Fluid Mechanics: Exploring Bernoulli’s Theorem, Boundary Layer Separation, and Fluid Properties
Fluid Mechanics
Bernoulli’s Theorem for Streamline Flow
where Cb = Coefficient of venturimeter and its value is less than 1
Bernoulli’s theorem states that in a steady flow of an incompressible and non-viscous fluid along a streamline, the sum of pressure energy, kinetic energy, and potential energy per unit mass remains constant.
Proof:
Pressure Energy:
The pressure energy per unit mass is given by (P/ρ), where (P) is the pressure and (ρ) is the density of the fluid. This represents the work done on the fluid to bring it to that point.
Kinetic Energy:
The kinetic energy per unit mass is (V^2/2), where (V) is the velocity of the fluid. This represents the energy associated with the motion of the fluid.
Potential Energy:
The potential energy per unit mass is (gh), where (g) is the acceleration due to gravity and (h) is the height of the fluid above a reference point. This represents the energy due to the elevation of the fluid.
Therefore, Bernoulli’s theorem can be expressed as:
Where:
- P is the static pressure of the fluid.
- ρ is the density of the fluid.
- v is the velocity of the fluid.
- g is the acceleration due to gravity.
- h is the height above a reference level.
By considering these three energy components, Bernoulli’s theorem can be applied to analyze and understand the behavior of fluid flow in various situations.
Surface Tension
Definition: Surface tension is a physical property of liquids that causes the surface of a liquid to behave like a stretched elastic membrane. It is the force per unit length acting along the surface of a liquid at the interface with another phase (liquid, gas, or solid) that tends to minimize the surface area.
Origin: Surface tension arises due to cohesive forces between molecules in the liquid. Molecules at the surface of a liquid experience an unbalanced attractive force towards the interior of the liquid because there are no liquid molecules above them to balance the attractive forces from below. This creates a net inward force, causing the surface to contract and resist external force.
Capillarity (Capillary Action)
Definition: Capillarity, or capillary action, is the ability of a liquid to flow in narrow spaces without the assistance of external forces (and often against gravity). It occurs due to the interaction between the adhesive forces (between the liquid and the solid surface) and the cohesive forces (within the liquid).
Origin: Capillarity results from the balance between adhesive forces (attraction between liquid molecules and the surface of the capillary) and cohesive forces (attraction between liquid molecules). When a thin tube is placed in a liquid, the liquid either rises or falls in the tube depending on the relative strength of these forces.
Where:
- γ is the surface tension of the liquid.
- θ is the contact angle between the liquid and the tube.
- ρ is the density of the liquid.
- g is the acceleration due to gravity.
- r is the radius of the capillary tube.
Boundary Layer Separation
Definition: Boundary layer separation refers to the phenomenon where the boundary layer of a fluid flow detaches from the surface it is flowing over, leading to a disruption in the flow pattern.
Effect of Pressure Gradient on Boundary Layer Separation: The pressure gradient plays a crucial role in boundary layer separation. A favorable pressure gradient tends to delay boundary layer separation, promoting attached flow along the surface. In contrast, an adverse pressure gradient accelerates boundary layer separation, leading to flow separation and turbulence.
Controlling Boundary Layer Separation
To manage and control boundary layer separation, various techniques are employed:
- Streamlining Shapes: Designing objects with shapes that promote favorable pressure gradients.
- Boundary Layer Suction: Removing the low-momentum fluid from the boundary layer.
- Vortex Generators: Creating small vortices that mix high-momentum fluid into the boundary layer.
- Surface Roughness: Introducing controlled roughness to transition the boundary layer to turbulent, which is more resistant to separation.
Conclusion
Boundary layer separation is a critical aspect of fluid dynamics with significant implications for engineering applications. Understanding the role of pressure gradients helps in designing surfaces and devices that minimize separation, reduce drag, and improve overall performance.
Fluid Properties
(i) Specific Weight: Specific weight is defined as the weight per unit volume of a substance. It is calculated as the product of the density of the substance and the acceleration due to gravity. The formula for specific weight (γ) is γ = ρ * g, where ρ is the density and g is the acceleration due to gravity.
(ii) Specific Gravity: Specific gravity refers to the ratio of the density of a substance to the density of a reference substance, usually water. It is a dimensionless quantity and is calculated as the ratio of the density of the substance to the density of water at a specified temperature. The formula for specific gravity (SG) is SG = ρ_substance / ρ_water, where ρ_substance is the density of the substance and ρ_water is the density of water.
(iii) Viscosity: Viscosity is a measure of a fluid’s resistance to deformation or flow. It describes the internal friction within a fluid as it moves. Viscosity can be categorized as dynamic (or absolute) viscosity, which is a measure of a fluid’s resistance to shear stress, and kinematic viscosity, which is the ratio of dynamic viscosity to density. The SI unit for dynamic viscosity is Pascal-seconds (Pa·s) or N·s/m².
(iv) Surface Tension: Surface tension is the property of a liquid that allows it to resist an external force due to the cohesive nature of its molecules at the surface. It is measured as the force per unit length required to break a liquid film. Surface tension arises due to the imbalance of intermolecular forces at the surface compared to within the liquid.
(v) Capillarity: Capillarity refers to the phenomenon where liquids move up or down in narrow tubes or porous materials due to intermolecular forces between the liquid and the tube material. This movement occurs against gravity and is influenced by factors like surface tension, tube diameter, and wetting properties. Capillarity is responsible for phenomena like capillary action in plants and the rise or fall of liquids in small spaces.