Applications of Equation of Continuity

INTRODUCTION

ln this paper we will throw light on equation of continuity and its interesting applications. Let us consider fluid flowing through some region of space from external source .Equation of continuity is derived using law of conservation of mass which states ―The total mass of fluid within a region in absence of source or sink remain conserved‖. Let be density of fluid at a point P. For calculation of density we need to know mass of small volume V of fluid containing point P. Consider arbitrary surface S bounding volume V drawn in a region where fluid flows. Total mass of fluid within volume V = Rate of increase of this mass in volume V = Mass of fluid leaving the surface S per unit time = By gauss divergence theorem, mass of fluid leaving the surface S per unit time = By law of conservation of mass, Rate of increase of mass in volume V = Mass of fluid entering S per unit time =

+ =0

=0 This is true for arbitrary dV Therefore, + =0 … (1) Equation (1) is known as equation of continuity.

SOME SPECIAL CASES

1). If the fluid flow is steady (i.e. flow properties are independent of time) .Then equation (1) becomes =0 2). If flow is considered incompressible (i.e. volume and density of fluid does not change with applied pressure).Then Equation (1) becomes =0 3). If velocity is of potential kind then = - ; Equation (1) will takes form - =0 Example: A spherical bubble of gas is formed as result of explosion under water that was at rest volume, is constant) .It is required to find radius R of gas bubble at any time t after explosion, taking radii at time t =0. Solution : Radius of gas bubble can easily be found using simple adiabatic law of pressure and equation of continuity. Consider concentric spherical surface in liquid region and velocity of fluid at any point in radial direction is Since = We have = = - At r = R, = By equation of continuity, volume of fluid entering V at per unit time = volume of fluid leaving V at S per unit time =

= = -

= If pressure at infinity is zero, using Bernoulli equation of motion We have pressure p at r =R given by p = By adiabatic law of pressure = constant We have =

= … (2)

Clearly this is second order differential equation in R (t) which can be solved easily. Substitute = s, equation (2) will take the form as = s = Since =0 when R= We have c = =

= = …(3)

Integrating equation (3) we get radius of gas bubble at any time t.

REFERENCES

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13). Shames ,1H Mechanics of fluids , Mc-Graw –Hill New York 1962. 14). Potter Mc and J.F Foss . Fluid Mechanics . GREAT lakes Press , 1982. 15). Kelley ,J.B ― The extended Bernoulli equation ― Amer J.Phys,18 pp. 202-204 ,1950 16). Currie I.G FundamentaL Mechanics of fluids ,Mc Graw –Hill :New York ,1974. 17). Lumey J.L ―Eularian and lagrangian Description in fluid Mechanics ― in illustrated experiments in fluid mechanics ,pp 3- National committee for fluid Mechanic Films .MIT : Cambridge ,1972

Assistant Professor, DAV College(Lahore), Ambala City