Advertisements
Advertisements
प्रश्न
Derive Poiseuille’s formula for the volume of a liquid flowing per second through a pipe under streamlined flow.
Advertisements
उत्तर
Consider a liquid flowing steadily through a horizontal capillary tube. Let v = `("V"/"t")` be the volume of the liquid flowing out per second through a capillary tube. It depends on (1) coefficient of viscosity (η) of the liquid, (2) radius of the tube (r), and (3) the pressure gradient `("P"/"l")`.
Then, `"v" ∝ η^"a""r"^"b"("P"/"l")^"c"`
v = `"k"η^"a""r"^"b"("P"/"l")^"c"` .............(1)
where, k is a dimensionless constant.
Therefore, [v] = `"volume"/"time" = ["L"^3"T"^-1], ["dP"/"dx"] = "Pressure"/"distance"`
`["Ml"^-2"T"^-2], [η] = ["M"^-1"T"^-1]` and `["r"] = ["L"]`
Substituting in equation (1)
`["L"^3"T"^-1] = ["ML"^-1"T"^-1]^"a" ["L"]^"b" ["ML"^-2"T"^-2]^"c"`
`"M"^0"L"^3"T"^-1 = "M"^("a" + "c") "L"^(-"a" + "b" - 2"c") "T"^(-"a" - 2"c")`
So, equating the powers of M, L, and T on both sides, we get
a + c = 0, −a + b −2c = 3, and −a −2c = −1
We have three unknowns a, b and c. We have three equations, on solving, we get
a = – 1, b = 4 and c = 1
Therefore, equation (1) becomes,
v = `"k"η^-1"r"^4("P"/"l")^1`
Experimentally, the value of k is shown to be `π/8`, we have
v = `(π"r"^4"P")/(8η"l")`
The above equation is known as Poiseuille’s equation for the flow of liquid through a narrow tube or a capillary tube. This relation holds good for the fluids whose velocities are lesser than the critical velocity (vc).
APPEARS IN
संबंधित प्रश्न
Fill in the blanks using the word(s) from the list appended with each statement
For solids with elastic modulus of rigidity, the shearing force is proportional to . . . , while for fluids it is proportional to . .. (shear strain / rate of shear strain)
In Millikan’s oil drop experiment, what is the terminal speed of an uncharged drop of radius 2.0 × 10–5 m and density 1.2 × 103 kg m–3? Take the viscosity of air at the temperature of the experiment to be 1.8 × 10–5 Pa s. How much is the viscous force on the drop at that speed? Neglect buoyancy of the drop due to air.
The unit of viscosity is
What is Reynold’s number? Give its significance.
A solid ball of density ρ1 and radius r falls vertically through a liquid of density ρ2. Assume that the viscous force acting on the ball is F = krv, where k is a constant and v its velocity. What is the terminal velocity of the ball?
The velocity of a small ball of mass 0.3 g and density 8 g/cc when dropped in a container filled with glycerine becomes constant after some time. If the density of glycerine is 1.3 g/cc, then the value of viscous force acting on the ball will be x × 10-4 N, and the value of x is ______.
[use g = 10 m/s2]
The coefficient of apparent expansion of mercury in a glass vessel is 153 × 10-6/°C and in a steel vessel is 144 × 10-6/°C. If α for steel is 12 × 10-6/°C, then that of glass is ______.
Define velocity gradient.
State and explain Newton's law of viscosity.
A force of 0.01 N is required to move a flat glass plate of area of 10 cm2 with a uniform velocity of 1 cm/ s over the surface of a liquid 2 mm thick. If the coefficient of viscosity of the liquid is 20 poise, the velocity gradient in the liquid layer is ______.
