# Pressure and velocity relationship in fluids the body

### Bernoulli's Equation

For example, when fluid passes over a solid body, the streamlines get closer together, the flow velocity increases, and the pressure decreases. Airfoils are. The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli's equation, named after its discoverer, the Swiss scientist Daniel. The main task in fluid dynamics is to find the velocity field describing the flow in a conservation of momentum (the Cauchy equation, Sec. ) . Long ranged external body forces that penetrate matter and act equally on all . The surface stresses (pressure and viscous effects) on any fluid element were introduced.

Time dependent flow is known as unsteady also called transient [6]. Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference.

For instance, laminar flow over a sphere is steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady.

Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow.

- Fluid dynamics

Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer time than the governing equations of the same problem without taking advantage of the steadiness of the flow field. Laminar vs turbulent flow[ edit ] Turbulence is flow characterized by recirculation, eddiesand apparent randomness. Flow in which turbulence is not exhibited is called laminar.

The presence of eddies or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via a Reynolds decompositionin which the flow is broken down into the sum of an average component and a perturbation component.

### List of equations in fluid mechanics - Wikipedia

It is believed that turbulent flows can be described well through the use of the Navier—Stokes equations. Direct numerical simulation DNSbased on the Navier—Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows.

Transport aircraft wings such as on an Airbus A or Boeing have Reynolds numbers of 40 million based on the wing chord dimension. Solving these real-life flow problems requires turbulence models for the foreseeable future. Reynolds-averaged Navier—Stokes equations RANS combined with turbulence modelling provides a model of the effects of the turbulent flow.

Such a modelling mainly provides the additional momentum transfer by the Reynolds stressesalthough the turbulence also enhances the heat and mass transfer. Another promising methodology is large eddy simulation LESespecially in the guise of detached eddy simulation DES —which is a combination of RANS turbulence modelling and large eddy simulation.

Subsonic vs transonic, supersonic and hypersonic flows[ edit ] While many flows e. New phenomena occur at these regimes such as instabilities in transonic flow, shock waves for supersonic flow, or non-equilibrium chemical behaviour due to ionization in hypersonic flows. In practice, each of those flow regimes is treated separately. Reactive vs non-reactive flows[ edit ] Reactive flows are flows that are chemically reactive, which finds its applications in many areas such as combustion IC enginepropulsion devices Rocketsjet engines etc.

Magnetohydrodynamics Magnetohydrodynamics is the multi-disciplinary study of the flow of electrically conducting fluids in electromagnetic fields. Examples of such fluids include plasmasliquid metals, and salt water. The fluid flow equations are solved simultaneously with Maxwell's equations of electromagnetism.

### Fluid dynamics and Bernoulli's equation

Relativistic fluid dynamics[ edit ] Relativistic fluid dynamics studies the macroscopic and microscopic fluid motion at large velocities comparable to the velocity of light. The governing equations are derived in Riemannian geometry for Minkowski spacetime.

Other approximations[ edit ] There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below. The Boussinesq approximation neglects variations in density except to calculate buoyancy forces. It is often used in free convection problems where density changes are small. Lubrication theory and Hele—Shaw flow exploits the large aspect ratio of the domain to show that certain terms in the equations are small and so can be neglected.

Slender-body theory is a methodology used in Stokes flow problems to estimate the force on, or flow field around, a long slender object in a viscous fluid. The shallow-water equations can be used to describe a layer of relatively inviscid fluid with a free surfacein which surface gradients are small.

Darcy's law is used for flow in porous mediaand works with variables averaged over several pore-widths. In rotating systems, the quasi-geostrophic equations assume an almost perfect balance between pressure gradients and the Coriolis force.

It is useful in the study of atmospheric dynamics. Terminology in fluid dynamics[ edit ] The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. The ball position is stable because if the ball moves sideways, its outer side moves into a region of lower velocity and higher pressure, whereas its inner side moves closer to the center where the velocity is higher and the pressure is lower.

The differences in pressure tend to move the ball back towards the center. Example 3 Suppose a ball is spinning clockwise as it travels through the air from left to right The forces acting on the spinning ball would be the same if it was placed in a stream of air moving from right to left, as shown in figure Spinning ball in an airflow.

A thin layer of air a boundary layer is forced to spin with the ball because of viscous friction. At A the motion due to spin is opposite to that of the air stream, and therefore near A there is a region of low velocity where the pressure is close to atmospheric.

At B, the direction of motion of the boundary layer is the same as that of the external air stream, and since the velocities add, the pressure in this region is below atmospheric.

The ball experiences a force acting from A to B, causing its path to curve. If the spin was counterclockwise, the path would have the opposite curvature. The appearance of a side force on a spinning sphere or cylinder is called the Magnus effect, and it well known to all participants in ball sportsespecially baseball, cricket and tennis players.

## List of equations in fluid mechanics

Stagnation pressure and dynamic pressure Bernoulli's equation leads to some interesting conclusions regarding the variation of pressure along a streamline. Consider a steady flow impinging on a perpendicular plate figure There is one streamline that divides the flow in half: Along this dividing streamline, the fluid moves towards the plate. Since the flow cannot pass through the plate, the fluid must come to rest at the point where it meets the plate. Bernoulli's equation along the stagnation streamline gives where the point e is far upstream and point 0 is at the stagnation point.

It is the highest pressure found anywhere in the flowfield, and it occurs at the stagnation point.

It is called the dynamic pressure because it arises from the motion of the fluid. The dynamic pressure is not really a pressure at all: Pitot tube One of the most immediate applications of Bernoulli's equation is in the measurement of velocity with a Pitot-tube. The Pitot tube named after the French scientist Pitot is one of the simplest and most useful instruments ever devised. It simply consists of a tube bent at right angles figure Pitot tube in a wind tunnel. By pointing the tube directly upstream into the flow and measuring the difference between the pressure sensed by the Pitot tube and the pressure of the surrounding air flow, it can give a very accurate measure of the velocity.