Fundamentals of Hydrostatics
Gas Properties
Compressibility
Bulk Modulus
Definition: The increase in pressure required to produce a unit relative change in volume.
$$ E=-\frac{dp}{dV/V} $$For a given mass of gas, volume is inversely proportional to density, i.e.,
$$ \frac{d \rho}{\rho}=-\frac{dV}{V} $$Substituting back yields
$$ E=\rho \frac{dp}{d \rho} $$Bulk modulus of water at room temperature: $2.1 \times 10^9N/m^2$
Under normal conditions, water can be considered an incompressible fluid.
Viscosity
Newton’s Law of Viscosity
The frictional resistance generated by fluid motion is proportional to the contact area
$$ \tau =\mu \frac{du}{d \vec{n}} $$$\tau$: Frictional stress, the frictional resistance per unit area
$\vec{n}$: Normal direction of the contact surface
$\mu$: Proportionality constant, known as the coefficient of viscosity of the fluid, with units of $N \cdot s/m^2$
$\frac{du}{d \vec{n}}$: Velocity gradient
The coefficient of viscosity varies for different fluid media and changes with temperature, being largely independent of pressure.
The coefficient of viscosity for gases increases with rising temperature.
Sutherland’s Formula
One of the approximate formulas describing the relationship between air viscosity coefficient and temperature is Sutherland’s Formula.
$$ \frac{\mu}{\mu_0}=(\frac{T}{288.15})^{1.5}\frac{288.15+C}{T+C} $$$\mu_0$: Air viscosity coefficient at temperature $288.15K$
$C$: Constant, with a value of $110.4K$
Kinematic Viscosity Coefficient
$$ \nu=\frac{\mu}{\rho} $$$\nu$: Kinematic viscosity coefficient, unit is $m^2/s$
$\mu$: Viscosity coefficient
$\rho$: Density
Thermal Conductivity
Definition: When there is a temperature gradient in a certain direction within a gas, heat will transfer from the higher-temperature region to the lower-temperature region. This property is referred to as the thermal conductivity of the gas.
The amount of heat transferred per unit time is proportional to the heat transfer area and the temperature gradient along the direction of heat flow, expressed as:
$$ q=-\lambda \frac{\partial T}{\partial \vec{n}} $$$q$: Heat per unit time through a unit area, unit: $kJ/(m^2 \cdot s)$
$\frac{\partial T}{\partial \vec{n}}$: Temperature gradient, unit: $K/m$
$\lambda$: Thermal conductivity coefficient, unit: $kJ/(m \cdot K \cdot s)$
The negative sign indicates that the direction of heat transfer is always opposite to the direction of the temperature gradient.
Fluid Classification
Continuum Hypothesis
Ideal Fluid
不考虑粘性，在这种模型中，流体微团不承受粘性力作用。常用于气体。
忽略粘性的气体称为理想气体。
Pressure Isotropy
The pressure at a point in an ideal fluid is independent of the orientation of the pressure surface; it is merely a continuous function of spatial coordinates.
Incompressible Fluid
不考虑气体压缩性或弹性，可认为体积弹性模数无穷大，或流体密度为常数。常用于液体。
求解不可压流体的流动规律，只需要服从力学定律，不需要考虑热力学关系。
对流速较低的气体，也可按不可压流体处理流动问题。
Adiabatic Fluid
A model that does not consider the heat transfer properties of a fluid, treating the thermal conductivity coefficient of the fluid as zero. Low-speed flowing air typically has very small thermal conductivity values and can be regarded as adiabatic.
A gas model that disregards the heat conduction effects between gas microelements is referred to as an adiabatic gas.
Perfect Gas
For any state, there exists a certain functional relationship between the pressure, density, and temperature of a gas
$$ p=p(\rho,T) $$Equation of State for a Perfect Gas
$$ p=\frac{\overline{R}}{m}\rho T $$$\overline{R}$: Universal gas constant, $8315m^2/(s^2 \cdot K)$
$m$: Relative molecular mass of a specific gas
When $R=\frac{\overline{R}}{m}$,
$$ p=\rho R T $$$R$ is the gas constant, approximately $287.035m^2/(s^2 \cdot K)$ for air
Forces on a Fluid Particle
Stress
Shear Stress (Friction)
Volume Force
- Gravity
- Electromagnetic Force
- Centrifugal Force
Static Equilibrium Equation
In a ==static fluid==, take a point $P$ with pressure $p$.
