3.2 Hydrostatics

3.2 Hydrostatics

Introduction

• Hydrostatics is the study of fluids at rest and the forces exerted by these fluids on immersed or bounding surfaces.

• A fluid at rest cannot sustain shear stress; forces are always normal (perpendicular) to any surface in contact with the fluid.

• This branch of fluid mechanics is fundamental to the design of dams, tanks, gates, ships, and hydraulic structures.

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1. Pressure and Head

• Pressure is defined as the normal force exerted by a fluid per unit area.

  $\boldsymbol{p = \frac{F}{A}}$

  where, $\boldsymbol{p}$ is the pressure, $\boldsymbol{F}$ is the normal force, and $\boldsymbol{A}$ is the area.

• Pressure Head is a convenient way to express pressure in terms of the height of a column of fluid that would produce the same pressure at its base.

• It links pressure to the specific weight of the fluid.

  $\boldsymbol{p = \rho g h}$

  where, $\boldsymbol{\rho}$ is the density of the fluid, $\boldsymbol{g}$ is the acceleration due to gravity, and $\boldsymbol{h}$ is the pressure head (height of the fluid column).

• Therefore, the pressure head is given by

  $\boldsymbol{h = \frac{p}{\rho g}}$

• Pressure at a point in a static fluid is independent of direction (isotropic). It is a scalar quantity.

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2. Pascal’s Law

• Statement: Pressure applied to a confined, incompressible fluid at rest is transmitted undiminished in all directions and acts with equal force on equal areas, and at right angles to the container walls.

• This principle is the foundation of hydraulic systems like jacks, lifts, and brakes.

  $\boldsymbol{\frac{F_1}{A_1} = \frac{F_2}{A_2} = p}$

  where a force $\boldsymbol{F_1}$ applied on a small area $\boldsymbol{A_1}$ creates a pressure $\boldsymbol{p}$ that results in a much larger force $\boldsymbol{F_2}$ on a larger area $\boldsymbol{A_2}$.

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3. Pressure-Depth Relationship

• In a static fluid with constant density, pressure increases linearly with depth below the free surface due to the weight of the overlying fluid.

• The absolute pressure at a depth $\boldsymbol{h}$ is given by

  $\boldsymbol{p = p_0 + \rho g h}$

  where, $\boldsymbol{p_0}$ is the pressure at the free surface (often atmospheric pressure, $\boldsymbol{p_{\mathrm{atm}}}$), and $\boldsymbol{\rho g h}$ is the gauge pressure.

• Gauge Pressure is the pressure relative to the atmospheric pressure.

  $\boldsymbol{p_{\mathrm{gauge}} = \rho g h}$

• Absolute Pressure is the sum of gauge pressure and atmospheric pressure.

  $\boldsymbol{p_{\mathrm{abs}} = p_{\mathrm{atm}} + p_{\mathrm{gauge}}}$

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4. Manometers

• Manometers are devices that use columns of liquid to measure pressure differences.

• They operate on the principle that a pressure difference supports a difference in liquid levels in connected columns.

• Simple U-Tube Manometer: Used to measure the pressure of a gas or liquid relative to the atmosphere or another point.

  • For a U-tube manometer with a differential fluid of density $\boldsymbol{\rho_m}$, the pressure difference is

    $\boldsymbol{p_1 - p_2 = (\rho_m - \rho)g \Delta h}$

    where $\boldsymbol{\rho}$ is the density of the fluid in the system and $\boldsymbol{\Delta h}$ is the height difference of the manometric fluid.

• Differential Manometer: Used to measure the difference in pressure between two points in a pipe or system.

• Inclined Manometer: Increases sensitivity for measuring very small pressure differences by amplifying the liquid movement along an inclined scale.

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5. Pressure Force and Center of Pressure on Submerged Bodies

5.1 Plane Surfaces

• The total hydrostatic force on a submerged plane surface is the product of the pressure at the centroid of the area and the total area.

