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Elasticity is the property of any material or substance getting stretched or compacted by external pressure. The applied pressure alters the shape or size of the material. When the pressure applied increases, the volume occupied by the material decreases, and pressure and volume are inversely proportional. The fluid elasticity or compressibility employs bulk modulus to calculate a numerical value. The numerical value gives how incompressible the material is to applied external pressure. The bulk modulus of water is 2.15 × 109 Pa.
Fluid Elasticity
Fluid elasticity refers to the elastic nature or property of a fluid substance. Fluid elasticity means how idle the liquid is on exerting pressure on the fluid to compact its volume or extend it. When more pressure applies, the fluid volume will decrease. When less pressure applies, the volume of the fluid increases; that is, it occupies more space. This change in the fluid volume due to the pressure is called fluid elasticity.
Solids generally employ all three moduli. At the same time, fluids employ only the bulk modulus. All three moduli are related to the elastic properties of a material. The three different moduli are as follows.
- Young’s modulus
- Shear modulus
- Bulk modulus
The law applying to the elasticity of materials is Hooke’s law. It states that when pressure is applied to compact material, the volume change of the material is proportional to the force acting on the material.
Bulk modulus
Generally, the bulk modulus is given by the ratio of applied pressure to the change in the volume of the substance. It is the numerical value of the stress divided by the strain on a material. As per the definition, the bulk modulus formula is as follows.
K = Pressure applied on the material / Change or decrease in the volume of the material.
K = P / (V0-Vn) / V0
Where,
K → Bulk modulus, B also denotes the Bulk modulus
P → Pressure applied to the elastic material
V0 → original volume of the material
Vn → the change or decrease in the volume of the material
Volume shrinks(a negative sign for volume change) when pressure is applied to the material. Thus, the formula can be expressed as
K = dP / -(dV / V)
K = -V(dP/dV)
SI unit of Bulk Modulus
The SI unit of Bulk modulus is N/m². It is also referred to as Pascal(Pa). The bulk modulus has a dimensional formula of [M L-¹ T-²] derived from the dimensional formulae of the bulk modulus components.
The bulk modulus of fluids
The pressure required is enormous to compress even a little amount of fluid in space. Thus mathematically, the numerical value of the bulk modulus of fluids is also high. For fluids, the bulk modulus value is calculated and determined in terms of density as it applies better than volume in the case of fluids. Therefore, to calculate the bulk modulus of fluids, the general formula alters by substituting volume in terms of density to write in terms of density. Here comes the concept of volume and density being inversely variable to each other.
Generally, the bulk modulus formula is
K = V(△P/△V)
Substituting volume in terms of density, we get
K = ⍴ (△P/△⍴)
Where,
K → Bulk modulus, B also represents the Bulk modulus
⍴ → the density of the material
△P → change or increase in pressure
△⍴ → change in the density of the material
Characteristics of bulk modulus
The bulk modulus works under specific parameters or characteristics. They are as follows.
- The pressure acting on the material should be stretching the material only within its elastic limit.
- It relates to the volumetric change and the applied pressure in the process.
- It also applies to liquids and gases, apart from solids.
Pressure in the Mariana Trench
Calculating the pressure and density to determine the compression in the Mariana Trench is one of the best examples of demonstrating bulk modulus for fluid elasticity with the help of the bulk modulus of water. Mariana trench is well-known as the deepest part of the ocean ever found, and is assumed to experience more pressure as we go deeper. That’s the prime reason why it is still unexplored.
The entire length of the seawater till the Mariana trench base is 10994m. The water pressure = P = density × acceleration due to gravity × height of the waterbody.
P1 = ⍴ × g × h
We can infer the density is 1070 kg/m² using the bulk modulus of water in kg/m². The bulk modulus of water in the Mariana Trench is lower, too. However, the pressure is higher for humans to withstand.
Conclusion
The elastic materials tend to resize themselves on experiencing applied pressure. Inversely, the change in the volume occupied by the material depends on the pressure applied to the material. So, when the pressure is enormous, the material occupies a lesser volume. When the pressure is minimal, the material fills up larger volumes.
The fluid elasticity employs the aid of the bulk modulus, but the other two. The bulk modulus says how far the fluid remains intact on applying how much pressure. The bulk modulus of water helps calculate the pressure in a particular waterbody.
