KIL 3002 Fluid Mechanics 2 Notes (2024) PDF

DIFFERENTIAL FORM OF LINEAR MOMENTUM BALANCE 1. Acquire the concepts of infinitesimal control volume in fluid flow analysis. 2. Derivation of linear momentum balance in differential form from an infinitesimal control volume. "#...

DIFFERENTIAL FORM OF LINEAR MOMENTUM BALANCE 1. Acquire the concepts of infinitesimal control volume in fluid flow analysis. 2. Derivation of linear momentum balance in differential form from an infinitesimal control volume. "# ! + # ∇# = !) − ∇+ + ∇ τ "$ A l t e r n a t i v e form of Cauchy’s first equation of motion MOTIVATIONS Types of control volume fluid - ↓ - i ~ S... '.... ( L S - T Knowledge of the velocity vector field is : - - & &.... wall nearly equivalent to solving any fluid problem. C & -. T " - ⑭..... -1 : - jo W. APPROACH INTEGRAL FORM DIFFERENTIAL FORM - - S properties ↓ One control volume for the whole INTEREST Overall features of the flow Flow details at any or every system system. Forces points in flow system. SCALE Macroscopic scale Microscopic scale METHOD Integral analysis. Differential analysis. REGION Control volume. Infinitesimal control volumes. Each control volume represents flow properties at a point DIFFERENTIAL FORM OF LINEAR MOMENTUM BALANCE Notes Integral form Net force Momentum accumulation rate Both integral and differential forms represent the linear momentum balance. " - #!./ + - !# # 0.1 = - !2./ − - 0+.1 + 3 "$ ! " ! " Net momentum outflow Differential form Net momentum outflow Gravitational force Viscous force: Force due to friction between fluid "# molecules. ! + # ∇# = !) − ∇+ + ∇ 4 "$ Momentum accumulation rate Pressure gradient DERIVATION OF DIFFERENTIAL FORM FROM AN INFINITESIMAL CONTROL VOLUME Infinitesimal control volume From the general form of linear momentum balance: Σ Momentum out – Σ Momentum in -. &(/1'&5 sum of.22()('./3&4 &1 @ABCDE + = &1 )&)*4/() )&)*4/().2/34F &4 53/ℎ34 2. 8. 1-&) 2. 8. 2. 8. 1. Body force (weight) !$ 2. Surface force (pressure and viscous force) !$!" To write the linear momentum balance in differential !# !" equation, each of the terms in the general form should be !# !$!# represented in differential form. 7 !" 6 5 %&'()*, % = !$!#!" G(-1.2*.-*., H = !$!# = !#!" = !$!" DERIVATION OF DIFFERENTIAL FORM Infinitesimal control volume ACCUMULATION RATE AND GRAVITATIONAL FORCE Rate of momentum accumulation and the gravitational force within control volume in differential form: MOMENTUM GRAVITY.7 Volume of c.v..5 " x – direction (!9# ).6.7.5 !2#.6.7.5.6 "$ 7 " Notations y – direction (!9$ ).6.7.5 !2$.6.7.5 6 "$ Velocity components: x – direction : 8! 5 " z – direction (!9% ).6.7.5 !2%.6.7.5 y – direction : 8" "$ z – direction : 8# DERIVATION OF DIFFERENTIAL FORM Infinitesimal control volume PRESSURE FORCE Force due to pressure at the surfaces of the control volume, pressure force: At x : ??@=>×B=>B.7 +#.7.5 C − DEFGHIJD.5.6 At x + dx : −+#&'#.7.5 7 6.5 5 Force due to pressure difference:.7 +#&'# 6 – direction +#.7.5 −+#&'#.7.5 +# 7 – direction +$.6.5 −+$&'$.6.5 6 !$ 6 +.6 5 – direction +%.6.7 −+%&'%.6.7 DERIVATION OF DIFFERENTIAL FORM Infinitesimal control volume MOMENTUM The momentum into and out of the surfaces of the infinitesimal control C − DEFGHIJD volume in differential form: For momentum towards x – directions through x –surfaces:.5 Surface Surface area Velocity normal to the surface.7 At x: !9#.7.5× 9# K = (!9# 9# )(.7.5 !! !! ( Velocity in the designated direction !$ At x + dx: !9#.7.5 × 9# K = (!9# 9# )(&)(.7.5 (&)( The net outflow of momentum towards x – directions through x –surfaces: (!9# 9# )(&)(.7.5 − (!9# 9# )(.7.5 Momentum out – momentum in DERIVATION OF 7 DIFFERENTIAL FORM 6 5.7 VISCOUS FORCE.5.6 Net momentum outflow K − EMB@NCDE C – surfaces: O − [email protected] 9$.6 9% (!9# 9# )(&)(.7.5 − (!9# 9# )(.7.5.5 9# 9# L – surfaces: 7 +.7 5 (!9# 9$ )*&)*.6.5 − (!9# 9$ )*.6.5.7.7 9# 9#.5 7 M – surfaces: 7 5 +.5 9% (!9# 9% )+&)+.6.7 − (!9# 9% )+.6.7 6 9$ 5 The net outflow of momentum towards The net outflow of momentum towards x – directions through y – surfaces x – directions through z –surfaces DERIVATION OF DIFFERENTIAL FORM Infinitesimal control volume VISCOUS FORCE (pull) ^ − EMB@NCDE N#$ There are two types of viscous forces tensile acting on a control volume: ↑ N#% act perpendicular -compressive N## N## (push( 1. Shear stress due to11 sliding motions. # N#% act parallel $ N#$ $ + !