ME 152A: Fluid Mechanics

Important equations:

du Viscosity: in a shearing flow τμ= where τ is the shear stress (force/area), μ is the dy absolute viscosity (kg/m/s) and du/ dy is the shearing rate in the fluid. Note the kinematic viscosity is ν = μρ/ , where ρ is the fluid density. Ideal gas law: p = ρRT where p is the pressure, R is the gas constant (287 J/kg/K for air) and T is the absolute temperature. dp dp Bulk Modulus: E =− = where V is volume. v dV// V dρ ρ p Isothermal compression/expansion: = constant. ρ p Isentropic compression/expansion: = constant, where kcc= / , c and c are the ρ k pv pv specific heat at constant pressure and volume, respectively.

dp E Speed of sound: c ==v ; for an isentropic process (and ideal gas) dρ ρ kp ck==RT . ρ Surface tension: σ = force/length. dp Hydrostatic equation: =−γ =ρg where g is the acceleration of gravity; for an dz incompressible fluid: p =+=γhp0ρ ghp +0 where h is the distance below the level where the pressure is p0 (often assumed to be the free surface where p0 = patm ).

Hydrostatic forces on a submerged planar surface: FhRc= γ A where hc is the distance of the centroid of the planar surface below the free surface and A is the area of the

I xyc I xc surface. The center of pressure acts through the point (,xRRyx )(=+c , +y c ) yAcc yA where (,xccy )are the coordinates of the centroid, I xc is the second moment about an axis parallel to the x-axis and passing through the centroid and I xyc is the product of inertia with respect to the x-and y-axes centered at the centriod.

Buoyancy: the buoyant force acting on a submerged or semi-submerged object is equal to the weight of the displaved fluid: FVgVB = γ= ρwhere V is the volume of the submerged part of the object and ρ is the fluid density.

Bernoulli Equation(incompressible flow): pV22 pV 11++=++gz 2 2gz ρρ2212 where positions 1 and 2 are along a streamline. pV2 ++=dn gz constant across the streamline ρ ∫ R where R is the radius of curvature of the streamline and positive n is directed toward the center of curvature. Substantial or material derivative(change seen by a fluid particle): D( )∂∂∂∂ ( ) JK ( ) ( ) ( )∂ ( ) =+⋅∇=++Vuv( ) +w Dtt∂∂∂∂ t x y∂z

Acceleration: JK JK JK JK JK K DVV∂ ∂∂ V V ∂ V auv==++ +w Dtt∂ ∂∂ x y ∂z Acceleration in streamline coordinates: KKK∂VV K2 K aasanV=+ = s + n sn ∂sR K K where V is the speed of the fluid s and n are unit vectors in the streamwise and normal directions, respectively, and R is the radius curvature of the streamline.

Reynolds Transport Theorem:

dB d ∂ JKK sys ==+∫∫∫ρρρbdV bdV bV⋅ ndS dt dt sys ∂t CV CS where the subscript sys refers to a collection of matter of fixed identity, CV refers to the K control volume, CS refers to the control surface, n is a unit vector outward normal to the control surface, B is an extensive property of the fluid and b in an intensive property (B/mass)

JK JK JJK JK Moving control volume: VVabs=+ CV W where V abs is the fluid velocity relative to a JK JJK fixed coordinate system, V CV is the control volume velocity and W is the fluid velocity relative to the control volume. dB ∂ JJKK sys =+∫∫ρρbdV bW⋅ ndS dt∂ t CV CS

Conservation of Mass(non-moving control volume):

∂ JKK ∫∫ρρdV+ V⋅= n dA 0 ∂t CV CS Conservation of Mass(moving control volume):

∂ JJKK ρρdV+ W⋅= n dA 0 ∂t ∫∫ CV JK JJCSKJK JJK where the absolute velocity is VWV=+CV with W is the fluid velocity relative to the JK control volume and V CV is the velocity of the control volume.

Conservation of linear momentum(non-moving control volume):

JK∂ JK JK JK K FVdVVVn=+⋅ρρ() dA ∑ CV ∫∫ ∂t CV CS

JK Conservation of linear momentum(control volume moving with constant velocityV CV ):

JK∂ JJKJJKJJKK FWdVWWn=+⋅ρρ() dA ∑ CV ∫∫ ∂t CV CS

JK Conservation of linear momentum(control volume moving with velocityVtCV ()): JK JK dV CV ∂ JJKJJKJJKK F −=mWρρdVW +⋅()Wn dA ∑ CV CV ∫∫ dt∂ t CV CS

Moment of momentum equation:

KJK ∂ KJK KJKJKK (rF×= )ρρ ()rVdV ×+×⋅ ()() rVVndA ∑ CV CV ∫∫ ∂t CV CS

For steady flow in a turbomachine shaft torque is:

TmrVmrVshaft= −+ in()( inθθ in out out out )

JK where V is positive if it is in the same direction as the blade velocityU and V is θ JK θ negative if it is in the opposite direction as U . The shaft power is:

 Wshaft==−+ωω T shaft m in()()()( rV inθθ in m out ω rV out out =−+ mUV  in in θ in m out UV outθ out ) The rate of shaft work per mass is:

W wUVU≡=−()( + V) shaft m inθθ in out out

Energy equation: ∂∂JKK  pV2 JKK Q += W eρρ dV + e V ⋅ n dA = e ρ dV +() u +++ gzρ V ⋅ n dA ∑ net net ∫∫ ∫∫ ∂∂ttCV CS CV CS ρ 2  2  where the energy (internal, potential and kinetic) per mass is eugzV=+ + /2 and Qnet  is positive if the heat flux is into the control volume and Wnet is positive if the work is done on the control volume.

For simple inlet/outlet steady pump or turbine flow:

⎡⎤⎛⎞22 ⎛⎞pp21 VV 21   mu ⎢⎥21−+ u ⎜⎟ − +⎜⎟ − +gz()21 − z = Qnet + W shaft ⎣⎦⎝⎠ρρ ⎝⎠22

 Shaft head: hWs==≡ shaft / gmh i ideal head gain/loss.

Actual head: 22 ⎡⎛⎞pp21⎛⎞ VV 2 1 ⎤ hhhasL=−=⎢⎜⎟ − +⎜⎟ − +()zz21 −⎥ ⎣⎝⎠γγ ⎝⎠22gg ⎦ where hL is the head loss due to friction and other irreversible processes.

Efficiency:η pai= hh/ for a pump and ηTia =hh / for a turbine.

Velocity diagrams for a radial flow(centrifugal) pump: