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♦️Revision Notes on Flow of Liquids and Viscosity♦️(2/3)
➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖
Viscosity:- Viscosity is the property of fluids by virtue of which they tend to destroy any relative motion between their layers.
Velocity gradient:- Velocity gradient is defined as the rate of change of velocity with respect to distance.
(a) Velocity gradient = dv/dr
(b) Dimension of velocity gradient = [dv/dr] = [T-1]
(c) Direction of velocity gradient is perpendicular to the direction of flow, directed in the direction of increasing velocity.
(d) Average velocity gradient:- Average velocity gradient is the difference between velocities of two layers separated a unit distance apart.
Average velocity gradient = Δv/Δr
Newton’s law of viscosity:-
In accordance to Newton’s law of viscosity, the viscous drag force depends upon the nature of fluid along with following factors:-
(a) F∝A (common area of two layers)
(b) F∝dv/dr (velocity gradient)
(c) So, F =ηA (dv/dr)
Here η is called coefficient of viscosity of fluid.
Coefficient of viscosity of fluid (ηv) or fugitive elasticity:-
ηv = shear stress/velocity gradient = (F/A)/(dv/dr)
Modulus of rigidity(ηr):-
ηr = shear stress/shear strain = (F/A)/(θ) = (F/A)/(dx/dr)
Here, θ = dx/dr = displacement gradient
Coefficient of viscosity (Absolute viscosity or Dynamic viscosity):-
F= ηA (dv/dr) if A = 1, dv = 1, dr =1, F = η
Co-efficient of viscosity of a fluid is defined as the tangential force per unit area which is required to maintain (or resist) a unit relative velocity between two layers a unit distance apart.
Or
Co-efficient of viscosity of a fluid is defined as the tangential force per unit area which is required to maintain a unit velocity gradient between its layers.
Unit of η:-
S.I:- η = 1 deca poise = 1 N sec/m2
Co-efficient of viscosity of a fluid is said to be one deca-poise if a tangential force of 1 N per meter square is required to maintain a relative velocity of 1 ms-1 between its layer 1 m apart.
C.G.S:- η = 1 poise = 1 dyn sec/cm2
Coefficient of viscosity of a fluid is said to be one poise if a tangential force of 1 dyn per square cm is required to maintain a relative velocity of 1 cms-1 between its layers 1 cm apart.
Relation between deca-poise and poise:-
1 deca-poise = 10 poise
Dimension formula for η:-
η = Fdr/Adv = [M1L-1T-1]
Fluidity:- Reciprocal of coefficient of viscosity of a fluid is called its fluidity.
Fluidity = 1/η
Unit of fluidity: poise-1
Dimension of fluidity: [M-1L1T1]
Kinematic viscosity:- Kinematic viscosity of a fluid is defined as the ration between its coefficient of viscosity to the density of fluid.
Kinematic viscosity = η/ρ
Units of kinematic viscosity:- C.G.S – 1 stoke = cm2 s-1
Kinetic viscosity of a fluid having its dynamic viscosity one poise and density one g cm-3 is said to be 1 stoke.
Dimensional formula of kinematic viscosity = η/ρ = [M0L2T-1]
Critical velocity (Reynold’s Number):- Critical velocity (vc) is the maximum velocity of the flow of liquid flowing in a streamlined flow.
vc = NR η/ρD
Here η is the coefficient of viscosity of liquid, ρ is the density of liquid and D is the diameter of the tube.
Reynold’s Number, NR = ρvcD/ η
Stokes law:- In accordance to Stoke’s law, force of viscosity F depend upon,
(a) Co-efficient of viscosity of fluid η
(b) Radius of the moving body r
(c) Velocity of body v
So, force of viscosity, F = 6π η r v
Terminal velocity:- v = 2/9 [r2 (ρ-σ)/η]
η = 2/9 [r2 (ρ-σ)g/v]
Variation of viscosity with a change in temperature and pressure:-
(a) Effect of temperature:-
η= A /(1+Bt)c
Here A, B and C are constants.
Again, ηv1/2 = Aec/vt
Here, A and C are constants and v is the relative velocity.
(b) Effect of pressure:- Co-efficient of viscosity of liquids increases due to an increase in pressure but there is no relation, so far, to explain the effect.
Change in viscosity of gases:-
(a) Effect of temperature:- Co-efficient of viscosity of a gas at a given temperature is given by,
η= η0AT1/2
Here T is the absolute temperature of gas.
