Fluid Dynamics

Fluid dynamics refers to a sub-discipline of fluid mechanics that revolves around fluid flow in motion. Furthermore, fluid dynamics comprises of some branches like aerodynamics and hydrodynamics. Fluid dynamics involves the calculation of various fluid properties, such as flow velocity, pressure, density, and temperature, as functions of space and time.

Introduction to Fluid Dynamics

Fluid dynamics offers a systematic structure to the practical disciplines that depend on the flow measurement. Furthermore, it helps in the derivation of empirical and semi-empirical laws. As such, fluid dynamics facilitate the solving of practical problems.

The fluid dynamics applications include understanding nebulae in interstellar space, predicting weather patterns, calculating force and moments on aircraft, determining the mass flow rate of petroleum through pipelines, and modelling fission weapon detonation.

How do we Measure Fluid Dynamics?

In order to solve fluid dynamics problems, use of three conservation laws takes place. Furthermore, one can write these laws in an integral or differential form. Moreover, the application of these conservation laws may be to a control volume, a region of the flow.

A control volume is a discrete volume in space via which one can assume the fluid to flow. Furthermore, experts use the integral formulations of the conservation laws to describe the change of energy, mass, or momentum, within the control volume.

Also, there is an application of the Stokes’ theorem by differential formulations of the conservation laws to yield an expression. The interpretation of this expression can be as the integral form of the law that is applied to an infinitesimally small volume within the flow.

To an extent, all fluids can be said to be compressible. Moreover, this means that a change in temperature or pressure will result in a change in density.

However, the changes in temperature and pressure, in most cases, are quite small such that the changes in density turn out to be negligible. In this case, the modelling of the flow can take place as an incompressible flow. Otherwise, one must make use of the more general compressible flow equations.

Formula of Fluid Dynamics

Equations in Fluid Dynamics: Bernoulli’s Equation

P/ρ + gz + v² = k

P/ρg + z + v²/2g = k

P/ρg + v²/2g + z = k

Here,

P/ρg is the pressure head or pressure energy per unit weight fluid

v²/2g refers to the kinetic head or kinetic energy per unit weight

z over here is the potential head or potential energy per unit weight

P is certainly the Pressure

ρ is the Density

K is the Constant

The Bernoulli equation happens to be different for isothermal and adiabatic processes.

dP/ρ + VdV + gdZ = 0

Also, ∫(dP/ρ + VdV + gdZ) = K

∫dP/ρ + V²/2 + gZ = K

Where,

Z is the elevation point

ρ is the density of the fluid

One can also write the equation as,

q + P = Po

Where,

q is the dynamic pressure

P_O is the total pressure

P is the static pressure

Derivation of the Formula of Fluid Dynamics

Consider a pipe with varying height and diameter via which the flowing of an incompressible fluid takes place. Furthermore, the relationship here is between the areas of cross-sections A, pressure p, the flow speed v, and the height from the ground y, at two different points 1 and 2.

Assumptions:

• The density of the incompressible fluid would always remain constant at both points.
• There is a conservation of energy of the fluid due to a lack of viscous forces in the fluid.

Therefore, the work taking place on the fluid is given as:

dW = F₁dx₁ – F₂dx₂

Also, dW = p₁A₁dx₁ – p₂A₂dx₂

So, dW will be = p₁dV – p₂dV = (p₁ – p₂)dV

An important point to remember here is that the work taking place on the fluid is because of the change in kinetic energy and conservation of gravitational force. Moreover, the expression of the change in the fluid’s kinetic energy is as:

dK = \(\frac{1}{2}m_{2}v_{2}^{2} – \frac{1}{2}m_{1}v_{1}^{2} = \frac{1}{2}pdV\left ( v_{2}^{2} – v_{1}^{2} \right )\)

The change in potential energy is expressed as:

dU = mgy₂ – mgy₁, which will be = ρdVg(y₂ – y₁)

Therefore, the energy equation is expressed as:

dW = dK + dU

(p₁ – p₂)dV = \(\frac{1}{2}pdV\left ( v_{2}^{2} – v_{1}^{2} \right )\)

+ ρdVg(y₂ – y₁)

(p₁ – p₂) = \(\frac{1}{2}p\left ( v_{2}^{2} – v_{1}^{2} \right )\)

+ ρg(y₂ – y₁)

Finally, on facilitating a rearrangement of the above equation, we get

\(p_{1} + \frac{1}{2}pv_{1}^{2} + pgy_{1} = p_{2} + \frac{1}{2}pv_{2}^{2} + pgy_{2}\)

Most noteworthy, this is Bernoulli’s equation.

FAQs For Fluid Dynamics

Question 1: What is meant by fluid mechanics?

Answer 1: Fluid mechanics, simply speaking, is a branch of physics that covers fluid dynamics. Furthermore, it deals with the mechanics of fluids and the forces that act on them.

Question 2: What properties of the fluid are studied in fluid dynamics?

Answer 2: The properties of fluid which are studied in fluid dynamics are flow velocity, pressure, density, and temperature.