$ 2. Normal stress due to change in $ " velocity normal to surface –1( pushing. N$$ K − EMB@NCDE Notations # + !# N%% P$% = stress acting on N%# N$# the Yluid on the N$% N%$ N$% " 3 − \(-1.2* N%$ N$# ] − !3-*2/3&4 # N%# N$$ N%% " + !" O − EMB@NCDE DERIVATION OF DIFFERENTIAL FORM ^ − EMB@NCDE O − EMB@NCDE N#$ N$$ VISCOUS FORCE # + !# N#% The sum of all the viscous forces on control N## N## N$# volume according to their directions: N#% N$% N$% $ N#$ $ + !$ N$# # C – direction: N$$ P&& _ !#!" − P&& _ !#!" + P"& _ !$!" − P"& _ !$!" + P#& _ !$!# − P#& _ !$!# *()* * +()+ + '()' ' N%% L – direction: N%# N%$ " P&" _ !#!" − P&" _ !#!" + P"" _ !$!" − P"" _ !$!" + P#" _ !$!# − P#" _ !$!# N%$ *()* * +()+ + '()' ' N%# M − direction: N%% " + !" P&# _ !#!" − P&# _ !#!" + P"# _ !$!" − P"# _ !$!" + P## _ !$!# − P## _ !$!# K − EMB@NCDE *()* * +()+ + '()' ' LINEAR MOMENTUM BALANCE sum of 1&-2*\.2/34F &4 2. 8. = :*/ &(/1'&5 &1 )&)*4/() + -./* &1 )&)*4/().22()('./3&4 53/ℎ34 2. 8. X-DIRECTION Substituting the differential expressions for all terms in the general form of linear momentum balance for C – direction : `& !#!" −`&(,& !#!" + aF& !$!#!" + P&& _ !#!" − P&& _ !#!" + P"& _ !$!" − P"& _ !$!" + P#& _ !$!# − P#& _ !$!# *()* * +()+ + '()' ' d = (a8& 8& )*()* !#!" − a8& 8& * !#!" + a8" 8& !$!" − a8" 8& + !$!" + a8# 8& '()' !$!# − a8# 8& ' !$!# + (a8& )!$!#!" +()+ d/ Rearranging : `& −`&(,& !#!" + aF& !$!#!" + P&& *()* − P&& * !#!" + P"& +()+ − P"& + !$!" + P#& '()' − P#& ' !$!# d = (a8& 8& )*()* − a8& 8& * !#!" + a8" 8& − a8" 8& !$!" + a8# 8& '()' − a8# 8& ' !$!# + (a8& )!$!#!" +()+ + d/ Dividing the linear momentum equation by OCOLOM: LINEAR MOMENTUM BALANCE X-DIRECTION Dividing the linear momentum equation by OCOLOM: `& −`&(,& P&& − P&& P"& +()+ − P"& + P#& − P#& *()* * '()' ' + aF& + + + !$ !$ !# !" (a8& 8& )*()* − a8& 8& a8" 8& − a8" 8& a8# 8& − a8# 8& d * +()+ + '()' ' = + + + (a8& ) !$ !# !" d/ 1 $ + ℎ − 1($) 1 - ($) = lim.→0 ℎ Taking the limit of.6 → 0,.7 → 0 and.5 → 0: Take note that this equation is for x-direction only. d` d d d d d d d − + aF& + P&& + P"& + P#& = a8& 8& + a8" 8& + a8# 8& + a8& d$ d$ d# d" d$ d# d" d/ LINEAR MOMENTUM BALANCE OVERALL Compact notations Other directions Writing in compact notations, the differential form of h = 8& + 8" + 8# linear momentum equation in: g* = P&& + P&" + P&# Repeating for other directions: ^ – direction: d d d ∇= + + d$ d# d" O – direction: d` d aF& − + ∇ g& = a8& + ∇ (a8& h) d$ d/ d` d aF" − + ∇ g" = a8" + ∇ (a8" h) d# d/ where 1 1 1 d d d K – direction: ∇ g& = 1& P&& + 1" P"& + 1# P#& and ∇ a8& h = a8& 8& + a8& 8" + a8& 8# d$ d# d" d` d aF# − + ∇ g# = a8# + ∇ (a8# h) d" d/ Taking the sum of 6, 7 and 5 − directions: d` d` d` aF& + aF" + aF# − + + + ∇ g& + ∇ g" + ∇ g# d$ d# d" d d d = a8& + a8" + a8# + ∇ a8& h + ∇ a8" h + ∇ (a8# h d/ d/ d/ LINEAR MOMENTUM BALANCE Compact notations OVERALL h = 8& + 8" + 8# g* = P&& + P&" + P&# d` d` d` d d d aF& + aF" + aF# − + + + ∇ g& + ∇ g" + ∇ g# ∇= + + d$ d# d" d$ d# d" d d d = a8& + a8" + a8# + ∇ a8& h + ∇ a8" h + ∇ (a8# h d/ d/ d/ The complete linear momentum balance equation in differential form, writing the overall linear momentum balance into a compact form: 1 ai − ∇` + ∇ g = 12 ah + ∇ ahh Cauchy’s Equation of Motion Alternatively, Cauchy’s Equation can also be written in the form of: dh ai − ∇` + ∇ g = a + h ∇h d/ DIFFERENTIAL ANALYSIS OF LINEAR MOMENTUM BALANCE SUMMARY Applications From the differential analysis of linear momentum balance, the Cauchy’s equation of motion is derived: Compressibility: "# Applicable for both compressible and !) − ∇+ + ∇ 4 = ! + # ∇# "$ incompressible fluid. This is the compact form of the equation. It should be expanded to be Rheology: used/solved. Applicable for Newtonian and non-Newtonian fluid. Symmetry: Applicable in any system symmetry.