flow:
    • Both cross-sectional area and velocity vary simultaneously over time.

Streamlines

  • Imaginary curves indicating fluid motion where fluid velocity is tangent at every point along the streamline.
  • Streamlines can form closed curves around solid objects in flow.
  • A streamtube is formed by streamlines that enclose a flow entity, ensuring fluid cannot escape its contour.

Laminar and Turbulent Flow

  • Laminar Flow:

    • Smooth, steady, and exhibits uniformity without mixing (e.g., low velocity taps).

  • Turbulent Flow:

    • Chaotic and involves mixing; typically occurs at high flow velocity.

Transition Zone:

  • A region in which laminar flow transitions to turbulent, characterized by ripples and oscillations.

Velocity Variations

  • Pipe Flow:
    • Velocity at the wall is zero, increases toward the centerline.
  • Open Channel Flow:
    • Velocity governed by depth; zero at the lowest point and highest near the surface.

Velocity Profiles

  • Laminar flow through pipes results in a parabolic velocity profile, while turbulent flow leads to a flatter, more uniform velocity profile.

Reynolds Experiment and Number

  • Explores flow types dependent on fluid density ($\rho$), viscosity ($\eta$), pipe diameter ($D$), and velocity ($V$).

  • Reynolds Number $N_{R}$:

    • Dimensionless quantity determining the flow characterization.
    • Defined as $N_{R} = \frac{\rho v D}{\eta}$.

  • Critical Values:

    • $N_{R} < 2000$: Laminar Flow
    • $2000 < N_{R} < 4000$: Transition Zone
    • $N_{R} > 4000$: Turbulent Flow

Reynolds Number Calculation Example

  • Given density $\rho = 80 kg/m^3$, viscosity $\eta = 0.7 N⋅s/m^2$, velocity $V = 6 m/s$, diameter $D = 0.25 m$;
    • Calculate $N_{R} = \frac{80 kg/m^3 \cdot 6 m/s \cdot 0.25 m}{0.7 kg/m^{-1} s^{-1}} = 171$ (Indicating laminar flow).