FLUID MECHANIC EQUATION M. Ragheb 11/2/2017 INTRODUCTION The early part of the 18 th -century saw the burgeoning of the field of theoretical fluid mechanics pioneered by Leonhard Euler and the father and son Johann and Daniel Bernoulli. We introduce the equations of continuity and conseration of momentum of fluid flow and the Naier-tokes and Euler equations as the fundamental relations goerning deterministic or mechanistic reactor safety analysis. THE MA CONERVATION OR CONTINUITY EQUATION The continuity equation of fluid mechanics expresses the notion that mass cannot be created nor destroyed or that mass is consered. It relates the flow field ariables at a point of the flow in terms of the fluid density and the fluid elocity ector, and is gien by:.( V ) 0 (1) We consider the ector identity resembling the chain rule of differentiation:.( V ). V V. (2) where the diergence operator is noted to act on a ector quantity, and the gradient operator acts on a scalar quantity. This allows us to rewrite the continuity equation as: V.. V 0 (3) UBTANTIAL DERIVATIVE We can use the substantial deriatie: D Dt ( V. ) Conetie Local Deriatie Deriatie (4) where the partial time deriatie is called the local deriatie and the dot product term is called the conectie deriatie.
In terms of the substantial deriatie the continuity equation can be expressed as: D. V 0 (5) Dt MOMENTUM CONERVATION OR EQUATION OF MOTION Newton s second law is frequently written in terms of acceleration and a force ectors as: F ma (6) A more general form describes the force ector as the rate of change of the momentum ector as: F d ( mv ) (7) dt Its general form is written in term of olume integrals and a surface integral oer an arbitrary control olume as a function of the pressure p, the body forces f and the iscous forces Fiscous : ( V ) d ( V. d) V pd f d F iscous d (8) where the elocity ector is: V uxˆ yˆ wzˆ (9) The Cartesian coordinates x, y and z components of the continuity equation are: x x iscous x y y iscous y ( u) d ( V. d ) u d f d ( F ) d ( ) d ( V. d ) d f d ( F ) d ( w) d ( V. d ) w d f d ( F ) z d z z iscous (10) In this equation the product: is a scalar and has no components. ( V. d) (11)
NAVIER TOKE EQUATION By using the diergence or Gauss s theorem the surface integral can be turned into a olume integral: ( V. d) u ( uv ). d.( uv ) d ( V. d) ( V ). d.( V ) d ( V. d) w ( wv ). d.( wv ) d (12) The olume integrals oer an arbitrary olume now yield: ( u).( uv ) f x ( F ) ( ).( V ) f y ( F ) x x iscous y y iscous ( w).( wv ) f ( F ) z z z iscous (13) These are known as the Naier-tokes equations. They apply to the unsteady, three dimensional flow of any fluid, compressible or incompressible, iscous or iniscid. In terms of the substantial deriatie, the Naier-tokes equations can be expressed as: Du f Dt x ( F ) D f Dt y ( F ) x x iscous y y iscous Dw f ( F ) Dt z z z iscous (14) EULER EQUATION For a steady state flow the time partial deriaties anish. For iniscid flow the iscous terms are equal to zero. In the absence of body forces the f x, f y, anf f z terms disappear. The Euler equations result as:
.( uv ) x.( V ) y.( wv ) z (15) CONERVATION OF MOMENTUM FOR INVICID COMPREIBLE FLOW For an iniscid flow without body forces, the momentum conseration equations of fluid mechanics are: Du Dt x D Dt y Dw Dt z (16) These equations can also be written as: ( u) V. ( u) x ( ) V. ( ) y ( w) V. ( w) z (17) For steady flow the partial time deriatie anishes, and we can write: V. ( u) x V. ( ) y V. ( w) z (18) Expanding the gradient term, we get:
u u u u w x y z x u w x y z y w w w u w x y z z (19) Rearranging, we get: u u u 1 u w x y z x 1 u w x y z y w w w 1 u w x y z z (20) TREAMLINE DIFFERENTIAL EQUATION The definition of a streamline in a flow is that it is parallel to the elocity ector. Hence the cross product of the directed element of the streamline and the elocity ector is zero: where: ds dx xˆ dy yˆ dz zˆ V u xˆ yˆ w zˆ ds V 0 (21) The cross product can be expanded in the form of a determinant as: xˆ yˆ zˆ ds V dx dy dz u w xˆ( wdy dz) yˆ( u dz wdx) zˆ ( dx u dy) 0 (22) The ector being equal to zero, its components must be equal to zero yielding the differential equations for the streamline f(x, y, z) = 0, as:
wdy dz 0 u dz wdx 0 dx u dy 0 (23) EULER EQUATION Multiplying the flow equations respectiely by dx, dy and dz, we get: u u u 1 u dx dx w dx dx x y z x 1 u dy dy w dy dy x y z y w w w 1 u dz dz w dz dz x y z z (24) Using the streamline differential equations, we can write: u u u 1 u dx u dy w dz dx x y z x 1 u dx dy w dz dy x y z y w w w 1 u dx dy w dz dz x y z z (25) The differentials of functions u = u(x, y, z), = (x, y, z), w = w (x, y, z) are: u u u du dx dy dz x y z d dx dy dz x y z w w w dw dx dy dz x y z (26) This allows us to write:
1 udu dx x 1 d dy y 1 wdw dz z (27) Through integration we can write: 1 2 1 d( u ) dx 2 x 1 2 1 d( ) dy 2 y 1 2 1 d( w ) dz 2 z (28) Adding the three last equations we get: 1 1 d u w dx dy dz 2 x y z 2 2 2 ( ) ( ) 1 2 1 d( V ) dp 2 (29) From the last equation we can write a simple form of Euler s equation as: dp VdV (30) Euler s equation applies to an iniscid flow with no body forces at steady-state. It relates the change in speed along a streamline dv to the change in pressure dp along the same streamline. BERNOULLI EQUATION, INCOMPREIBLE FLOW Considering the case of incompressible flow, we can use limit integration to yield: p2 V2 dp VdV p1 V1 2 2 p2 p1 V2 V1 2 1 2 1 2 p1 V1 p2 V2 constant 2 2 (31)
The relation between pressure p and speed V in an iniscid incompressible flow was enunciated in the form of Bernoulli s equation, first presented by Euler: 1 (32) 2 2 p V constant This equation is the most famous equation in fluid mechanics. Its significance is that when the elocity increases, the pressure decreases, and when the elocity decreases, the pressure increases. The dimensions of the terms in the equation are kinetic energy per unit olume. Een though it was deried from the momentum conseration equation, it is also a relation for the mechanical energy in an incompressible flow. It states that the work done on a fluid by the pressure forces is equal to the change of kinetic energy of the flow. In fact it can be deried from the energy conseration equation of fluid flow. The fact that Bernoulli s equation can be interpreted as Newton s second law or an energy equation illustrates that the energy equation is redundant for the analysis of iniscid, incompressible flow. REFERENCE 1. John D. Anderson, Jr., Fundamentals of Aerodynamics, 3 rd edition, McGrawHill, 2001. EXERCIE 1. From Euler s equation: dp VdV Derie the expression for Bernoulli s law suggesting that the sum of the static and kinetic pressures is a constant between two points at steady-state in an iniscid flow without body forces. 2. A wind rotor airfoil is placed in the air flow at sea leel conditions with a free stream speed of 10 m/s. The density at standard sea leel conditions is 1.23 kg/m 3 and the pressure is 1.01 x 10 5 Newtons / m 2. At a point along the rotor airfoil the pressure is 0.90 x 10 5 Newtons / m 2. By applying Bernoulli s equation estimate the wind speed at this point.