♦️Revision Notes on Flow of Liquids and Viscosity♦️(2/3)
➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖
Viscosity:- Viscosity is the property of fluids by virtue of which they tend to destroy any relative motion between their layers.
Velocity gradient:- Velocity gradient is defined as the rate of change of velocity with respect to distance.
(a) Velocity gradient = dv/dr
(b) Dimension of velocity gradient = [dv/dr] = [T-1]
(c) Direction of velocity gradient is perpendicular to the direction of flow, directed in the direction of increasing velocity.
(d) Average velocity gradient:- Average velocity gradient is the difference between velocities of two layers separated a unit distance apart.
Average velocity gradient = Δv/Δr
Newton’s law of viscosity:-
In accordance to Newton’s law of viscosity, the viscous drag force depends upon the nature of fluid along with following factors:-
(a) F∝A (common area of two layers)
(b) F∝dv/dr (velocity gradient)
(c) So, F =ηA (dv/dr)
Here η is called coefficient of viscosity of fluid.
Coefficient of viscosity of fluid (ηv) or fugitive elasticity:-
ηv = shear stress/velocity gradient = (F/A)/(dv/dr)
Modulus of rigidity(ηr):-
ηr = shear stress/shear strain = (F/A)/(θ) = (F/A)/(dx/dr)
Here, θ = dx/dr = displacement gradient
Coefficient of viscosity (Absolute viscosity or Dynamic viscosity):-
F= ηA (dv/dr) if A = 1, dv = 1, dr =1, F = η
Co-efficient of viscosity of a fluid is defined as the tangential force per unit area which is required to maintain (or resist) a unit relative velocity between two layers a unit distance apart.
Or
Co-efficient of viscosity of a fluid is defined as the tangential force per unit area which is required to maintain a unit velocity gradient between its layers.
Unit of η:-
S.I:- η = 1 deca poise = 1 N sec/m2
Co-efficient of viscosity of a fluid is said to be one deca-poise if a tangential force of 1 N per meter square is required to maintain a relative velocity of 1 ms-1 between its layer 1 m apart.
C.G.S:- η = 1 poise = 1 dyn sec/cm2
Coefficient of viscosity of a fluid is said to be one poise if a tangential force of 1 dyn per square cm is required to maintain a relative velocity of 1 cms-1 between its layers 1 cm apart.
Relation between deca-poise and poise:-
1 deca-poise = 10 poise
Dimension formula for η:-
η = Fdr/Adv = [M1L-1T-1]
Fluidity:- Reciprocal of coefficient of viscosity of a fluid is called its fluidity.
Fluidity = 1/η
Unit of fluidity: poise-1
Dimension of fluidity: [M-1L1T1]
Kinematic viscosity:- Kinematic viscosity of a fluid is defined as the ration between its coefficient of viscosity to the density of fluid.
Kinematic viscosity = η/ρ
Units of kinematic viscosity:- C.G.S – 1 stoke = cm2 s-1
Kinetic viscosity of a fluid having its dynamic viscosity one poise and density one g cm-3 is said to be 1 stoke.
Dimensional formula of kinematic viscosity = η/ρ = [M0L2T-1]
Critical velocity (Reynold’s Number):- Critical velocity (vc) is the maximum velocity of the flow of liquid flowing in a streamlined flow.
vc = NR η/ρD
Here η is the coefficient of viscosity of liquid, ρ is the density of liquid and D is the diameter of the tube.
Reynold’s Number, NR = ρvcD/ η
Stokes law:- In accordance to Stoke’s law, force of viscosity F depend upon,
(a) Co-efficient of viscosity of fluid η
(b) Radius of the moving body r
(c) Velocity of body v
So, force of viscosity, F = 6π η r v
Terminal velocity:- v = 2/9 [r2 (ρ-σ)/η]
η = 2/9 [r2 (ρ-σ)g/v]
Variation of viscosity with a change in temperature and pressure:-
(a) Effect of temperature:-
η= A /(1+Bt)c
Here A, B and C are constants.
Again, ηv1/2 = Aec/vt
Here, A and C are constants and v is the relative velocity.
(b) Effect of pressure:- Co-efficient of viscosity of liquids increases due to an increase in pressure but there is no relation, so far, to explain the effect.
Change in viscosity of gases:-
(a) Effect of temperature:- Co-efficient of viscosity of a gas at a given temperature is given by,
η= η0AT1/2
Here T is the absolute temperature of gas.