Construct a Cartesian coordinate system where the pressure at any point in the fluid is
$$ p(x,y,z) $$Build a rectangular parallelepiped centered at $P$ with edges parallel to the coordinate axes and lengths $dx, dy, dz$.
Observing the $x$-axis direction, the forces on the two faces are respectively
$$ [p(x_0,y_0,z_0)+(\frac{\partial p}{\partial x})(\frac{dx}{2})]dx dy $$$$ [p(x_0,y_0,z_0)-(\frac{\partial p}{\partial x})(\frac{dx}{2})]dx dy $$The body force on the fluid element in the $x$-axis direction is
$$ f_x \rho dx dy dz $$Here, $f_x$ is the component of the body force per unit mass in the $x$-axis direction.
Since the fluid is static, the fluid element is in equilibrium.
The force balance equation in the $x$-axis direction is
$$ [p(x_0,y_0,z_0)-(\frac{\partial p}{\partial x})(\frac{dx}{2})]dx dy-[p(x_0,y_0,z_0)+(\frac{\partial p}{\partial x})(\frac{dx}{2})]dx dy+f_x \rho dx dy dz=0 $$Simplifying gives
$$ \frac{\partial p}{\partial x}=\rho f_x $$$$ \frac{\partial p}{\partial y}=\rho f_y $$$$ \frac{\partial p}{\partial z}=\rho f_z $$$\because$ The total differential of $p$ is
$$ dp=\frac{\partial p}{\partial x}dx+\frac{\partial p}{\partial y}dy+\frac{\partial p}{\partial z}dz $$$\therefore$
$$ dp=\rho(f_x dx+f_y dy+f_z dz) $$Define the ==body force potential function==
$$ \varOmega=\varOmega(x,y,z) $$Its total differential is
$$ d \varOmega=\frac{\partial \varOmega}{\partial x}dx+\frac{\partial \varOmega}{\partial y}dy+\frac{\partial \varOmega}{\partial z}dz $$Where $\frac{\partial \varOmega}{\partial x}=f_x$, $\frac{\partial \varOmega}{\partial y}=f_y$, $\frac{\partial \varOmega}{\partial z}=-f_z$.
From the above relations, we obtain
$$ dp=-\rho d \varOmega $$Integrating both sides over $x, y, z$ gives
$$ p=-\rho \varOmega+C(\text{constant}) $$$$ C=p+\rho \varOmega $$When the pressure $p_a$ at a known point A, the difference in body force potential $\varOmega_a-\varOmega$ between two points, and the density $\rho$ of the static fluid (uniform everywhere) are known, the pressure at any point can be determined from its body force potential $\varOmega$:
$$ p=p_a+\rho (\varOmega_a-\varOmega) $$Corollary: An isobaric surface in the fluid must also be an equipotential surface of the body force.
Atmosphere
Atmospheric Layers
Lower Atmosphere
- Altitude: Sea level – 85 km
- Characteristics: Uniform composition, nitrogen accounts for 78.1% of the total volume, oxygen accounts for 21% of the total volume
Troposphere
- Altitude
- Equator: 16~18 km
- Mid-latitudes: 10~12 km
- Polar regions: 7~10 km
- Mass: Accounts for 75% of the total atmospheric mass
- Characteristics: Features vertical air currents, storms, and thunderstorms. Temperature decreases rapidly with increasing altitude.
Convective Boundary Layer
Transition layer, with a thickness ranging from several hundred meters to one or two kilometers.
Stratosphere
- Altitude: Troposphere ~32 km
- Mass: Accounts for about one-quarter of the atmospheric mass
- Characteristics: No weather phenomena, horizontal air flow, constant temperature (average about 216.65 K)
Mesosphere
- Altitude: 32~85 km
- Mass: 1/3000
- Temperature: First rises then falls, can drop below 106 K at 85 km.
