Quick Revision of Fluid Mechanics Equations
Fluid mechanics is the branch of physics that studies the behavior of fluids at rest and in motion. It deals with the forces that fluids exert on other objects and how fluids respond to those forces. The fluids studied can be either liquids or gases.
This subject is important in many fields, including engineering, meteorology, oceanography, and aerospace. It is used in many industrial processes, such as internal combustion engines, turbine design, and many others.
What is a fluid?
Fluids are defined by their ability to flow and change shape, as opposed to solids, which maintain a fixed shape and volume.
There are two main types of fluids: liquids and gases. Liquids, such as water and oil, are a type of fluid that has a definite volume but no definite shape, meaning they will take the shape of their container. Gases, such as air and natural gas, have no definite shape or volume, meaning they will expand to fill their container.
Fluids can also be classified as incompressible or compressible. An incompressible fluid is a fluid that cannot be compressed, such as water. A compressible fluid is a fluid that can be compressed, such as a gas. The behavior of a fluid depends on its compressibility. For example, the speed of sound in a gas is affected by its compressibility.
Fluids are also classified as Newtonian or non-Newtonian. A Newtonian fluid is a fluid that behaves in a predictable way, following the laws of physics, such as liquids like water and oil. Non-Newtonian fluids are fluids that do not behave in a predictable way, such as toothpaste and blood. They can exhibit different behavior under different flow conditions.
In summary, understanding the properties and behavior of fluids is important in many industries and natural phenomena. The behavior of a fluid can be affected by its compressibility, viscosity and its ability to follow the laws of physics.
Here are 10 most important equations in fluid mechanics:
1.
Newton's Law of Viscosity
It states that the shear stress (the force per unit area) in a fluid is proportional to the rate of change of velocity of the fluid with respect to distance in the direction perpendicular to the direction of flow. Mathematically, it is represented as :
τ = μ * (du/dy)
where τ is the shear stress, μ is the viscosity of the fluid, and (du/dy) is the rate of change of velocity of the fluid in the direction perpendicular to the flow. The proportionality constant μ is known as the coefficient of viscosity.
This law applies to fluids that behave in a predictable way, following the laws of physics, such as liquids like water and oil, which are known as Newtonian fluids.
The viscosity of a fluid is a measure of its resistance to flow. Fluids with high viscosity are more resistant to flow, while fluids with low viscosity flow more easily.
2.
Pascal's Law
Pascal's law states that the pressure applied to an enclosed fluid is transmitted equally in all directions and at right angles to the container walls. Mathematically, it is represented as :
P1 = P2 = P3 = .... = Pn
where P1, P2, P3, ..., Pn are the pressures at different points within the fluid. This means that if an external force is applied to a fluid at one point, it will be transmitted throughout the entire fluid.
This law is the basis of many hydraulic systems, such as brakes, lifts, and construction equipment. It is also used in many industrial processes such as oil and gas extraction.
3.
Hydrostatic Law
Hydrostatic law, also known as Pascal's law of fluids, states that the pressure exerted by a fluid at rest is transmitted equally in all directions and at right angles to the surface on which it acts. This means that if an external force is applied to a fluid at one point, it will be transmitted throughout the entire fluid. The pressure at a given point in a fluid at rest is determined by the weight of the fluid above it. Mathematically, it is represented as:
P = ρgh
where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid above the point in question. This law is widely used in fluid mechanics and many engineering applications such as pumps, valves, and pipelines.
4.
Continuity Equation
In fluid mechanics, the continuity equation is a fundamental principle that states that the mass flow rate of a fluid through any given cross-sectional area must remain constant. Mathematically, it is represented as:
∑(mass flow rate)in = ∑(mass flow rate)out
or
A1V1 = A2V2 = A3V3 = .... = AnVn
where A1, A2, A3, ..., An are the cross-sectional areas and V1, V2, V3, ..., Vn are the velocities of the fluid at different points within the fluid.
This equation states that the mass of fluid flowing into any given point must be equal to the mass of fluid flowing out of that point. It is based on the principle of conservation of mass. It is a general statement of the conservation of mass, which applies to all systems, including fluids, gases and solids.
The continuity equation is also related to other principles in fluid mechanics such as the conservation of mass, the equation of motion and the energy equation.
It is used in many engineering applications such as fluid flow in pipes, pumps, and valves, and in the analysis of fluid systems. It is also used in the design and optimization of fluid systems.
5.
Bernoulli's Equation
This is a fundamental principle in fluid dynamics that describes the relationship between the pressure, velocity, and height of a fluid as it flows through a pipe or over a surface. The equation states that "in an ideal incompressible fluid when the flow is steady and continuous, the sum of the pressure energy, kinetic energy and potential energy is constant at any point along a streamline". Bernoulli's equation is given by:
P + 1/2 * ρu^2 + ρgh = constant
where, P is the pressure of the fluid, ρ is the density of the fluid, u is the velocity of the fluid, g is the acceleration due to gravity, h is the height of the fluid above a reference point.
