Evaluation of Euler’s Equation of Motion

INTRODUCTION

In this paper we will discuss equation of motion for ideal invicid fluid and its applications. Equation of motion is most fundamental equation in fluid mechanics that uses Newton‘s 2nd law of motion according to which the rate of change of momentum of particle can be equated to total force applied .We assume that flow is homogenous and incompressible. The flow properties are independent of time and follows continuum hypothesis. The flow is along stream line and velocity of fluid is uniform over cross-section. We select a portion of fluid having volume enclosing material surface .Then the rate of variation of linear momentum of selected portion is equated force that acts on that portion of fluid. Rate of variation of linear momentum in volume V = = Here is velocity of fluid having density. We know, in general that when we deal with a portion of fluid both surface as well as body forces comes into play. Total surface force on = dS Then by Gauss divergence theorem, total surface force in volume V becomes = Here p is pressure acting at a point in fluid region. If we denote as body force per unit mass Then total body force on = By Newton‘s second law of motion we have

= +

Since is arbitrary, we have 0 …

(1)

+- =0 is eulars equation of motion for ideal invicid fluid. Since is total acceleration, therefore

= + ( = + (1/2 2) - (

Thus equation (1) becomes

+ (1/2 2) - ( ) =+ … (2)

If we assume the flow to be of potential (i.e. = - and =0) find under the action of conservative body forces (i.e. = - ) then integration of equation (2) gives well-known equation called Bernoulli‗s equation of motion.

+ (1/2 2) =- +(1/2 2) ++ =0 …(3)

Taking dot product with equation (3) become +1/2 2+ + =c (t)… (4) Bernoulli equation is laid down, the actual problems can be quickly solved. Example: If a tube of small uniform bore having radii a centered at O forms 1/4th part of circle. (Denote tube by AB and B is considered below A). When end B is closed, tube is completely filled with liquid of certain density Observe that in starting = if end B is opened immediately (here u denotes velocity) and pressure at a point P drops to - Solution to above problem become easy with help of Euler‘s equation of motion When tube is closed pressure at ends A and B is atmospheric pressure (i.e. ) Since = and = u By Euler‘s equation of motion +u=-+) +u =g+ - + ) ; then comparing coefficients of +u =g - (Clearly from figure = and s = a ) +u = - +u = - +u = - … (5) At point A, we have s = 0; u = 0; y = 0; p = ; t = 0 Integrating (5) with respect to s + = gy -+c (t)… (6) Hence c(t) = +0 = ga –+ Since =+u Thus, at t=0 = At point P s =a ; u=0; t=0; y = a By equation (6) +0 = ga -+c (0) Therefore p = - The above example illustrate that how physical problems can be solved quickly with help of Euler‘s and Bernoulli‘s equation of motion.

1. Glauert, M.B 1957 Proc.Roy.Soc.A 242 ,1080 2. Goldstein ,S.(Ed) 1938 Modern development in fluid dynamics , vol 1 and 2 . oxford university 3. Hadmard ,J.1911 comptes rendes 152, 1735 4. Lamb ,H.1911 Phil. Mag.(6),21,112. 5. Shames ,1H Mechanics of fluids , Mc-Graw –Hill New York 1962. 6. Potter Mc and J.F Foss . Fluid Mechanics . GREAT lakes Press , 1982. 7. Kelley ,J.B ― The extended Bernoulli equation ― Amer J.Phys,18 pp. 202-204 ,1950 8. Currie I.G FundamentaL Mechanics of fluids ,Mc Graw –Hill :New York ,1974. 9. 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 10. Andrade ,E.N. da C. 1939 Proc. Phys. Soc.51 ,784 11. Apelt , C.J 1961 Aero. Res. Coun., Rep. and Mem. No. 3175. 12. Blasius , H. 1908 Z. Maths. Phys. 56,I. Mathematics of Department, Assistant Professor, D.A.V. College (Lahore), Ambala City E-mail:gurpreetkaur12021997@gmail.com