CHAPTER-14 Fluids

Ch 14-2, 3 Fluid Density and Pressure  Fluid: a substance that can flow  Density  of a fluid having a mass m and a volume V is given by :  = m/V (uniform density) Density Units: kg/m 3  Density of a compressible material such as gases depends upon the pressure P, where P is given by: P=F/A

Ch 14-4 Fluid at Rest Hydrostatic Pressure: (Pressure due to fluid at rest) The pressure at a point in a fluid in static equilibrium depends upon the depth of that point but not on any horizontal dimension of the fluid or container. Consider the imaginary water cylinder with horizontal base, with weight mg, enclosed between two depths y 1 and y 2.The cylinder has a volume V, face area A and height y 1 -y 2, Water is in static equilibrium. Three forces F 1, F 2 and mg acts such that F 2 -F 1 -mg=0; F 2 =F 1 +mg But p 1 =F 1 /A; p 2 =F 2 /A; m=  Vg=  A(y 1 -y 2 ) p 2 =p 1 +  g(y 1 -y 2 )

Ch 14-4 Absolute Pressure and Gauge Pressure p 2 =p 1 +  g(y 1 -y 2 ) If p 2 =p; p 1 =p 0 and y 1 =0 ; y 2 =-h p=p 0 +  gh  p =p-p 0 =  gh P is Absolute pressure  p is Gauge pressure

Ch 14-4 Absolute Pressure and Gauge Pressure

Ch 14-6 Pascal’s Principle  Pascal’s Principle : A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.  A change in pressure  P at input of hydraulic lever is converted to change in pressure  P at output of hydraulic lever  P=F i /A i =F o /A o ; F o = A o (F i /A i ),F o  F i  If input piston moves through a distance di, then the output piston moves through a smaller distance do because V=A i d i =A o d o  With a hydraulic lever, a given force applied over a given distance can be transformed to a greater force applied over a smaller distance

Ch 14-7 Archimedes’ Principle  Archimedes’ Principle:  Buoyant Force F B : Upward force on a fully or partially submerged object by the fluid surrounding the object, magnitude of the force F B equal to weight of the displaced fluid m f g=  V f g. V f is volume of the displaced fluid.  Floating Object: When an object floats in a fluid, the magnitude of F B is equal to magnitude of the gravitational force F g (=mg). Then F B = F g =mg= m f g=  V f g  A floating object displaces its own weight of fluid  For objects submerged in a fluid, its Apparent weight W app is less than true weight W Apparent weight W app = W-F B

Ch 14-8 Ideal Fluid in Motion Four assumptions related to ideal flow:  Steady flow: the velocity of the moving fluid at any fixed point does not change with time  Incompressible Flow: Fluid has constant density  Nonviscous flow: an object can move through the fluid at constant speed- no resistive force within the fluid to moving objects through it  Irrotational flow: Objects moving through the fluid do not rotate about an axis through its center of mass

Ch 14-9 Equation of Continuity  Equation of Continuity: A relation between the speed v of an ideal fluid flowing through a tube of cross sectional area A in steady flow state  Since fluid is incompressible, equal volume of fluid enters and leaves the tube in equal time  Volume  V flowing through a tube in time  t is  V = A  x =Av  t Then  V = A 1 v 1  t =A 2 v 2  t  A 1 v 1 =A 2 v 2  R V =A 1 v 1 =constant (Volume flow rate)  R m =  A 1 v 1 =constant (Mass flow rate)

Ch Bernoulli’s Equation Bernoulli’s Equation: If y 1,v 1 and p 1 are the elevation, speed and pressure of the fluid entering the tube and and y 2,v 2 and p 2 are the elevation, speed and pressure of the fluid leaving the tube Then p 1 + (  v 1 2 )/2+  gy 1 =p 2 + (  v 2 2 )/2+  gy 2 or p+ (  v 2 )/2+  gy = constant