Upper Atmosphere
- Altitude: Above 85 km
- Characteristics: Non-uniform composition, directly absorbs solar radiation
Thermosphere
- Altitude: 85~500 km
- Temperature: Increases with altitude, reaching up to 1370 K during the day at 500 km.
- Characteristics: Directly exposed to solar shortwave radiation
Outer Atmosphere
- Altitude: 500+ km, where the atmosphere gradually merges with interstellar space
- Mass: $1/10^{11}$
- Characteristics: The atmosphere is too thin to be defined by temperature. Air molecules can escape into space.
Upper Atmosphere and Ionosphere
- The upper atmosphere is dissociated into electrons and ions by solar shortwave radiation, forming the ionosphere.
- Above 100 km altitude, the air becomes a good conductor.
- Above 150 km, the air is too thin to transmit sound.
D Layer
- 高度：60~80 km
E Layer
- 高度：100~120 km
$F_1$ Layer
- Height: 180~220 km
$F_2$ Layer
- Altitude: 300~350 km
Flow Field
Flow field: The space filled with moving fluid
Flow parameters: Physical quantities used to characterize fluid motion, such as velocity, density, pressure, etc.
Fluid mechanics methods: Lagrangian method, Eulerian method
Lagrange Method
Focuses on particles (motion)
- Studies the variation of motion parameters and trajectories of individual particles in the flow field over time.
- Synthesizes the changes in motion parameters of all fluid particles to derive the overall flow field motion characteristics.
Eulerian Method
Focuses on spatial points (fixed)
- Studies how the motion parameters of fluid particles change over time as they pass through fixed spatial points.
- By combining the variation of motion parameters at all spatial points in the flow field, the overall flow field motion can be understood.
In the Eulerian method, the motion parameters of the flow field are generally functions of spatial coordinates and time.
Taking velocity as an example:
$$ v = v(x, y, z, t) $$The four variables are independent.
In a general three-dimensional space, a Cartesian coordinate system is established, and scalar parameters are decomposed into the $x$, $y$, and $z$ directions for separate analysis.
$$ v_x = v_x(x, y, z, t) $$Differentiating to obtain the acceleration components:
$$ a_x = \frac{d v_x}{dt} = \frac{\partial v_x}{\partial t} + \frac{\partial v_x}{\partial x}\frac{dx}{dt} + \frac{\partial v_x}{\partial y}\frac{dy}{dt} + \frac{\partial v_x}{\partial z}\frac{dz}{dt} $$$\because$ $\frac{dx}{dt} = v_x$, $\frac{dy}{dt} = v_y$, $\frac{dz}{dt} = v_z$
$\therefore$
$$ a_x = \frac{\partial v_x}{\partial t} + v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} + v_z \frac{\partial v_x}{\partial z} $$From this, it can be seen that acceleration is a function of time and position:
$$ a_x = a_x(t, x, y, z) $$Local Acceleration
The first term on the right side of the equation represents the rate of change of velocity of a fluid particle at a fixed spatial point over time, known as local acceleration. (Relationship between velocity and time)
Local acceleration is caused by the temporal variability of velocity in the flow field.
Convective Acceleration
The latter three terms reflect the rate of change in velocity as a fluid particle moves along the velocity vector direction from one point in space to an adjacent point at the same instant, known as convective acceleration. (Relationship between velocity and displacement)
Convective acceleration is caused by the non-uniformity of the flow field.
Unsteady Flow Field
In the flow field, the physical quantities at least at one spatial point vary with time.
Steady Flow Field
In a steady flow field, the physical quantities at any spatial point do not change with time.
Pathline
The collection of all spatial points traversed by a marked moving fluid particle in a flow field over a period of time is called the pathline of that fluid particle.
Streamlines
In a flow field, curves that are tangent to the velocity vector at every spatial point are called streamlines.
Streamlines are curves formed by different fluid particles at the same instant, indicating the direction of velocity for those particles at that moment.
Characteristics:
In a steady flow field, the streamlines do not change over time.
Unsteady flow fields have streamlines that vary with time.
In a steady flow field, the streamline passing through a given spatial point coincides with the trajectories of all fluid particles passing through that point.
Generally, streamlines do not intersect (at the same instant and spatial point, there cannot be two velocity directions).
At points where the velocity is zero, streamlines can intersect. Such points are typically called stagnation points.
At points where the velocity is infinite, streamlines can intersect. Such points are typically called singularities.
Streamlines can be tangent to each other, and beyond the point of tangency, the two lines coincide.
Every point in the flow field has a streamline passing through it. The collection of all streamlines is called the streamline pattern or simply the flow pattern.
Streamline Differential Equation
Let the velocity at a point $M(x,y,z)$ on a streamline be $\vec{v}$, and the infinitesimal segment length of the streamline at point $M$ be $ds$, decomposed into $v_x,v_y,v_z$ and $dx,dy,dz$ in the Cartesian coordinate system.
At any point on the streamline, the direction of velocity is the same as the tangent direction of the streamline, so
$$ \cos(\vec{v},\vec{i})=\frac{v_x}{v}=\frac{dx}{ds} $$Here, $\vec{i}$ is the unit normal vector in the $x$-axis direction, and similarly for the $y$ and $z$ axes.
$$ \frac{dx}{v_x}=\frac{dy}{v_y}=\frac{dz}{v_z} $$The above equation is the differential equation of the streamline.
When the velocity distribution is known, the shape of the streamline passing through any point in the flow field can be determined.
Stream Tube
In a flow field, a closed curve C that is not a streamline is considered. Drawing streamlines through every point on C, the collection of these streamlines forms a tubular surface known as a stream tube.
Analysis of Fluid Particle Motion
Motion Forms
Rigid Body Motion
- Translational Motion
- Rotational Motion About an Axis
Fluid Motion
- Translational motion
- Rotational motion about an axis
- Deformation motion
- Linear deformation
- Shear deformation
Two-Dimensional Analysis
![Pasted image 20240902212258.png][1]
Consider an arbitrary rectangular fluid element ABCD in the flow field, with side lengths of $\delta_x,\delta_y$ respectively, both being infinitesimal quantities.
Let $v_x,v_y$ be the component velocities of the fluid element at point A, and assume the component velocities are continuous functions of spatial coordinates. The velocities at points B and D can be expressed using Taylor series expansions around point A.
$\because$ The side lengths of the fluid element are sufficiently small
$\therefore$ Higher-order infinitesimals can be neglected
$$ v_{Bx}=v_x+\frac{\partial v_x}{\partial x}\delta_x $$$$ v_{By}=v_y+\frac{\partial v_y}{x}\delta_x $$When the fluid element moves, in addition to the overall motion, point B also moves relative to point A.
The relative velocity in the $x$-direction is $v_{Bx}-v_x=\frac{\partial v_x}{\partial x}\delta_x$, and similarly for the $y$-direction $\frac{\partial v_x}{\partial x}\delta_x$.
The relative velocity of D with respect to A is $v_{Dx}-v_x=\frac{\partial v_x}{\partial y}\delta_y,v_{Dy}-v_y=\frac{\partial v_y}{\partial y}\delta_y$.
Linear Deformation Motion
![Pasted image 20240902212840.png][2]
The relative velocities $\frac{\partial v_x}{\partial x}\delta_x$ and $\frac{\partial v_y}{\partial y}\delta_y$ represent the linear deformation rates of rectangle ABCD’s edges. During time interval $dt$:
$$ AB'=AB+\frac{\partial v_x}{\partial x}\delta_x dt $$$$ AD'=AD+\frac{\partial v_y}{\partial y}\delta_y dt $$The relative rate of area change is:
$$ \frac{d(\delta S)}{\delta S \cdot dt}=\frac{AB' \cdot CD'-AB \cdot CD}{AB \cdot CD \cdot dt} $$After neglecting higher-order terms:
$$ \frac{d(\delta S)}{\delta S \cdot dt}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y} $$Extending to three-dimensional space similarly yields:
$$ \frac{d(\delta V)}{\delta V \cdot dt}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z} $$Angular Deformation Motion
![Pasted image 20240902214219.png][3]
The relative velocities $\frac{\partial v_y}{\partial x}\delta x,\frac{\partial v_x}{\partial y}\delta y$ represent the rotation of edges AB and AD around point A.
Defining counterclockwise rotation as positive,
The angular velocity of edge AB is
$$ \frac{d\alpha_1}{dt}=\frac{\partial v_y}{\partial x}\delta_x / \delta_x=\frac{\partial v_y}{\partial x} $$Similarly, the angular velocity of edge AD is
$$ \frac{d \alpha_2}{dt}=-\frac{\partial v_x}{\partial y} $$Microelement Rotation Angular Velocity about the $z$-axis
Definition: The average value of the angular velocities of two mutually perpendicular lines in the projection of a microelement on the $xOy$ plane rotating about the $z$-axis. (Half the sum of angular velocities)
$$ \epsilon_z=\frac{1}{2}(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}) $$Angular Deformation Rate
Definition: Half of the change in angle per unit time between two mutually perpendicular lines in the projection of a fluid element on the $xOy$ plane. (Half of the angular velocity difference)
$$ \gamma_z=\frac{1}{2}\left(\frac{\partial v_y}{\partial x}+\frac{\partial v_x}{\partial y}\right) $$When extended to three-dimensional space, the angular velocities and angular deformation rates of the fluid element along the three axes can be similarly derived.
Omitted.
Divergence
Definition: The sum of the directional derivatives of each velocity component in its respective direction is called the three-degree of the velocity vector.
$$ div \vec{v}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z} $$Physical meaning: It quantifies the relative volume change rate of a fluid element during motion.
==Assumption==: The density of the fluid remains unchanged (the fluid motion is considered incompressible).
The volume flow rate emitted from a point is defined as
$$ \lim_{\delta V \to 0}\frac{Volume outflow - Volume inflow}{\delta V \cdot dt}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z} $$This equals the net volume outflow per unit time from a unit volume control volume at a certain point in space, which is also equal to the relative volume change rate of the fluid element during motion.
Curl
Definition: Twice the angular velocity of rotation.
$$ \vec{\omega}=curl \vec{v}=(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z})\vec{i}+(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x})\vec{j}+(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y})\vec{k} $$Velocity Potential
In fluid mechanics, fluid motion can be classified based on whether fluid particles have rotational motion:
- Rotational flow
- Irrotational flow
When the flow is considered irrotational, $\omega=0$,
$$ \begin{cases} \frac{\partial v_z}{\partial y}=\frac{\partial v_y}{\partial z} \\ \frac{\partial v_x}{\partial z}=\frac{\partial v_z}{\partial x} \\ \frac{\partial v_y}{\partial x}=\frac{\partial v_x}{\partial y} \\ \end{cases} $$The above system of equations is the necessary and sufficient condition for $v_xdx+v_ydy+v_zdz$ to constitute the total differential of some function $\phi(x,y,z)$. That is,
$$ d \phi=v_xdx+v_ydy+v_zdz=\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz $$$\phi$ is called the velocity potential or velocity potential function.
$$ \begin{cases} v_x=\frac{\partial \phi}{\partial x}\\ v_y=\frac{\partial \phi}{\partial y}\\ v_z=\frac{\partial \phi}{\partial z}\\ \end{cases} $$When using cylindrical coordinates,
$$ \phi=\phi(r,\theta,z) $$$$ \begin{cases} v_r=\frac{\partial \phi}{\partial r}\\ v_\theta=\frac{\partial \phi}{\partial \theta}\\ v_z=\frac{\partial \phi}{\partial z}\\ \end{cases} $$Scalars
Pressure
Density
Temperature
Viscosity Coefficient
Vector
Flow Velocity
Shear Stress
Ideal Gas Equation of State
$$ pV=nRT $$$R=8.314J \cdot mol^{-1} \cdot K^{-1}$ is the universal molar gas constant.
$$ p=\frac{n \cdot M}{V}\frac{R}{M} T $$$$ n(\text{amount of substance}) \cdot M(\text{molar mass})=m(\text{mass}) $$$$ p=\rho R' T $$$$ R'=\frac{R}{M} $$$R'$ is the ==specific gas constant==.