  $\boldsymbol{F = p_c \cdot A = (\rho g h_c) \cdot A}$

  where, $\boldsymbol{p_c}$ is the pressure at the centroid, $\boldsymbol{h_c}$ is the depth of the centroid, and $\boldsymbol{A}$ is the submerged area.

• The center of pressure is the point where the total resultant hydrostatic force acts. It is always below the centroid for an inclined or vertical plane surface.

• The vertical location (from the free surface along the plane) is given by

  $\boldsymbol{h_{\mathrm{cp}} = h_c + \frac{I_{c,xx}}{A h_c}}$

  where, $\boldsymbol{I_{c,xx}}$ is the second moment of area (moment of inertia) of the surface about its centroidal axis parallel to the free surface.

5.2 Curved Surfaces

• Determining the force on a curved surface requires resolving the force into horizontal and vertical components.

• Horizontal Component: Equal to the hydrostatic force on the vertical projection of the curved surface. Its line of action passes through the center of pressure of the projected area.

• Vertical Component: Equal to the weight of the fluid contained in the volume vertically above the curved surface up to the free surface. It acts through the centroid of that fluid volume.

5.3 Practical Applications

• Analysis of forces on dam faces, sluice gates, tank walls, and submerged hatches.

• Design of hydraulic structures to ensure stability against overturning and sliding.

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6. Pressure Diagrams

• Pressure diagrams are graphical representations of pressure distribution on a submerged surface.

• For a fluid of constant density, the pressure varies linearly with depth.

• On a vertical wall, the diagram is a triangle, with zero pressure at the surface and maximum pressure at the base.

• On an inclined or horizontal surface, the diagram is a rectangle if the pressure is uniform.

• The total force on the surface is equal to the area of the pressure diagram.

• The location of the resultant force acts through the centroid of the pressure diagram.

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7. Buoyancy

• Archimedes' Principle: A body fully or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces.

• The buoyant force is given by

  $\boldsymbol{F_b = \rho_f g V_{\mathrm{disp}}}$

  where, $\boldsymbol{\rho_f}$ is the density of the fluid, and $\boldsymbol{V_{\mathrm{disp}}}$ is the volume of fluid displaced by the body.

• This force acts vertically upward through the center of buoyancy, which is the centroid of the displaced fluid volume.

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8. Stability of Floating/Submerged Bodies

• Stability refers to the tendency of a body to return to its original equilibrium position after being given a small angular displacement.

8.1 Submerged Bodies

• Stability depends on the relative positions of the center of gravity ($\boldsymbol{G}$) and the center of buoyancy ($\boldsymbol{B}$).

• Stable Equilibrium: $\boldsymbol{G}$ is below $\boldsymbol{B}$. A small tilt creates a restoring couple.

• Unstable Equilibrium: $\boldsymbol{G}$ is above $\boldsymbol{B}$. A small tilt creates an overturning couple.

• Neutral Equilibrium: $\boldsymbol{G}$ coincides with $\boldsymbol{B}$.

8.2 Floating Bodies

• For floating bodies, $\boldsymbol{B}$ shifts position as the body tilts because the shape of the displaced volume changes.

• The point where the vertical line through the new center of buoyancy intersects the original vertical axis through $\boldsymbol{G}$ and $\boldsymbol{B}$ is called the metacenter ($\boldsymbol{M}$).

• Metacentric Height ($\boldsymbol{GM}$) is the distance between $\boldsymbol{G}$ and $\boldsymbol{M}$. It is the key measure of stability.

  • Stable: $\boldsymbol{M}$ is above $\boldsymbol{G}$ (Positive $\boldsymbol{GM}$). A righting moment is generated.

  • Unstable: $\boldsymbol{M}$ is below $\boldsymbol{G}$ (Negative $\boldsymbol{GM}$). An overturning moment is generated.

  • Neutral: $\boldsymbol{M}$ coincides with $\boldsymbol{G}$ (Zero $\boldsymbol{GM}$).

• A larger positive metacentric height implies greater initial stability but may result in a quicker, less comfortable roll for ships.

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