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- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/10%3A_Inviscid_Flow_or_Potential_Flow/10.5_Unsteady_State_Bernoulli_in_Accelerated_Coordinates\(U_{0y}\,y - U_{0y}\,x\) \(U_{0y}\,x+U_{0x}\,y\) \(\dfrac{Q}{2\,\pi}\,\ln\,r\) \(\dfrac{Q}{2\,\pi}\,\ln\,z\) \(-\dfrac{Q}{2\,\pi}\,\ln\,r\) \(-\dfrac{Q}{2\,\pi}\,\ln\,z\) \dfrac{\dfrac{r^2+{r_0}^2}{2...\(U_{0y}\,y - U_{0y}\,x\) \(U_{0y}\,x+U_{0x}\,y\) \(\dfrac{Q}{2\,\pi}\,\ln\,r\) \(\dfrac{Q}{2\,\pi}\,\ln\,z\) \(-\dfrac{Q}{2\,\pi}\,\ln\,r\) \(-\dfrac{Q}{2\,\pi}\,\ln\,z\) \dfrac{\dfrac{r^2+{r_0}^2}{2\,r\,r_0\, \cos \theta} + 1} {\dfrac{r^2+{r_0}^2}{2\,r\,r_0\, \cos \theta} - 1}\right)\) \(\dfrac{Q_0}{2\,\pi} \left( \tan^{-1} \dfrac{y}{x-r_0} - \tan^{-1} \dfrac{y}{x+r_0} \right)\) \(-\dfrac{\Gamma}{2\,\pi}\,\ln\,r\) \(-\dfrac{i\,\Gamma}{2\,\pi}\,\ln\,z\)
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/11%3A_Compressible_Flow_One_Dimensional/11.7%3A_Fanno_Flow/11.7.06%3A_Working_Conditions/11.7.6.2%3A_Fanno_Flow_Supersonic_BranchThe total maximum available for supersonic flow \(\mathbf{b}-\mathbf{b'}\), \(\left(\dfrac{4\,f\,L}{D}\right)_{max}\), is only a theoretical length in which the supersonic flow can occur if nozzle is ...The total maximum available for supersonic flow \(\mathbf{b}-\mathbf{b'}\), \(\left(\dfrac{4\,f\,L}{D}\right)_{max}\), is only a theoretical length in which the supersonic flow can occur if nozzle is provided with a larger Mach number (a change to the nozzle area ratio which also reduces the mass flow rate). In the last range \(c−\infty\) the end is really the pressure limit or the break of the model and the isothermal model is more appropriate to describe the flow.
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/00%3A_Front_Matter/01%3A_TitlePageFluid Mechanics Bar-Meir
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/11%3A_Compressible_Flow_One_DimensionalContributors and Attributions Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Pott...Contributors and Attributions Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/11%3A_Compressible_Flow_One_Dimensional/11.4_Isentropic_Flow/11.4.3%3A_The_Properties_in_the_Adiabatic_Nozzle\] Differentiation of the equation state (perfect gas), \(P = \rho R T\), and dividing the results by the equation of state (\(\rho\, R\, T\)) yields \] Rearranging equation (38) so that the density, ...\] Differentiation of the equation state (perfect gas), \(P = \rho R T\), and dividing the results by the equation of state (\(\rho\, R\, T\)) yields \] Rearranging equation (38) so that the density, \(\rho\), can be replaced by the static pressure, \(dP/\rho\) yields {dA \over A} + {d\rho \over \rho}\, {dP \over dP} \] Equation (41) is a differential equation for the pressure as a function of the cross section area.