For ideal air $R'=287J/(kg \cdot K)$
Aerodynamics and Moments
Aerodynamic Force $R$: Resultant
The force exerted by air on an object
- Pressure $p$: Pressure
- Shear stress $\tau$: Shear stress
The resultant of pressure and shear stress is the force exerted by air on the object, known as the aerodynamic force.
Wind-axis system
- Lift $L$: Lift, the vertical component
- Drag $D$: Drag, the horizontal component
Freestream
$$ V_{\infty} $$Freestream refers to the undisturbed incoming flow in front of the aircraft, i.e., the natural flow of air without interference from the aircraft or other objects.
The directions of lift and drag are determined by the freestream direction.
Angle of Attack (AOA)
$$ \alpha $$Angle of attack (abbreviated as AOA, commonly represented by the Greek letter α) is a term in aerodynamics. It refers to the angle between the chord line of an airfoil and the direction of the free stream (or relative wind flow). For an aircraft, the angle of attack is defined as the angle between the aircraft’s longitudinal axis and the direction of the relative wind. A positive angle of attack occurs when the airfoil is tilted upward, while a negative angle of attack occurs when it is tilted downward.
Body Axis System
- Normal force $N$: Perpendicular to the airfoil direction
- Axial force $A$: Parallel to the airfoil direction
力矩 $M$：Moment
The moment that makes the aircraft pitch up is positive, and the moment that makes the aircraft pitch down is negative.
Dynamic Pressure $q$
The dynamic pressure generated by free stream $V_{\infty},\rho_{\infty}$
$$ q_{\infty}=\frac{1}{2}\rho_{\infty}V_{\infty}^2 $$Unit is $Pa$, same as pressure
Characteristic Geometric Dimension $S$
For three-dimensional objects, it represents the area; for two-dimensional objects, it represents the perimeter.
Dimensionless Parameters
Three-dimensional objects commonly use uppercase $C$, while two-dimensional objects commonly use lowercase $c$.
Lift Coefficient
$$ C_L=\frac{L}{q_{\infty}S} $$Drag Coefficient
$$ C_D=\frac{D}{q_{\infty}S} $$Normal Force Coefficient
$$ C_N=\frac{N}{q_{\infty}S} $$Axial Force Coefficient
$$ C_A=\frac{A}{q_{\infty}S} $$Aerodynamic Coefficients
$$ C_R=\frac{R}{q_{\infty}S} $$Moment Coefficient
$$ C_M=\frac{\vec{M}}{\vec{r} \times \vec{q_{\infty}}S} $$Pressure Coefficient
$p$：某点静压 (Static pressure at a point)
$p_{\infty}$：自由来流静压 (Freestream static pressure)
$$ C_p=\frac{p-p_{\infty}}{q_{\infty}} $$Friction Coefficient
$\tau$: Shear stress at a point, which is the derivative of shear stress with respect to area. Dimensions are the same as pressure.
$$ C_f=\frac{\tau}{q_{\infty}} $$Two Centers
Center of Pressure (COP)
Center of Pressure (COP): The point of intersection between the line of action of the resultant force of fluid pressure on a plane or curved surface within the fluid and that plane or surface. The aerodynamic force $R$ produces zero moment $\vec{0}$ about this point.
Aerodynamic Center (AC)
Aerodynamic Center (English: aerodynamic center, abbreviated as AC) in aerodynamics refers to a fixed point on an airfoil where the pitching moment does not vary with changes in angle of attack, i.e.,
$$ \frac{d C_M}{d \alpha}=0 $$The Difference Between Aerodynamic Center and Center of Pressure
The center of pressure is a special point where the force system is synthesized, resulting in zero net moment at that point, with the center of pressure located behind the aerodynamic center. On the other hand, the aerodynamic center is the point where the net moment remains unchanged.
The position of the center of pressure changes with variations in the angle of attack. As the angle of attack increases, the lift increases, causing the center of pressure to move forward. This simultaneously reduces the distance between the center of pressure and the aerodynamic center. The increased lift multiplied by the shortened moment arm precisely equals the unchanged moment, which is exactly what the definition of the aerodynamic center requires.
When will I have a drink and discuss the details again?