Fluid dynamics is the study of fluids in motion. Since the motion of a real fluid is complex, the treatment is simplified by making a few assumptions. We consider an ideal fluid which is non-viscous and incompressible. The flow of the ideal fluid is assumed to be steady and irrotational. Steady flow means the velocity of the fluid at a given point remains constant in time, and variation in velocity from point to point must be smooth.
The path followed by a fluid element under steady flow is called a streamline. The line can either be straight or curved such that the tangent drawn to it at any point gives the direction of flow of the fluid at that point. Steady flow is also called streamline flow or laminar flow. Example is water flowing through a tube at low speed.
On the other hand, if the fluid moves at high speed or the bounding surfaces cause abrupt changes in its velocity, the flow becomes irregular and complex. Such a flow is called a turbulent flow. Example is flow of water in a stream near boulders.
According to equation of continuity, the product of the cross-sectional area and the speed of an ideal fluid in steady flow remains constant at all points along a tube. The equation is based on the assumption that no fluid can flow across the wall of the tube, and there is no "source" or "sink" inside the tube which can create or destroy any fluid.
When a fluid flows steadily through a pipe of varying cross section and elevation, the pressure within the pipe changes along its length. The way the pressure depends on the fluid's speed and the pipe's elevation is governed by Bernoulli's principle. One way of writing Bernoulli's equation is:
P + 1/2 ρv2 + ρgy = constant
Put in words, the sum total of the pressure, kinetic energy per unit volume and gravitational potential energy per unit volume of an ideal fluid in steady flow remains constant along a streamline.
There are interesting applications of Bernoulli's equation. One of them is to determine the speed of efflux, that is the speed with which a liquid emerges from a hole made at the wall of its container. Venturi tube is a horizontal constricted pipe used to measure the flow speed of an incompressible fluid. Pitot tube is another device to measure fluid speed. The ascent of an aeroplane is made possible by a net upward force — known as dynamic lift — on its wings. The shape and orientation of the wing are designed such that the speed of air is greater above the wing and hence, by Bernoulli's principle, air pressure is higher under the wing.
Unlike ideal fluids, a real fluid possesses a property called viscosity. When a real fluid flows through a pipe, the velocity of the fluid layer gradually increases from zero at the pipe wall to a maximum at the centre. The force of internal friction between the layers is known as viscous force. Newton's formula applicable to laminar flow of a viscous fluid is:
Ft = η A dv/dy
where Ft is the tangential force acting on fluid surface of area A, dv/dy is speed gradient, and η is coefficient of viscosity of the given fluid. The dimensions of viscosity coefficient are [ML-1T-1], and its unit is newton second per square metre (N-s/m2).
Poiseuille's law on laminar viscous flow in a pipe is:
φ = πΔP R4 / 8ηl
In this expression, φ is volume flux (in m3/s), ΔP is pressure difference between two ends of the pipe of length l and radis R, η is coefficient of viscosity of the fluid.
The minimum speed of a fluid in a pipe above which the fluid ceases to be steady is called the critical speed of the fluid. Reynolds number, NR , suggests whether the flow of a viscous fluid in a pipe would be steady or turbulent. Experiments show that if NR does not exceed 2000, the flow is steady; if NR exceeds 3000, the flow is turbulent; if NR falls between 2000 and 3000, the flow may be either steady or turbulent depending on the shape of the pipe entrance and the distance from it.
When a solid body moves through a viscous fluid, it experiences a resistive drag force exerted by the fluid on it. For a slow-moving spherical body, this drag force is given by Stoke's law as follows:
FD = 6πηrv
On the right-hand side, η is the coefficient of viscosity of the fluid and r is the radius of the body moving with speed v relative to the fluid. Under the combined effects of this drag force, force of gravity and force of buoyancy, the sphere eventually falls with a constant terminal speed.