Advanced Insights into Fluid Mechanics
Delve into the intricate world of fluid mechanics with Dr. Afsaneh Mojra from K. N. Toosi University of Technology. Explore the two primary methods of deriving governing equations for fluid motion, understanding the significance of basic conservation laws, and unraveling the complexities of field variables like density and temperature in fluids. Dive deep into the fundamental principles that govern fluid behavior and properties, guiding you through advanced concepts and equations in the field of fluid mechanics.
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Presentation Transcript
Advanced Fluid Mechanics Dr. Afsaneh Mojra Department of Mechanical Engineering K. N. Toosi University of Technology
There are two methods to derive the governing equations for motion of a fluid: Statistical Approach and Continuum Approach Basic Conservation Laws Dr. Afsaneh Mojra
Basic Conservation Laws Dr. Afsaneh Mojra
???????????? ?????: ? = ?(?,?,?,?) ???????? ?????: ? = ?(?,?,?,?) ???????? ?????: ? = ?(?,?,?,?) Basic Conservation Laws Dr. Afsaneh Mojra
: a field variable such as density or temperature of the fluid ?? =?? ???? +?? ??=?? ???? +?? ??+?? ?? ?? ??= ?, ??=?? ???? +?? ??+?? ?? ?? ??= ?, ??+ ??? ???? ?? ?? ?? ?? ??+?? ?? ??= ? ??+ ??? ?? ?? ?? 0, ?? ??+ ??? ?? Basic Conservation Laws Dr. Afsaneh Mojra
? ?,?,?,? = ? ? ?,?,?,? + ? ? ?,?,?,? + ? ? ?,?,?,? ?????????=?????????? ??=??(?????????? ,?????????? ,?????????? ,?) =?? ?? ?? ?????????? ?? ?????????? ?? ?????????? ?? =?? ?? ??+ ?? ?? ?? + + ?? ?????????? ?????????? ?????????? ?????????(?,?,?,?) =?? ??=?? ??+ ??? ??+ ??? ??+ ??? ?? Basic Conservation Laws Dr. Afsaneh Mojra
??=?? ??+ ??? ??+ ??? ??+ ??? ?? ??=?? ??+ ??? ??+ ??? ??+ ??? ?? ??=?? ??+ ??? ??+ ??? ??+ ??? ?? Basic Conservation Laws Dr. Afsaneh Mojra
: : ?? : ? = ? ? :any property of fluid ? ?? 1 ?? ? ? ?? = lim ? ? + ?? ?? ? ? ?? ?? 0 ? ? ? ? ? ?+?? Basic Conservation Laws Dr. Afsaneh Mojra
Basic Conservation Laws Dr. Afsaneh Mojra
1 ?? ? ? + ?? ?? ? ? + ?? ?? ? ?? ? ? ? ? ?? = lim ? ?+?? +1 ?? ?? 0 ? ? ? ? + ?? ?? ? ? ?? ? ? ? ? ? ?? 1 ?? ?? ???? ? ? ?? = lim ? ? + ?? ?? + ?? 0 ? ? ? ?+?? ? ? ? ? ? ?? ?? ???? ? ? ?? = lim ? ? + ?? ?.??? + ?? 0 ? ? ? ? ? ? ?? ???? = ? ? ?.??? + ? ? ? ? Basic Conservation Laws Dr. Afsaneh Mojra
Use of Gauss theorem: ? ? ?.??? = . ?? ?? ? ? ? ? ? ?? ?? ??+ . ?? ??? = ?? ? ? ? ?? ?? ??+ ? ??? = ??? ?? ??? ? ? Basic Conservation Laws Dr. Afsaneh Mojra
Consider a specific mass of fluid whose volume V is arbitrarily chosen. If this given fluid mass is followed as it flows, its size and shape will be observed to change but its mass will remain unchanged. ??? Basic Conservation Laws Dr. Afsaneh Mojra
