3.3.1 Expanded forms of the Continuity Equation. θ (ρv θ)+ z (ρv z)=0. θ (ρv θ sin θ)+ 1

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1 Ke Equations Ke Equations Epanded foms of the Continuit Equation Rectangula coodinates: t (v (v z (v z0 Clindical coodinates: t 1 Spheical coodinates: (v 1 t 1 2 (2 v 1 θ (v θ z (v z0 θ (v θ sin θ 1 φ (v φ0 Table 3.1: The equation of continuit in the thee pima coodinate sstems.

2 Ke Equations 59 Rectangula coodinates: Clindical coodinates: v v v z z 0 Spheical coodinates: 1 (v 1 v θ θ v z z (2 v 1 θ (v θ sin θ 1 v φ φ 0 Table 3.2: The equation of continuit fo fluids of constant densit in the thee pima coodinate sstems.

3 Ke Equations Epanded foms of the Equation of Motion ( v t v v v v v z ( v t v v v v v z ( vz t v v z v v z v z v P ( z τ τ τ z g z (3.29 v P ( z τ τ τ z g z (3.30 v z P ( z z τz τ z τ zz g z z (3.31 Table 3.3: The equation of motion in ectangula coodinates.

4 Ke Equations 61 ( v t v v P v θ ( 1 v θ v2 θ v z (τ 1 v z τ θ θ τ θθ τ z z g (3.32 ( vθ t v v θ 1 P θ v θ θ v v θ v θ v z z v θ ( 1 2 (2 τ θ 1 τ θθ θ τ θz z g θ (3.33 ( vz t v v z P z v θ ( 1 v z θ v z (τ z 1 v z z τ θz θ τ zz z g z (3.34 Table 3.4: The equation of motion in clindical coodinates.

5 Ke Equations 62 ( v t v v P v θ v θ ( 1 2 (2 τ 1 1 τ φ φ τ θθ τ φφ v φ v φ v2 θ vφ 2 θ (τ θ sin θ g (3.35 ( vθ t v v θ 1 P θ v θ θ v θ ( 1 2 (2 τ θ 1 1 τ θφ φ τ θ v φ v θ φ v v θ v2 φ cot θ θ (τ θθ sin θ cot θ τ φφ g θ (3.36 ( vφ t v v φ v θ 1 P φ v φ θ v φ v φ φ v φv v θv φ ( 1 2 (2 τ φ 1 τ θφ θ 1 τ φφ φ cot θ τ φ 2cotθ τ θφ g φ (3.37 Table 3.5: The equation of motion in spheical coodinates.

6 Ke Equations 63 ( v t v v v v v z v P ( 2 z µ v 2 v 2 2 v g 2 z 2 (3.38 ( v t v v v v v z ( vz t v v z v v z v z v P ( 2 z µ v 2 v z P ( 2 z z µ v z 2 2 v 2 2 v z 2 2 v g z 2 ( v z g z 2 z (3.40 Table 3.6: The Navie-Stokes equation in ectangula coodinates, fo fluids of constant densit and constant viscosit µ.

7 Ke Equations 64 ( v t v v P µ v θ v θ v2 θ v z [ ( 1 (v ( vθ t v v θ v θ 1 ( vz t v v z P z µ P θ µ [ 2 v 2 θ 2 v θ z 2 v z v θ 2 v θ θ v v θ ( 1 (v θ ] v θ v z θ v z [ ( 1 v z 2 ] v θ 2 θ 2 v g z 2 (3.41 v θ v z z v θ θ 2 g θ (3.42 v z z v z θ 2 ] 2 v z g z 2 z (3.43 Table 3.7: The Navie-Stokes equation in clindical coodinates, fo fluids of constant densit and constant viscosit µ.

