Summer 2013
Lecture 1 Conservation Equations
Moshe Matalon
Conservation equations for a pure substance
We focus on a material volume Vm(t) (made of the same set of material n n points). d ⇢dV =0 dt Vm(t1) Vm(t2) VZ(t) Mass Conservation
The time rate of a change of a quantity (x,t) over a given volume V,moving with velocity VI is d @ dV = dV + (V n) dS dt @t I · VZ(t) VZ(t) SZ(t) This is the multidimensional analogue of Leibnitz’s theorem from calculus; it is a kinematic relation, not a law of fluid mechanics. Moshe Matalon
Moshe Matalon University of Illinois at Urbana-Champaign 1 Summer 2013
When VI = v,i.e.,theCVmoveswiththefluidvelocity,theresultisthe Reynolds’ transport theorem
D @ dV = dV + v n dS Dt @t · VZ(t) VZ(t) S(Zt)
D @ dV = + ( v) dV Dt @t r· ZV(t) ZV(t) ⇢ D D dV = + v dV Dt Dt r· ZV(t) ZV(t) ⇢
and the common notation for the “time rate of change” is D/Dt,theconvective or material derivative. For example:
time rate of change of the D⇢ @⇢ = + v ⇢ density of a fluid particle Dt @t ·r acceleration of a fluid Dv @v = + v v particle Dt @t ·r
Moshe Matalon
Conservation of mass
D @⇢ ⇢dV =0 + ⇢v dV =0 Dt @t r· ZV(t) ) ZV ⇢ @⇢ + ⇢v =0 @t r·
@⇢ + v ⇢ +⇢ v =0 D⇢ @t ·r r· + ⇢ v =0 ) Dt r· D⇢/Dt | {z } 1 D⇢ v = r· ⇢ Dt rate of compression or rate of dilation
D⇢ =0 v =0 Dt )r·
Moshe Matalon
Moshe Matalon University of Illinois at Urbana-Champaign 2 Summer 2013
Conservation of momentum
Newton’s second law can be applied directly to a material C.V. The time rate of change of momentum in V is the sum of the forces acting on V.
D ⇢v dV = ⇢f dV + t dS Dt ZV(t) ZV(t) ZS(t) body force surface force per unit volume per unit area | {z } | {z } t(x,t)=n · is the stress tensor; = { ij} is a symmetric tensor, i.e ij = ji. D ⇢v dV = ⇢f dV + n dS Dt · ZV(t) ZV(t) ZS(t) @(⇢v) + (⇢vv) ⇢f dV =0 @t r· r· ZV(t) ⇢ ⇢vv is the momentum flux tensor, Moshe Matalon contains nine components ⇢vivj
@(⇢v) + (⇢vv)=⇢f + @t r· r· Dv ⇢ = ⇢f + Dt r·
Constitutive relation = pI + ⌃ stress tensor Newtonian Fluid pressure viscous stress tensor 2 ⌃ = µ[( v)+( v)T]+( µ)( v)I r r 3 r· bulk shear viscosity viscosity The first term is a measure of the deformation of the fluid element and the second term is the rate of change of its volume
Dv ⇢ = p + ⌃ + ⇢f Dt r r·
Moshe Matalon
Moshe Matalon University of Illinois at Urbana-Champaign 3 Summer 2013
Conservation of energy Applied to a material control volume Vm Total energy (internal + kinetic) e = e + 1 v v T 2 · D rate of work done rate of internal ⇢e dV = + Dt T on V by external forces energy flux across S ZV(t) heat flux vector positive for outward transport D ⇢e dV = ⇢f v dV + t v dS q n dS Dt T · · · ZV(t) ZV(t) ZS(t) ZS(t) + n ( v) dS + · · q dV ZS(t) r· ZV(t) + ( v) dV r· · ZV(t)
@(⇢e ) T + (⇢e v) ⇢f v ( v)+ q dV =0 @t r· T · r· · r· ZV(t) ⇢ Moshe Matalon
@(⇢e ) T + (⇢e v)=⇢f v + ( v) q @t r· T · r· · r·
De ⇢ T = ⇢f v + ( v) q Dt · r· · r· rate of work done on rate of heat the fluid by body transferred to the fluid; and viscous forces | {z } D(e + 1 v2) | {z } positive for ⇢ 2 outward transfer Dt D( 1 v2) ⇢ 2 = ⇢f v + v ( ) Dt · · r·
De ⇢ =[ ( v) v ( )] q Dt r· · · r· r · : v = p v + r r· | {z }
De ⇢ = p v + q Dt r· r·
Moshe Matalon
Moshe Matalon University of Illinois at Urbana-Champaign 4 Summer 2013
Thermodynamics State of the gas is uniquely determined by two variables: p = p(⇢,T ),e= e(p, T ), etc.
Other thermodynamic variables are the enthalpy (per unit mass) h = e + p/⇢ and the entropy (per unit mass) s defined from the relations
1 Tds = dh ⇢ dp 1 Tds = de + pd(⇢ ) Ideal gas: