Viscocity

MECHANICAL PROPERTIES OF MATTER Viscosity is the property of the fluid (liquid or gas) by virtue of which it opposes the relative motion between its adjacent layers. It is the fluid friction or internal friction. The internal tangential force which try to retard the relative motion between the layers is called viscous force. NEWTON'S LAW OF VISCOSITY Suppose a liquid is flowing in streamlined motion on a horizontal surface OX. The liquid layer in contact with the surface is at rest while the velocity of other layers increases with increasing distance from the surface OX. The highest layer flows with maximum velocity. Let us consider two parallel layers PQ and RS at distances y and y + y from OX. Thus the change in velocity in a perpendicular distance y is vx. The rate of change of velocity with distance perpendicular to the direction of flow i.e. v x , is called velocity–gradient. According to Newton, the viscous force F acting between two layers of a y liquid flowing in streamlined motion depends upon following two factors : (i) F  Contact–area of the layers i.e. F  A (ii) F  Velocity–gradient  v x v v x  v x  A x where  is a between layers i.e. F  . Therefore F A F =  y y y  y  constant called coefficient of viscosity of the liquid. In above formula if A = 1 and v x = 1, then F = . i.e. the coefficient of viscosity of a liquid is defined as the viscous y force per unit area of contact between two layers having a unit velocity gradient. • SI Units : Ns or deca poise m2 • CGS units : dyne–s/cm2 or poise (1 decapoise = 10 poise) • Dimension : M1L–1T–1 42 Ex. A plate of area 2 m2 is made to move horizontally with a speed of 2 m/s by applying a horizontal tangential force over the free surface of a liquid. If the depth of the liquid is 1m and the liquid in contact with the plate is stationary. Coefficient of viscosity of liquid is 0.01 poise. Find the tangential force needed to move the plate. Sol. Velocity gradient = v 2 0 m/ s = =2 = 2s–1 y 10 m From, Newton's law of viscous force, F  A v=2m/s F 1m v = (0.01 × 10–1) (2) (2) = 4 × 10–3 N y Ex. A man is rowing a boat with a constant velocity v0 in a river the contact area of boat is ‘A’ and coefficient of viscosity is . The depth of river is ‘D’. Find the force required to row the boat. Sol. F – FT = m a As boat moves with constant velocity a = 0 so F = FT Ex. But FT =  A v0  0 v0 dv dv but = = dz dz D D then F = FT = Av0 D A cubical block (of side 2m) of mass 20 kg slides on inclined plane lubricated with the oil of viscosity  = 10–1 poise with constant velocity of 10 m/s. Find out the thickness of layer of liquid. (g = 10 m/s2) Sol. F = F =  A 30° dv dv v = mg sin = dz dz h 20 × 10 × sin 30° =  × 4 × 10 h h = 4 × 10–3 m = 4 mm Ex. As per the shown figure the central solid cylinder starts with initial angular velocity 0. Find out the time after which the angular velocity becomes half. Sol. F = A F= dv dv R1  0 , where = dz dz R 2  R 1 2   R13  l 2  R1 l  R1 and  = FR1 = R 2  R1 R 2  R1 MECHANICAL PROPERTIES OF MATTER  = 2  R13 l MR12  d  2   R13  l    dt  = R 2  R1 2 R 2  R1 0 2 or –  0 t d 4 R1 l m (R  R )l n2 =  dt  t = 4 2  l R1  m (R 2  R1 ) 0 1 DEPENDENCY OF VISCOSITY OF FLUIDS 1. On Temperature of Fluid (a) Since cohesive forces decrease with increase in temperature as increase in K.E. Therefore with the rise in temperature, the viscosity of liquids decreases. (b) The viscosity of gases is the result of diffusion of gas molecules from one moving layer to other moving layer. Now with increase in temperature, the rate of diffusion increases. So, the viscosity also increases. Thus, the viscosity of gases increases with the rise of temperature. 2. On Pressure of Fluid (a) The viscosity of liquids increases with the increase of pressure. (b) The viscosity of gases is practically independent of pressure. 3. On Nature of Fluid STOKE'S LAW AND TERMINAL VELOCITY Stoke showed that if a small sphere of radius r is moving with a velocity v through a homogeneous medium (liquid or gas), coefficient of viscosity  then the viscous force acting on the sphere is Fv = 6rv. It is called Stoke's Law. Terminal Velocity When a solid sphere falls in a liquid then its accelerating velocity is controlled by the viscous force of liquid and hence it attains a constant velocity which is known as terminal velocity (vT). As shown in figure when the body moves with constant velocity i.e. terminal velocity (zero acceleration) the net upward force (upthrust Th + viscous force Fv) balances the downward force (weight of body W) Therefore Th + Fv = W  2 r 2 (   ) 4 3 4 g r g + 6rvT = r3g vT = 9  3 3 where r = radius of body,  = density of body,  = density of liquid,  = coefficient of viscosity Some applications of Stoke’s law : (a) The velocity of rain drops is very small in comparison to the velocity of a body falling freely from the height of clouds. (b) Descending by a parachute with lesser velocity. (c) Determination of electronic charge with the help of Milikan's experiment. Ex. Sol. A spherical ball is moving with terminal velocity inside a liquid. Determine the relationship of rate of heat loss with the radius of ball.  2 gr 2 (0  l ) 2  Rate of heat loss = power = F × v = 6  rv × v = 6 rv = 6pr   9  2 Therefore rate of heat loss  r5 Ex. A drop of water of radius 0.0015 mm is falling in air. If the coefficient of viscosity of air is 1.8 × 10–5 kg m–1 s–1. What will be the terminal velocity of the drop. Density of air can be neglected. 2 Sol.  15  1 0 4  3 2   1 0  9.8 2 1 00 0 2 r (   )g   vT    2 .72  10 4 m/ s 9  9  1 .8  10 5 REYNOLDS NUMBER (Re) The type of flow pattern (laminar or turbulent) is determined by a non–dimensional number called Reynolds number (Re). Which is defined as Re = vd where  is the density of the fluid having viscosity  and flowing with mean  speed v. d denotes the diameter of obstacle or boundary of fluid flow. Although there is not a perfect demarkation for value of Re for laminar and turbulent flow but some authentic references take the value as Re < 1000 Type of flow laminar > 20 00 often turbulent between 1 00 0 to 2 00 0 may be laminar or turbulent on gradually increasing the speed of flow at certain speed transition from laminar flow to turbulent flow takes place. This speed is called critical speed. For lower density and higher viscosity fluids laminar flow is more probable.