8 Ke Equations 65 ( v t v v v θ P ( µ 2 2 sin θ v θ v φ v φ v2 θ vφ 2 2 v 2 v 2 2 v θ 2 θ 2 v 2 θ cot θ v φ g (3.44 φ ( vθ t v v θ v θ 1 P θ µ 2cosθ 2 sin 2 θ ( v θ θ ( vφ t v v φ v θ v φ θ 1 ( P φ µ 2 v φ 2cosθ v θ 2 sin 2 θ φ v φ v θ φ v v θ v θ 2 sin 2 θ v2 φ cot θ 2 v θ 2 v 2 θ v φ g θ (3.45 φ v φ v φ φ v φv v θv φ v φ 2 sin 2 θ 2 v 2 sin θ φ cot θ g φ (3.46 Table 3.8: The Navie-Stokes equation in spheical coodinates, fo fluids of constant densit and constant viscosit µ.

9 Ke Equations The Rate of Defomation Tenso Rectangula coodinates: γ 2 v v v z v vz v v z v 2 v vz γ γ γ z γ γ γ z γ z γ z γ zz v z v z vz vz 2 vz z Clindical coodinates: γ 1 v θ v z 2 v vz ( vθ 1 v θ 2 ( 1 v θ z v θ θ 1 ( vθ v v z θ γ γ θ γ z γ θ γ θθ γ θz γ z γ zθ γ zz v vz z v θ 1 v z z θ 2 vz z 1 v θ 1 2 v v φ Spheical coodinates: γ ( vθ ( vφ sin θ ( vθ v θ v ( θ vφ sin θ 1 1 v θ 2 ( 1 θ γ γ θ γ φ γ θ γ θθ γ θφ γ φ γ φθ γ φφ sin θ v θ φ 2 ( 1 1 v ( vφ ( φ vφ θ sin θ 1 v θ φ v φ v v θ cot θ φ Table 3.9: The ate of defomation tenso in the thee pima coodinate sstems.

10 Table 3.10: The equation of eneg in tems of eneg and momentum flues Ke Equations 67 Rectangula coodinates: ( T C V t v T v ( P T T [ ( v τ v T v T z z ( v v v z z [ ( v τ z z v z [ q q q ] z z v τ τ v τ zz ( v τ z z v z Clindical coodinates: ( T C V t v T v [ θ T θ v T 1 z z (q 1 ( ( P 1 T T (v 1 v θ θ v z z [ ( ] v τ τ 1 vθ θθ θ v v z τ zz z [ {τ θ ( vθ 1 ] ( v vz τ z θ v ( 1 τ θz z Spheical Coodinates: ( T C V t v T v θ T θ v φ T φ [ 1 ( 2 1 q 2 θ (q θ sin θ 1 ( ( P 1 ( T 2 1 v T 2 [ ( v 1 τ τ v θ θθ θ v [ ( vθ τ θ 1 v θ v ( θ vφ τ φ 1 τ θφ ( 1 v φ θ 1 v θ φ cot θ ] q φ φ θ (v θ sin θ 1 ( 1 v φ τ φφ φ v v φ ] ] v z z ] q θ θ q ] z z (3.47 (3.48 v z θ v } θ z v φ φ v ] θ cot θ v φ v φ (3.49

11 Ke Equations 68 Rectangula coodinates: ( T C P t v T v T v T z k z 2 ( 2 ( 2 v v vz 2µ ( z v µ ( v 2 [ 2 ] T 2 T 2 2 T 2 z 2 ( v z v 2 ( z v z v z Clindical coodinates: ( T C P t v T v [ ( θ T θ v T 1 z k T z ( 2 [ ( ] 2 ( 2 v 1 vθ 2µ θ v vz z ( vθ µ z 1 2 ( v z vz θ v 2 z [ 1 v θ 2 ( ] T 2 θ 2 T 2 z 2 ( ] 2 vθ (3.51 Spheical Coodinates: ( T C P t v T v θ T θ v φ T φ [ ( 1 k 2 T 1 ( sin θ T 1 2 ] T 2 2 sin θ θ θ 2 sin 2 θ φ 2 2 ( v 2µ ( 1 v θ θ v 2 ( 1 v φ φ v v 2 θ cot θ [ µ ( vθ 1 ] 2 [ v 1 v θ φ ( ] 2 vφ [ ( sin θ vφ 1 ] 2 v θ (3.52 θ sin θ φ Table 3.11: The equation of eneg fo Newtonian fluids of constant densit and themal conductivit in the thee pima coodinate sstems.