Lecture 2C: Equation of Fluid Statics

Overview

• Derive the equation of fluid statics.
• Discuss the simplified case for incompressible fluid.

Key Concepts

• Fluid Element: Consider an infinitesimal fluid element like a free body diagram; in hydrostatics, the fluid is at rest, therefore, no acceleration.
• Forces in Hydrostatics:
  • No viscous/shear forces in a fluid at rest.
  • Only normal forces can act (pressure forces).

Derivation of Equation

Body Forces

• Gravity Force: The only body force, expressed as (-\rho g , dx , dy , dz , \mathbf{k}) where (\rho) is density, (g) is gravity, and (\mathbf{k}) is the unit vector in the z-direction.

Surface Forces

• Pressure Forces: Act inward and normal on surfaces.
• Pressure is a function of space (x, y, z) and time (t).
• Use of Truncated Taylor Series to express pressure at a point:
  • (P(x+\Delta x) \approx P(x) + \frac{\partial P}{\partial x} \Delta x)
  • Ignore higher-order terms for infinitesimal elements.

Summation of Forces

• X-Direction:
  • No body forces; sum of surface forces: (-\frac{\partial P}{\partial x} dx , dy , dz = 0) yielding (\frac{\partial P}{\partial x} = 0).
• Y-Direction:
  • Similar to X-direction: (\frac{\partial P}{\partial y} = 0).
• Z-Direction:
  • Includes both body and surface forces; results in (\frac{\partial P}{\partial z} = -\rho g) (Equation 1).

Conclusions

• Pressure Variation:
  • Pressure does not vary in the x or y directions in hydrostatics.
  • Pressure varies in the z direction, decreases as you go up, increases as you go down.

Fluid Statics Equation

• General Form: (\frac{dP}{dz} = -\rho g) (Equation 2).
• For incompressible fluids: (P_2 - P_1 = \rho g (z_2 - z_1)).

Simplified Case: Incompressible Fluid

• Integration: For constant (\rho) and (g), integrate to find pressure difference: [P_{below} = P_{above} + \rho g \Delta z]
• Application: Useful for determining pressure differences in fluids at rest.

Important Notes

• For non-constant density, integration is required.
• Reminder: Pressure increases with depth (e.g., swimming scenario).

• End of Lecture
• Reminder to subscribe for more educational content.