Since the volume V was arbitrarily chosen, the only way in which the above equation can be satisfy for all possible choices of V is for the integrand to be zero. For incompressible fluid : Basic Conservation Laws Dr. Afsaneh Mojra
= = External force: 1- Body force: Gravity, Electromagnetic forces 2- Surface force: Pressure, Viscous forces P: Resultant surface force per unit area f: Resultant body force per unit mass ? ?? ? ??? ???? = ??? + ? ? ? Basic Conservation Laws Dr. Afsaneh Mojra
? ?? ????? = along j direction: ??????? + ????? ? ? ? RTT & Gauss theorem ? ?? ? ??? + ????? ?? ??? ? ???? ??? = ?? + ????? ? ? ????? =???? ? ?? ? ??? + + ??? ??? ??? Basic Conservation Laws Dr. Afsaneh Mojra
???? ??+ ?? ??? ??? =???? ??? ?? ??+ ?? ? ??? + ??? + ??? ??? Continuity=0 ???? ??+ ??? ??? ??? =???? ??? + ??? Shear force Temporal acceleration Body force Convective or local acceleration Basic Conservation Laws Dr. Afsaneh Mojra
The rate of change of the total energy of the fluid= rate at which work is being done on the fluid+ rate at which heat is being added by conduction ?? +1 2??.? ??:total energy (Internal & kinetic) contained in volume (V) ? ?.????:total work (body & surface work) ?.??? + ? ? ?.???:conductive heat leaving the fluid ? Basic Conservation Laws Dr. Afsaneh Mojra
? ?? ?? +1 2??.? ?? = ?.??? + ?.???? ?.??? ? ? ? ? Gauss theorem: ? ?.??? = ????????? = ??????? ??? ? ? ? ??? ??? ?????? = ?.??? = ?? ? ? ? Basic Conservation Laws Dr. Afsaneh Mojra
? ?? ?? +1 ? ?? +1 RTT: 2????? + 2????? ?? ?? ??? ? ??? ??? ? ??????? + ??????? = ?? ??? ? ? ? ????? + ????? ??? ? ?? ?? +1 ? ?? +1 ? 2????? + 2????? ?? = ??? ??? ??? II I ? ?? = ??? ?? +1 I 2????? ??+ ?? ??+ ??? 1 2???? +1 ?? ?? 2???? ?? Basic Conservation Laws Dr. Afsaneh Mojra
II ? ?? +1 ? ?? ??? +1 ? ? 1 2???? 2????? ?? = ? ??? + ??? 2???? ??? + ??? ??? ??? ??? ??? ? ?? +1 2????? ?? = ??? ?? ??? 1 ?? ??+ ??? ? 1 2???? ??+ ??? 2???? ??? ??? ? ?? ?? +1 ? ?? +1 2????? + ?? ??? 2????? ?? ? ??? I+II ??? 1 2???? + ??? = ??? + ?? 1 2???? ??+ ??? ?? ??? ??+ ????? ??? ??? = ??? ?? ??? ??+ ??? + ??? Basic Conservation Laws Dr. Afsaneh Mojra
??? ??+ ????? ??? ??? ????? + ????? ??? ??? ??+ ??? ?? ??? ? + ??? = ??? ??? ???? ??? ??? ??? ? ????? = ?? + ??? ??? ??? ??+ ????? ??? ??? ???? ??? ??? ??? + ????? ??? ??? ??+ ??? ?? ??? + ??? = ?? + ??? ??? Momentum Eq. ??? ??? ???? ??+ ??? =???? ??? + ??? Basic Conservation Laws Dr. Afsaneh Mojra
