Definition of viscosity:
When a layer of fluid slips or tends to slip on another layer in contact, the two layers exert tangential forces on each other. The directions are such that the relative motion between the layers is opposed. This property of fluid to oppose relative motion between its layers is called viscosity. The forces between two layers which oppose relative motion between them are termed as forces of viscosity.
EQUATION OF CONTINUITY:
consider the flow of liquid in a pipe as shown above. There are two cross sections of area A1 and A2.Let speed of liquid be v1 at A1 and v2 at A2. Let d be density density of liquid . And mass of liquid m1 flowing through area A1 in time t be
m1= A1. v1. d .t.
similarly, mass m2 flowing through A2
m2=A2.v2.d.t
since the liquid in between two cross sections is not absorbed nor the liquid is created , m1=m2.
so; A1.v1.d.t=A2.v2.d.t
so; A1.v1 = A2.v2
means Av is constant.
BERNOULLIS EQUATION:
The liquid being incompressible, the volume of liquid entering from A1 and leaving from A2 is same.
hence; A1.S1=A2.S2=m/d (where m is mass of liquid entering and leaving and d is density of liquid) _________________(*)
At A1 , pressure is p1 . As work done = force x displacement;
so work done by pressure to move liquid to move it from A1 to A2 is =w=F1xs1 – F2xs2
=(p1)(A1)(s1)-(p2)(A2)(s2).(as F=P x A )____________(1)
Let work done for lifting weight w from h1 to h2 be ‘w’.
so; w1=mgh2-mgh1______________________(2)
gain or loss of kinetic energy is
w2=(1/2)m(v2)2 – (1/2)m(v1)2__________________(3)
w=w1+w2
so; w=w1+w2
from equation 1,2 and 3;
(p1)(A1)(s1)-(p2)(A2)(s2)= (mgh2)-(mgh1)+(1/2)m(v2)2-(1/2)m(v1)2
from (*); A1S1=A2S2=m/d;
so;
(p1)(m/d)-(p2)(m/d)=mg(h2-h1)+(1/2m)[(v2)2-(v1)2]
(m/d)(p1-p2)=mg(h2-h1)+(1/2m)[(v2)2-(v1)2]
multiplying by d , we get:
m(p1-p2)=mg d(h2-h1)+(1/2)md[(v2)2-(v1)2]
dividing by m, we get:
p1-p2=d g h2 – d g h1 + (1/2) d [(v2)2-(v1)2]
p1 + d g h1 +(1/2) d (v1)2 =p2 + d g h2 + (1/2) d (v2)2
HENCE; WE OBSERVE THAT,
p + d g h + (1/2) d (v)2 is constant.
APPLICATIONS OF BERNOULLIS THEOREM
1 . VENTURIMETER
from Bernoullis equation,
p + d g h + (1/2) d v2 is constant.
so for points 1 and 2
p1 + d g h1 + (1/2) d (v1)2= p1 + d g h2 + (1/2) d (v2)