✍️Revision Notes on Vectors
➖➖➖➖➖➖➖➖➖➖➖➖
The length or the magnitude of the vector = (a, b, c) is defined by w = √a2+b2+c2
A vector may be divided by its own length to convert it into a unit vector, i.e. ? = u / |u|. (The vectors have been denoted by bold letters.)
If the coordinates of point A are xA, yA, zA and those of point B are xB, yB, zB then the vector connecting point A to point B is given by the vector r, where r = (xB - xA)i + (yB – yA) j + (zB – zA)k , here i, j and k denote the unit vectors along x, y and z axis respectively.
Some key points of vectors:
1) The magnitude of a vector is a scalar quantity
2) Vectors can be multiplied by a scalar. The result is another vector.
3) Suppose c is a scalar and v = (a, b) is a vector, then the scalar multiplication is defined by cv = c (a, b) = (ca, cb). Hence each component of vector is multiplied by the scalar.
4) If two vectors are of the same dimension then they can be added or subtracted from each other. The result is gain a vector.
If u, v and w are three vectors and c, d are scalars then the following results of vector addition hold true:
1) u + v = v + u (the commutative law of addition)
2) u + 0 = u
3) u + (-u) = 0 (existence of additive inverses)
4) c (du) = (cd)u
5) (c + d)u = cu + d u
6) c(u + v) = cu + cv
7) 1u = u
8) u + (v + w) = (u + v) + w (the associative law of addition)
Some Basic Rules of Algebra of Vectors:
1) a.a = |a|2 = a2
2) a.b = b.a
3) a.0 = 0
4) a.b = (a cos q)b = (projection of a on b)b = (projection of b on a) a
5) a.(b + c) = a.b + a.c (This is also termed as the distributive law)
6) (la).(mb) = lm (a.b)
7) (a ± b)2 = (a ± b) . (a ± b) = a2 + b2 ± 2a.b
8) If a and b are non-zero, then the angle between them is given by cos θ = a.b/|a||b|
9) a x a = 0
10) a x b = - (b x a)
11) a x (b + c) = a x b + a x c
Any vector perpendicular to the plane of a and b is l(a x b) where l is a real number.
Unit vector perpendicular to a and b is ± (a x b)/ |a x b|
The position of dot and cross can be interchanged without altering the product. Hence it is also represented by [a b c]
1) [a b c] = [b c a] = [c a b]
2) [a b c] = - [b a c]
3) [ka b c] = k[a b c]
4) [a+b c d] = [a c d] + [b c d]
5) a x (b x c) = (a x b) x c, if some or all of a, b and c are zero vectors or a and c are collinear.
Methods to prove collinearity of vectors:
1) Two vectors a and b are said to be collinear if there exists k ? R such that a = kb.
2) If p x q = 0, then p and q are collinear.
3) Three points A(a), B(b) and C(c) are collinear if there exists k ? R such that AB = kBC i.e. b-a = k (c-b).
4) If (b-a) x (c-b) = 0, then A, B and C are collinear.
5) A(a), B(b) and C(c) are collinear if there exists scalars l, m and n (not all zero) such that la + mb+ nc = 0, where l + m + n = o
Three vectors p, q and r are coplanar if there exists l, m ? R such that r = lp + mq i.e., one can be expressed as a linear combination of the other two.
If [p q r] = 0, then p, q and r are coplanar.
Four points A(a), B(b), C(c) and D(d) lie in the same plane if there exist l, m ? R such that b-a = l(c-b) + m(d-c).
If [b-a c-b d-c] = 0 then A, B, C, D are coplanar.
Two lines in space can be parallel, intersecting or neither (called skew lines). Let r = a1 + μb1 and r = a2 + μb2 be two lines.
They intersect if (b1 x b2)(a2 - a1) = 0
The two lines are parallel if b1 and b2 are collinear.
The angle between two planes is the angle between their normal unit vectors i.e. cos q = n1 . n2
If a, b and c are three coplanar vectors, then the system of vectors a', b' and c' is said to be the reciprocal system of vectors if aa' = bb' = cc' = 1 where a' = (b xc) /[a b c] , b' = (c xa)/ [a b c] and c' = (a x b)/[a b c] Also, [a' b' c'] = 1/ [a b c]
Dot Product of two vectors a and b defined by a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is given by a1b1 + a2b2 + ..., + anbn .
