Fluid Dynamics

Notes

Angle of Contact and Capillary

Angle of contact is the angle which is tangent to a liquid surface at the point of contact makes with the solid surface inside the liquid. It's value is acute (900) for liquid which do not wet the walls of container. According to Newton's third law of motion, the tube also exerts on equal and opposite reaction force (R) can be resolved into two compounds i.e Tcosθ which acts vertically upward and Tsinθ which acts horizontally. The horizontal components Tsinθ acting on molecules at the point of contact cancel each other whereas vertical component added up. Hence a liquid experiences a net upward force on the liquid $$F = T\cos\theta \times 2\pi r \dots (i) $$

Viscosity

Newton studied the viscous force acting between two layers of liquid assuming it flow to be laminated and found that It is directly proportional to the area of layers in contact. i.e.F∝A…(i) i.e.F∝A…(i) It is directly proportional to the velocity gradient between the layers FF∝Advdx where the constant of proportionality'η' is known as the coefficient of viscosity of a liquid. Hence, coefficient of viscosity of a liquid is defined as the viscus drag or viscus force acting per unit area of the layer having unit velocity gradient perpendicular to the direction of the flow of the liquid .

Poiseuille's Formula and Stroke's Law

Stroke studied the motion of a small spherical body through a viscous medium and viscous force (F) acting on the body is found to depend upon following factors: The coefficient of viscosity of the medium. i.e.Vηa…(i) i.e.Vηa…(i) The radius of a spherical body. i.e.V∝rb…(ii) i.e.V∝rb…(ii) The terminal velocity required by the body in the medium. i.e.V∝vc…(iii) As time passes the velocity of falling body increases and viscous force acting on it also increases in that proportion. A state will come when the body starts moving with uniform velocity called terminal velocity in the medium. Terminal velocity means the net force acting on the body becomes zero.

Stream-line and Turbulent Flow, Energy of a Liquid and Bernoulli's Theorem

If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different velocities which do not mix with each other then the flow of liquid is called laminar flow. In general laminar flow is a streamline flow and this note provides us an information about Bernoulli's Theorem .