Physics > Mechanical Properties of Fluids

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  • The average pressure  is defined as the normal force acting per unit area.

  • For a fluid of

    Mass  occupying volume. The dimensions of density are

  • Pascal's Law- The pressure in a fluid at rest is the same at all points if they are at the same height.
  • The pressure P, at depth below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount The excess of pressure, , at depth h is called a gauge pressure at that point.
  • Whenever external pressure is applied on any part of a fluid contained in a vessel, it is transmitted undiminished and equally in all directions. This is the Pascal's law for transmission of fluid pressure.
  • The path taken by a fluid particle under a steady flow is a streamline. It is defined as a curve whose tangent at any point is in the direction of the fluid velocity at that point.
  • Equation of continuity:

    Av gives the volume flux or flow rate and remains constant throughout the pipe of flow. Thus, at narrower portions where the streamlines are closely spaced, velocity increases and its vice versa.

  • Steady flow is achieved at low flow speeds. Beyond a limiting value, called critical speed, this flow loses steadiness and becomes turbulent.
  • Bernoulli's Equation:

    When a fluid is at rest i.e. its velocity is zero everywhere, Bernoulli's equation becomes

     

  • Aerofoil or lift on aircraft wing and the Magnus effect (dynamic lift of the ball due to spinning) can be explained using Bernoulli's theorem.
  • The coefficient of viscosity (pronounced 'eta') for a fluid is defined as the ratio of shearing stress to the strain rate.

    The SI unit of viscosity is poiseiulle (Pl).

  • When a body falls through a fluid it drags the layer of the fluid in contact with it. A relative motion between the different layers of the fluid set and as a result the body experiences a retarding force. It is seen that the viscous force is proportional to the velocity of the object and is opposite to the direction of motion. The other quantities on which the force  depends on viscosity  of the fluid and radius  of the sphere.

     (Stoke's Law).

  • Reynolds defined a dimensionless number, whose value gives one an approximate idea whether the flow would be turbulent. This number is called the Reynolds Re.

    where  is the density of the fluid flowing with a speed  d stands for the dimension of the pipe, and  is the viscosity of the fluid.  is a dimensionless number and therefore, it remains same in any system of units. It is found that flow is streamline or laminar for  less than 1000. The flow is turbulent for   The flow becomes unsteady for   between 1000 and 2000.

  • Liquids have no definite shape but have a definite volume; they acquire a free surface when poured in a container. These surfaces possess some additional energy. This phenomenon is known as surface tension and it is concerned with only liquid as gases do not have free surfaces.

    Or,

    This quantity S is the magnitude of surface tension.

  • The contact angle between water and glass is acute. Thus the surface of water in the capillary is concave. This means that there is a pressure difference between the two sides of the top surface. This is given by

    Thus the pressure of the water inside the tube, just at the meniscus (air-water interface) is less than the atmospheric pressure. They must be at the same pressure, namely P0 + h ρ g = Pi = PA

    where ρ is the density of water and h is called the capillary rise.

     

Sample Examples

 

Question

At a depth of  m in an ocean (a) what is the absolute pressure? (b) What is the gauge pressure? (c) Find the force acting on the window of area depth, the interior of which is maintained at sea-level atmospheric pressure.

Answer

 Here

(a) Absolute pressure

=  

=

(b) Gauge pressure is

The pressure outside the submarine is  and the pressure inside it is Pa. Hence, the net pressure acting on the window is gauge pressure, . Since the area of the window is , the force acting on it is

 

Question

What is the pressure on a swimmer  below the surface of a lake?

Answer

Here

=

=

 

Question

The density of the atmosphere at sea level is . Assume that it does not change with altitude. Then how high would the atmosphere extend?

Answer

In reality the density of air decreases with height. So does the value of g.

  

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