**Applicable for incompressible, non-viscous, steady flow fluids**.
This equation is important in the study of fluid flow, as it can be used to predict the behavior of fluids in a variety of situations, including the flow of air over an airplane wing, the flow of water through a pipe, and the flow of fluids in pumps and turbines. For example, in the case of an airplane wing, the air flowing over the top of the wing is moving faster than the air flowing underneath the wing, which results in a lower pressure above the wing and a higher pressure below the wing. This creates lift, which is what allows the airplane to fly.
6.
Navier Stokes Equation
The Navier-Stokes equations are a set of partial differential equations that describe the motion of a fluid, including its velocity, pressure, and temperature. These equations are based on the conservation of mass, momentum, and energy.
They take into account the effects of viscosity, which is the resistance of a fluid to flow, and turbulence, which is the chaotic, irregular motion of a fluid. The equations also include the effects of forces such as gravity and pressure gradients, and can be used to model fluid flow in a wide range of situations, including laminar flow, turbulent flow, and boundary layer flow. These equations can be represented in vector form as:
∂u/∂t + (u.∇)u = -1/ρ ∇p + μ/ρ ∇²u + g
where, u is the velocity vector of the fluid t is time ρ is the density of the fluid p is the pressure of the fluid μ is the dynamic viscosity of the fluid f is any external forces acting on the fluid.
The Navier-Stokes equations are nonlinear and highly complex. They are known to be difficult to solve in general, especially when the fluid is turbulent or the flow is three-dimensional. Due to the complexity of these equations, numerical methods are often used to approximate solutions for practical problems.
**Applicable for fluids with non-zero viscosity and they can't be applied to ideal fluids which have zero viscosity.**
The Navier-Stokes equations are widely used in fields such as aerospace engineering, mechanical engineering, civil engineering, and meteorology to predict the behavior of fluids under various conditions. They are also used in the design of aircraft, ships, automobiles, and other vehicles.
7.
Hagen Poiseuille Equation
It is a fundamental principle in fluid dynamics that describes the flow of a liquid through a cylindrical pipe.
**It is based on the assumption that the fluid is Newtonian, incompressible, and flow is laminar. Valid for a fully developed flow, a flow in which the velocity profile does not vary along the pipe axis**
The Hagen-Poiseuille equation states that the flow rate of a liquid through a pipe is directly proportional to the fourth power of the diameter of the pipe and the pressure gradient, and inversely proportional to the viscosity of the liquid and the length of the pipe. Mathematically, it can be represented as:
ΔP/L = (128*μ*Q) /(π*(d^4))
where, Q is the flow rate of the liquid, π is the mathematical constant pi, μ is the viscosity of the liquid, d is the diameter of the pipe, ΔP is the pressure gradient and L is the length of the pipe.
The Hagen-Poiseuille equation is widely used in fields such as chemical engineering, mechanical engineering, and biomedical engineering to predict the flow of fluids through pipes. It is also used in the design of pipelines, pumps, and other equipment that handle liquids. The equation is particularly useful for small diameter pipes and low-Reynolds number flows, where the flow is laminar and viscous forces dominate.
8.
Stokes Law
Stokes' law describes the drag force experienced by a small, spherical object moving through a fluid at low Reynolds numbers (Re <= 0.2). here, the viscous force are much more prominent than the inertial forces. At higher Reynolds numbers, the drag force is dependent on the object's shape and the turbulence of the fluid flow, and the drag force is described by different equations such as the Navier-Stokes. The total drag force is given by:
F = 3πμdu
where, F is the drag force, μ is the viscosity of the fluid, d is the radius of the object, u is the velocity of the object.
This equation is often used in the study of particle dynamics in liquids, such as the motion of small particles in suspensions, the behavior of microorganisms in fluids, and the study of droplets in sprays.
9.
Kozeny Carman Equation
The Kozeny-Carman equation, in fluid mechanics, relates the permeability of a porous medium to its microstructure, represented by the porosity and intrinsic permeability. The equation is widely used in the study of porous media, such as in the oil and gas industry, geology, and environmental engineering. It is an important tool for the prediction of fluid flow in porous media and can be used to estimate the permeability of reservoirs, aquifers, and other subsurface formations. The equation is given by:
ΔP/L = (150*V*μ*((1-E)^2))) /((φ^2)*(d^2)*(E^3))
where, ΔP is the pressure gradient, L is the length of the pipe, V is the superficial velocity, E is the porosity of the porous medium, μ is the viscosity of the fluid, φ is the sphericity factor and d is the diameter of the particle.
** This equation is applicable for flow through beds at particles with Reynolds number about 1.0.**
10.
Burke Plummer Equation
This equation is applicable in the same conditions as Kozeny-Carman equation but for the cases where Reynolds number is greater than 1000. The equation is given by:
ΔP/L = (1.75*(V^2)*ρ*(1-E)) /(φ*d*(E^3))
where, ΔP is the pressure gradient, L is the length of the pipe, V is the superficial velocity, E is the porosity of the porous medium, ρ is the density of the fluid, φ is the sphericity factor and d is the diameter of the particle.
Hope I have covered all important equations of fluid Mechanics. But if you feel that there are other equations also that are must to study, then you are most welcome to share your thoughts in the comment box.
Thankyou!
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