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/00%3A_Introduction/1.5%3A_Viscosity/1.5.2%3A_Non-Newtonian_FluidsThe general relationship for simple Bingham flow is \[\tau_{xy} = -\mu \cdot \pm \tau_{0} \quad if \lvert\tau_{yx}\rvert > \tau_{0}\] \[\frac{dU_{x}}{dy} = 0 \quad if\lvert\tau_{yx}\rvert < \tau_{0}\]...The general relationship for simple Bingham flow is \[\tau_{xy} = -\mu \cdot \pm \tau_{0} \quad if \lvert\tau_{yx}\rvert > \tau_{0}\] \[\frac{dU_{x}}{dy} = 0 \quad if\lvert\tau_{yx}\rvert < \tau_{0}\] There are materials that simple Bingham model does not provide adequate explanation and a more sophisticate model is required.
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/04%3A_Fluids_Statics/4.3%3A_Pressure_and_Density_in_a_Gravitational_Field/4.3.6_Liquid_PhaseWhile for most practical purposes, the Cartesian coordinates provides sufficient treatment to the problem, there are situations where the spherical coordinates must be considered and used. Derivations...While for most practical purposes, the Cartesian coordinates provides sufficient treatment to the problem, there are situations where the spherical coordinates must be considered and used. Derivations of the fluid static in spherical coordinates are \[ \nabla \bullet \left( \dfrac{1}{\rho}\nabla P \right) + Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/05%3A_The_Control_Volume_and_Mass_Conservation/5.3%3A_Continuity_Equation/5.3.1%3A_Non_Deformable_Control_VolumeWhen the control volume is fixed with time, the derivative in equation (??) can enter the integral since the boundaries are fixed in time and hence, Continuity with Fixed b.c. \int_{V_{c.v.}} \dfrac{d...When the control volume is fixed with time, the derivative in equation (??) can enter the integral since the boundaries are fixed in time and hence, Continuity with Fixed b.c. \int_{V_{c.v.}} \dfrac{d\,\rho}{dt} dV = -\int_{S_{c.v.}} \rho\,U_{rn} \, dA Equation (??) is simpler than equation (??). Contributors and Attributions Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/04%3A_Fluids_Statics/4.6%3A_Buoyancy_and_Stability/4.6.1%3A_Stability/4.6.1.4%3A_Stability_of_None_Systematical_or_%60%60Strange''_BodiesThe body weight doesn't change during the rotation that the green area on the left and the green area on right are the same (see Figure 4.46). After the tilting, the upper part of the body is above th...The body weight doesn't change during the rotation that the green area on the left and the green area on right are the same (see Figure 4.46). After the tilting, the upper part of the body is above the liquid or part of the body is submerged under the water. For the case of \(b \lt 3a\) the calculation of moment of inertia are similar to the previous case. The moment of inertia is calculated around this point (note the body is ``ended'' at end of the upper body).
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/zz%3A_Back_Matter
- https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/08%3A_Differential_Analysis/8.7%3A_Examples_for_Differential_Equation_(Navier-Stokes)/8.7.1%3A_Interfacial_Instability-\dfrac{\sin\theta\,\left( g\,h\,\rho_{g}\,\left( 2\,\rho_{g}\,\nu_{\ell}\,\rho_{\ell}+1\right) +a\,g\,h\,\nu_{\ell}\right) } \dfrac{\sin\theta\,\left( g\,h\,\rho_{g}\,\left( 2\,a\,\rho_{g}\,\nu_{\ell...-\dfrac{\sin\theta\,\left( g\,h\,\rho_{g}\,\left( 2\,\rho_{g}\,\nu_{\ell}\,\rho_{\ell}+1\right) +a\,g\,h\,\nu_{\ell}\right) } \dfrac{\sin\theta\,\left( g\,h\,\rho_{g}\,\left( 2\,a\,\rho_{g}\,\nu_{\ell}\,\rho_{\ell}-1\right) -a\,g\,h\,\nu_{\ell}\right) }{\rho_{g}\,\left( 3\,\dfrac{\mu_{g}}{\mu_{\ell}}+{a}^{2}\,\left( \dfrac{\mu_{g}}{\mu_{\ell}}-1\right) -2}{\dfrac{\mu_{g}}{\mu_{\ell}}}