??? ??? ??? ??? ??? ??+ ??? ?? ??? = ??? Local change of internal energy Temporal change of internal energy Reversible Irreversible Basic Conservation Laws Dr. Afsaneh Mojra
???? ??+ ??? ??? ??? =???? ??? + ??? ??? ??? ??? ??? ??? ??+ ??? ?? ??? = ??? : ?,?,?,? = ?,?,? , ? = ??,??,??,??? 3 3 :?????:17 : 5 2 + Basic Conservation Laws Dr. Afsaneh Mojra
In order to obtain a complete set of equations: 1. Stress tensor ??? deformation tensor??? 2. Heat flux ? temperature gradient??? ??= ??? ??? Stress tensor ??? deformation tensor??? : Constitutive Equations Basic Conservation Laws Dr. Afsaneh Mojra
?? ?? = ? ? ? ,? ? ?? 0 Taylor series about the point (0,0): ?? ?? ?? ?? = ? 0,0 + ? ? 0,0 0 ?? ?? + ? ? 0,0 +... ?? Basic Conservation Laws Dr. Afsaneh Mojra
?? ?? ?? ?? ?? ?? = ? 0,0 ?? + ? ? 0,0 + ? ? 0,0 +... ?? = ? 0,0 ?? 0 ????????? ?? = ? 0,0 ?? ?? ?? ?? = ??? 1(? ????????? ?? ? ? ? ????????? ?? ? ? ) 1 2??, 1 2?? ? 1 2??, 1 ? ???? ?? ? ? = ?? ??.?? = ? 2?? .?? Basic Conservation Laws Dr. Afsaneh Mojra
?1 2??, 1 2?? ??+... ? 1 2??, 1 2?? ??+... ?? = tan 1 ?? Taylor series about the point (0,0): ?? = tan 1 ?(0,0) +1 ????)(0,0) 1 ??(1+... ????)(0,0) 1 ??(1+... ????)(0,0)+... ?? 2??( 2??( ) ?(0,0) 1 ????)(0,0)+... ?? 2??( 2??( ) ?? ?? ?? 0,0 +... ?? ?? ?? ?? = tan 1 = tan 1 0,0 +... ?? ?? Basic Conservation Laws Dr. Afsaneh Mojra
?? ?? 0,0 +... ??+... ?? ??=?? ?? = 0,0 +... ?? So, in the limit as ??,??,?? & ?? tend to zero, .=?? =?? ?? . ? 0,0 & ? 0,0 ?? 1 2 1 2 =1 ?? ?? ?? ?? ??+?? . . ? ? 2 ?? =1 . . ? + ? 2 ?? Basic Conservation Laws Dr. Afsaneh Mojra
??? ??? 1 2 ??? ??? Rate of rotation: Rate of shearing: +??? ??? 1 2 ??? ??? ??? ??? +??? ??? ???=1 ??? ??? +1 ??? ??? =??? ??? Deformation rate tensor: 2 2 Basic Conservation Laws Dr. Afsaneh Mojra
???= ????+ ??? ?:????????,?:? ??? ?????? ???=1 ??? ??? +??? ??? For a Newtonian Fluid: 2????? There are no preferred direction in the fluid, so that the fluid properties are point functions. Most general isotropic tensor of rank 4 is of the form: ????? = ???????+ ? ??????+ ?????? + ? ?????? ?????? ? = 0 isotropic ????? = ???????+ ? ??????+ ?????? Basic Conservation Laws Dr. Afsaneh Mojra
1 2??????? ??? ??? +??? ??? ??? ??? ?: :?? ? 0 ? = ? ???= 1 ???? +??? ??? 1 2? ?????? ??? ??? +??? ??? :?? ? 0 ? = ? & ? = ? ???= ???= 1 1 ??? ??? ??: 2? +??? ??? 1 2? ?????? ??? ??? +??? ??? :?? ? 0 ? = ? & ? = ? ???= ???= 1 1 ??? ??? ???: 2? +??? ??? ??? ??? ??? ??? ???= ???? + ? : +??? ??? ??? ??? ??? ??? ???= ????+ ???? + ? Basic Conservation Laws Dr. Afsaneh Mojra
? & ?:????? ?? ?????????? Consider a simple shear flow of an incompressible fluid in which the velocity components are defined by: ) ? = ?(? ? = ? = 0 For a Newtonian Fluid: ?12= ?21= ??? ?: dynamic viscosity ?: 2nd viscosity coefficient ?? ?11= ?22= ?33= ? ?13= ?31= ?23= ?32= 0 Basic Conservation Laws Dr. Afsaneh Mojra
average normal stress: ? =1 3(?11+ ?22+ ?33) : ? =1 ? + ???? + 2??? + ? + ???? + 2??? + ? + ???? + 2??? 3 ??? ?? ??? ?? ??? ?? = ? + ? +2 ??? ??? 3? ? ? = (? +2 3?)??? ??? ? = ? +2 3?: ?:???? ????????? Basic Conservation Laws Dr. Afsaneh Mojra
? = 2 For monatomic gases: 3? ? = 0 For incompressible fluid: Basic Conservation Laws Dr. Afsaneh Mojra
Momentum Eq.: ???? ??+ ??? ??? ??? =???? ??? + ??? +??? ??? ??? ??? ??? ??? ???= ????+ ???? + ? ???? ??? ??? Basic Conservation Laws Dr. Afsaneh Mojra
???? ??? +??? ??? ? ??? ??? ??? ??? = ( ????+ ???? + ? ) ??? ???? ??? +??? ??? = ?? ? ???? ??? ? ??? ??? ?? ? 0 ? = ? ???= 1 + + ? ??? ??? ??? : ???? ??+ ??? ??? ??? +??? ??? = ?? ? ???? ??? ? ??? ??? + + ? + ??? ??? ??? ??? Basic Conservation Laws Dr. Afsaneh Mojra
??? ??? For incompressible fluid: = 0 ?2?? ?????? ?2?? ?????? +??? ??? ? ??? ??? ? ??? ??? ? = ? + = ? ??? ??? ?2?? ?????? ???? ??+ ??? ??? ??? = ?? + ? + ??? ??? For inviscid fluid: ???? ??+ ??? ??? ??? = ?? + ??? Euler Equation ??? Basic Conservation Laws Dr. Afsaneh Mojra
??? ??? ??? ??? ??? ??+ ??? ?? ??? Energy Eq.: = ??? +??? ??? ??? ??? ??? ??? ??? ??? ???= ????+ ???? + ? ??? ??? ?? ? 0 ? = ? ???= 1 ??? ??? +??? ??? ??? ??? ??? ??? ??? ??? ??? ??? +??? ??? ??? = ????+ ???? + ? 2 = ???? ??? ??? ??? ??? + ? + ? ??? reversible transfer of energy due to compression Basic Conservation Laws Dr. Afsaneh Mojra
Dissipation function: Conversion of the mechanical energy into the thermal energy 2 +??? ??? ??? ??? ??? ??? ??? ??? = ? + ? +??? ??? ??? ??? ??? ??? +1 ??? ??? ??? ??? ? = ? +??? ??? ??? ??? ??? ??? 1 2 +??? ??? =? 2 2 +??? ??? ??? ??? =1 2 ? ??? ??? = ???? ??? + ? ??? Basic Conservation Laws Dr. Afsaneh Mojra
: ??? ??+ ??? ?? ??? = ???? ? ??? ??? + + ? ??? ??? Basic Conservation Laws Dr. Afsaneh Mojra
?? ??+ ? Continuity: ??? = 0 ??? ???? ??+ ??? ??? ??? +??? ??? = ?? ? ???? ??? ? ??? ??? + + ? + ??? Momentum: ??? ??? ??? 2 +??? ??? ??? ??? ??? ??+ ??? ?? ??? = ???? ? ??? ??? ??? ??? ??? ??? Energy: + + ? + ? ??? ??? State Eq.: ? = ? ?,? ? = ??? ? = ??? State Eq.: ? = ? ?,? : ?,?,?,? = ?,?,? ,?:?????:7 : 7 Basic Conservation Laws Dr. Afsaneh Mojra
?? ??= 0,?? ??= 0 ??? = 0 0 = ?? + ??? ?? = ???? ??? ??? ?? ??= ?? ??? ??= ? ??? ??? ?? ? ??? ??? ??? ??= = ? +? ??? ? (? ??=?? ??+1 ?? ?? ? ?? ??) ?2 ? Basic Conservation Laws Dr. Afsaneh Mojra
= (?) ? ??=? ?? ??= ?? ?? ?? ?? : ?? ??= ? ??? ??? ??? ??? Basic Conservation Laws Dr. Afsaneh Mojra
