THE EQUATION OF MOTION IN RECTANGULAR COORDINATES (x, y, z) In terms of T a-component +D 5tu,5+ux p a (++出)+ y-component (数+吗++) y a)+昭v(B) dt Z-component P5+vx2+U,2+U p au In terms of velocity gradients for a Newtonian fluid with constant p and u: A-component +mao+vato. du z +Pg(D) y-componenr(a+uaz+uy可+ ay dy P6, p z-component ar a2+ym+υ 十 2y+z2 +;(
The three fundamental equations of conservation EQUATION OF Locol change Change b I Chonge by Chonge by CONSERVATION OF convection diffusion production 0 Boundary condition ac MASS c at Moss transfer a km.△c ENERGY 0 Heot transfer=ha△r MOMENTUM :0 Sheor force Surfoce tension force= 7( CORRESPONDING Diffusive QUANTITIES Unit Production Boundary (per unit of volume) transport transfer MASS D km△c ENERGY cppr h△T MOMENTUM T or TE-I
System of dimensionless groups(numerics) RotiooftermsI: I N:I:I:TIN:IY:IK: x:mM:Y Mass " 如回四 L Dam kmt Sh IL Energy cppL CppT m回杀m图器网回酬 Momentum evL Rel l几 Well fa正 MEANING OF SYMBOLS NUMERICS(see Gen Ref) surface per unit of volume T= surfoce tension Bm Bingham c s concentration n= viscosity Bo Bodenstein specific heat 入= heat conductivity Da Damkohler p* density Fa s Fanning electric charge T= shear stress Fo Fourier E modulus of elasticity w= angular frequency Me a Merkel fe= electric field per unit of volume Ny s Nusselt g s gravitational acceleration reaction rate per unit of volume h s heat transfer coefficient first order r=kc Po s Poiseuille k r reaction rate constant second order rskc2 etc Re Reynolds moss transfer coefficient heat production rate per unit of volume Sh s Sherwood 7- length per unit of volume force per unit of volume St a Stonton L- characteristic length grovitotionot f=gP We s Weber pressure centrifugal f-的Lp pressure grodient f-A P/L elocity surface tension f= 7/22 length coordinate electric f* ofet
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Temperature distribution in drag flow restart: with(detools): ode: =diff(T (y),y, y)=-2/ki de: ="T(y) k Forced (Dirichlet")boundary conditions >T f: =simplify(dsolve(fode, T(0)=0, T(1)=0), T(y)))i I Oy( Digits: =4: k: =1: 2: =1: eql: =rhs(T f): Natural (Cauchy") boundary conditions >T n:=simplify (dsolve((ode, T(0)=0), T(y)))i Tn=T(n bc n: =subs(y=l, diff(rhs (r n) y))=-subs(y=l, rhs(T n))i solve(subs(9=1, bc n),C1)i >c1:=3/4:eq2:=rhs(Tn): plot(eql, eq2), y=0.1, thickness=3)i 028 0.2 0.24 022 02 018 016 0.14 0.12 008 006 0.04 002 0 04 06 08
人veti十rok七 T P=\8t u a deT 以 ay2 P di d
I-d heat transport by diflusion and advection restart: with(Detools): ode: Pe*diff(T(x), x)=ciff(T (x),x, x)i de:=Pe-T(x) T(x) TT: = simplify(dsolve(ode, T(0)=0, T(1)=1), T(x)))i TT: =T(r_-I+e[pex I+e eql: = subs(Pe=l, rhs(TT)): eq5: = subs(Pe=5, rhs(TT)):eg10: =subs(pe=10,r hs(乎T)): plot((eql, eq5, eq10), x=0. 1, thickness=3)i 0.8 0.8 04 0 02 04 08
